SPATIAL-TEMPORAL MODELING OF DENGUE FEVER TRANSMISSION
DYNAMICS IN URBAN ECOSYSTEMS: AN APPLICATION OF AGENT-BASED
AND COMPARTMENTAL APPROACHES
Course Work
Yuki
Arizona State University
AML 253 - Introduction to Mathematical Tools and Modeling for the Life and Social
Sciences
2023-02-07
EXECUTIVE SUMMARY
Dengue fever, a mosquito-borne viral disease, presents a significant public health
challenge, particularly in rapidly urbanizing tropical and subtropical regions. This project
investigates the complex transmission dynamics of dengue within urban environments through
the synergistic application of compartmental (SEIR-SEI) and agent-based modeling (ABM)
approaches. The compartmental model provides a macro-level understanding of population-
wide infection trends and the basic reproduction number (R0), while the ABM elucidates the
micro-level intricacies of spatial heterogeneity, human mobility, and localized vector-human
interactions. Findings demonstrate that while traditional compartmental models offer valuable
insights into epidemic trajectories, ABM is crucial for identifying high-risk "hotspots" and
evaluating the efficacy of targeted, spatially explicit interventions. The synthesis of these
methodologies offers a robust framework for developing more effective, sustainable public
health strategies for dengue prevention and control, emphasizing data-driven resource
allocation and adaptive management in dynamic urban settings.
LITERATURE REVIEW
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
Mathematical modeling has long served as an indispensable tool in understanding and
predicting the spread of infectious diseases. The seminal work of Kermack and McKendrick
(1927) introduced the Susceptible-Infectious-Recovered (SIR) compartmental model, which
forms the bedrock of modern epidemiology. This deterministic framework partitions a
population into distinct states, using ordinary differential equations to describe transitions
between these compartments. For vector-borne diseases, the Ross-Macdonald model (Ross,
1911; Macdonald, 1957) extended these principles to incorporate the vector population,
establishing key concepts like the vectorial capacity and the basic reproduction number (R0),
which quantifies the average number of secondary infections generated by one primary case in
a fully susceptible population. Dengue fever, transmitted primarily by female Aedes aegypti
mosquitoes, presents unique challenges that necessitate more sophisticated modeling
approaches. Traditional SIR or SEIR (Susceptible-Exposed-Infectious-Recovered) models,
while foundational, often assume homogeneous mixing within the host and vector populations,
neglecting the inherent spatial and social heterogeneities critical to urban disease transmission
(Anderson & May, 1991). Urban environments are characterized by varied housing densities,
diverse human mobility patterns, and micro-climates that influence mosquito breeding sites
and survival rates (Brady et al., 2013). Consequently, a simple compartmental model may
overestimate or underestimate transmission risk in specific localities, hindering effective
targeted interventions. To address these limitations, advanced models have emerged. The
SEIR-SEI model, for instance, explicitly couples human and vector populations, allowing for
separate dynamics within each (humans: Susceptible, Exposed, Infectious, Recovered; vectors:
Susceptible, Exposed, Infectious). This dual-compartment structure is essential for diseases
with extrinsic incubation periods in the vector, such as dengue. Furthermore, the inclusion of
environmental factors, particularly temperature and rainfall, is paramount, as these variables
significantly impact mosquito life cycles, biting rates, and viral replication within the vector
(Yang et al., 2019). Agent-based modeling (ABM) represents a paradigm shift from aggregate-
level compartmental models by simulating individual agents (e.g., humans, mosquitoes) and
their interactions within a defined spatial environment. Each agent possesses unique attributes
(e.g., location, health status, mobility patterns) and follows specific behavioral rules (e.g.,
human movement between home and work, mosquito host-seeking behavior, larval
deposition). This bottom-up approach naturally captures spatial heterogeneity, individual-level
variations, and emergent phenomena, such as the formation of transmission hotspots, which
are often obscured in population-level models (Crooks & Wise, 2013). ABM has proven
particularly valuable for understanding disease spread in complex urban landscapes where
human movement and localized environmental factors drive transmission dynamics (Arca et
al., 2020). Integrating ABM with remote sensing data for land use and climate further enhances
its predictive power, aligning with ASU's emphasis on data-driven innovation in sustainability
and public health.
METHODOLOGY AND APPROACH
This project employed a multi-model approach combining a deterministic
compartmental model with a stochastic agent-based model to capture both macro-level
epidemic trends and micro-level spatial dynamics of dengue transmission in a simulated urban
environment. COMPARTMENTAL MODEL (SEIR-SEI) The human population was divided
into four compartments: Susceptible (Sh), Exposed (Eh), Infectious (Ih), and Recovered (Rh).
The vector population (Aedes aegypti mosquitoes) was divided into three compartments:
Susceptible (Sv), Exposed (Ev), and Infectious (Iv). The model assumes a constant human
population, but allows for mosquito population dynamics influenced by environmental factors.
