Please see the attached journals and answer the below mentioned journals

2017 IEEE International Conference on Bioinformatics and Biomedicine (BIBM)

978-1-5090-3050-7/17/$31.00 ©2017 IEEE 1425

JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, OCTOBER 2017 1

A simplified mathematical-computational model of the immune response to the yellow fever vaccine

C.R.B. Bonin a∗, G.C. Fernandes b R.W. dos Santos a and M.Lobosco a

Abstract—An effective yellow fever vaccine has been available since 1937. However, some issues regarding its use remain open, such as the minimum dose that can provide immunity against the disease. Mathematical-computational tools can be useful to assist the search for answers to some of these open issues. In this context, this study presents a simplified mathematical- computational model of the human immune response to the vaccination against yellow fever that takes into account important cells of the immune system. The model was able to qualitatively reproduce some experimental results reported in the literature, such as the amount of antibodies and viremia along time, as well as to reproduce distinct behaviors of the immune response reported in the literature. This is the first step towards an ideal scenario where it will be possible to simulate distinct situations related to the use of the yellow fever vaccine, such as its application in immunodeficient individuals, different vaccination strategies, duration of immunity and the need for a booster dose.

Index Terms—Computational vaccinology, Yellow fever, Math- ematical modeling, Computational modeling, Immune system, Ordinary Differential Equations

I. INTRODUCTION

Since 1937 an effective vaccine against Yellow Fever (YF)

has been available [1]. Despite this, outbreaks still occur in

countries in Africa, Central and South America, where the

population has low vaccination coverage. An outbreak was

recorded between 2015 and 2016 in Angola. From December