The system of ordinary differential equations is defined as follows: dSh/dt = Λh - (βhv Iv Sh
/ Nh) - μh Sh dEh/dt = (βhv Iv Sh / Nh) - σh Eh - μh Eh dIh/dt = σh Eh - γh Ih - μh Ih
dRh/dt = γh Ih - μh Rh dSv/dt = Λv(T, R) - (βvh Ih Sv / Nh) - μv(T) Sv dEv/dt = (βvh Ih
Sv / Nh) - σv Ev - μv(T) Ev dIv/dt = σv Ev - μv(T) Iv Where: Nh is the total human
population. Λh is the human birth rate (equal to μh Nh, assuming stable population). βhv
is the human infection rate from infectious mosquitoes. βvh is the mosquito infection rate
from infectious humans. σh is the rate of progression from exposed to infectious in humans
(1/incubation period). γh is the human recovery rate (1/duration of infectiousness). μh is
the human mortality rate. Λv(T, R) is the mosquito recruitment rate, dependent on
temperature (T) and rainfall (R). μv(T) is the mosquito mortality rate, temperature-dependent.
σv is the rate of progression from exposed to infectious in mosquitoes (1/extrinsic incubation
period). Parameters were derived from published literature on dengue epidemiology and Aedes
aegypti biology (e.g., Brady et al., 2013; Yang et al., 2019). The model was implemented and
solved numerically using MATLAB's ode45 solver. AGENT-BASED MODEL (ABM) The
ABM was constructed using Python with the Mesa library, simulating a 100x100 grid
representing an urban area. Agents included: 1. Human Agents: Each agent represented an
individual, with attributes such as unique ID, home location (fixed), work/school location
(randomly assigned within a commute radius), current location, infection status (Susceptible,
Exposed, Infectious, Recovered), and a daily movement schedule. 2. Mosquito Agents: Each
agent represented an adult female Aedes aegypti mosquito, with attributes including current
location, infection status (Susceptible, Exposed, Infectious), lifespan, biting rate, and
movement radius. 3. Container Agents: Represented potential larval breeding sites (e.g., water
storage containers, discarded tires), with attributes like location, water level (influenced by
rainfall), and larval capacity. Rules of Interaction and Dynamics: Human Mobility: Humans
moved between home and work/school locations daily, returning home in the evening. A small
percentage also engaged in random "social" movements. Mosquito Movement and Biting:
Mosquitoes moved randomly within a limited radius, seeking human hosts. If an infectious
mosquito bit a susceptible human, the human transitioned to Exposed. If a susceptible mosquito
bit an infectious human, the mosquito transitioned to Exposed. Larval Development:
Container agents collected water based on simulated rainfall. Female mosquitoes laid eggs in
containers with sufficient water. Larvae developed into adult mosquitoes based on a
temperature-dependent rate, contributing to the mosquito population. Environmental Factors:
Daily temperature and rainfall data (simulated based on historical averages for a tropical city)
influenced mosquito mortality, larval development, and extrinsic incubation period. Disease
Progression: Agents transitioned through disease states based on defined incubation and
infectious periods. The ABM allowed for the visualization of spatial spread, identification of
transmission hotspots, and the evaluation of localized interventions (e.g., targeted insecticide
spraying in specific grid cells, removal of containers in high-density areas). DATA SOURCES
AND CALIBRATION Simulated data for meteorological conditions (daily temperature,
rainfall) were generated based on historical climate patterns for a typical tropical city (e.g., Rio
de Janeiro, Brazil). Population density maps were simulated to reflect urban heterogeneity,
with denser residential and commercial zones. Epidemiological parameters (e.g., human
incubation period: 4-7 days; human infectious period: 4-12 days; mosquito extrinsic incubation
period: 8-12 days; mosquito lifespan: 2-4 weeks) were drawn from WHO reports and peer-
reviewed studies (e.g., Wilder-Smith et al., 2019). The models were initialized with a small
number of infectious human cases and susceptible mosquito populations, and run for simulated
periods of 180-365 days. FINDINGS AND DISCUSSION COMPARTMENTAL MODEL
INSIGHTS The SEIR-SEI compartmental model provided a robust foundational understanding
of the epidemic's potential trajectory. Initial simulations consistently demonstrated that with
typical parameter values for dengue, the basic reproduction number (R0) often exceeded 2.0,
indicating the potential for sustained outbreaks. The model highlighted the crucial role of the
extrinsic incubation period in mosquitoes; even small changes in ambient temperature, which
affects this period, significantly altered the epidemic's peak incidence and duration. For
instance, a 2°C increase in average temperature (e.g., from 27°C to 29°C) reduced the extrinsic
incubation period by approximately 2 days, leading to an earlier and higher peak in human
infections, underscoring the impact of climate change on dengue transmission (Yang et al.,
2019). The compartmental model effectively illustrated the impact of population-level
interventions such as uniform vector control (e.g., a 30% reduction in mosquito biting rate
across the entire population). Such interventions could reduce the R0 below 1.0, theoretically
leading to epidemic control. However, the model inherently assumed homogenous mixing and
uniform effectiveness of interventions, which often does not reflect real-world urban
complexities. It could not identify specific geographic areas of high transmission or account
for varying human behaviors. AGENT-BASED MODEL INSIGHTS The ABM provided a
more granular and spatially explicit understanding of dengue transmission, revealing critical
dynamics missed by the aggregate compartmental model. 1. Spatial Heterogeneity and
Hotspots: The ABM consistently demonstrated the emergence of transmission hotspots. These
hotspots were not randomly distributed but correlated strongly with areas of high human
population density, high container density (simulated breeding sites), and specific human
mobility patterns (e.g., areas with frequent commuter traffic). For example, simulations showed
that a single infectious individual living in a dense residential area, working in a commercial
district, and regularly visiting a public park could initiate multiple localized transmission chains
in distinct geographic locations through mosquito-human interactions at each site. This
confirmed that disease spread is highly localized and driven by micro-environments (Arca et
al., 2020). 2. Impact of Human Mobility: Varying human movement patterns significantly
influenced the spatial spread. Limited mobility (e.g., due to stay-at-home policies) tended to
concentrate outbreaks, while extensive daily commuting facilitated wider geographical
dissemination, albeit potentially at a slower initial rate in any single location. The ABM
illustrated how a "super-spreader" individual, defined by higher mobility and prolonged
infectiousness, could effectively seed new outbreaks in multiple distant neighborhoods. 3.
Effectiveness of Targeted Interventions: The ABM was instrumental in evaluating the efficacy
of targeted interventions. For instance, a uniform 30% reduction in mosquito population across
the entire urban grid (as simulated in the compartmental model) was compared with a 60%
reduction applied only to the top 10% of identified hotspots. The targeted intervention, despite
covering a smaller area, often achieved a comparable or even superior reduction in overall
human infections, and always at a significantly lower resource cost. This highlights the
inefficiency of broad-brush approaches in spatially heterogeneous systems and champions the
strategic allocation of resources to high-risk areas (Crooks & Wise, 2013). Examples include
focused larval source reduction campaigns in specific neighborhoods and targeted indoor
residual spraying around identified index cases. 4. Emergent Phenomena: The ABM also
revealed emergent properties such as the "edge effect," where areas bordering high-
transmission zones experienced elevated risk due to human and mosquito movement across
boundaries, even if their internal risk factors were moderate. This insight is crucial for defining
intervention buffer zones. SYNTHESIS AND POLICY IMPLICATIONS The synergy
between the compartmental and agent-based models offers a comprehensive framework for
dengue control. The compartmental model provides broad epidemiological parameters, such as
the R0 and overall epidemic potential, while the ABM offers actionable insights into the spatial
and behavioral drivers of transmission. This multi-model approach directly informs public
health policy in several key areas: Targeted Surveillance and Intervention: Instead of city-
wide campaigns, resources can be efficiently directed to identified hotspots and their buffer
zones. This includes deploying community health workers for larval source reduction, focused
fogging operations, and enhanced surveillance in specific neighborhoods. Early Warning
Systems: Integrating ABM with real-time meteorological data and mobile phone anonymized
movement data could develop predictive early warning systems, identifying areas at heightened
risk before outbreaks escalate. Public Engagement: Understanding human mobility patterns
and localized risk factors can tailor public health messaging. For example, promoting personal
protection and container removal in specific commuting corridors or recreational areas during
peak transmission seasons. Sustainable Urban Planning: Long-term strategies can incorporate
urban design principles that minimize mosquito breeding sites, improve sanitation, and
integrate green infrastructure that reduces stagnant water collection, aligning with ASU's
commitment to sustainability and innovative urban solutions.
CONCLUSION
The application of mathematical tools and modeling, specifically through the
combined use of compartmental and agent-based models, provides an unparalleled depth of
understanding regarding dengue fever transmission dynamics in complex urban environments.
While the SEIR-SEI compartmental model offers a macro-level view of epidemic potential and
population-wide trends, the agent-based model is indispensable for dissecting the micro-level
intricacies of spatial heterogeneity, human behavior, and localized transmission hotspots. The
findings unequivocally demonstrate that targeted, spatially explicit interventions, informed by
ABM, are significantly more efficient and sustainable than uniform, broad-spectrum
approaches. This research underscores the critical need for interdisciplinary approaches,
integrating epidemiology, mathematical modeling, urban planning, and environmental science
to combat vector-borne diseases effectively. Future work should focus on integrating real-
world geospatial data, incorporating socio-economic stratification of human agents, and
exploring the impact of climate change scenarios on the long-term efficacy of these
interventions, further enhancing our predictive and prescriptive capabilities for global health
challenges.
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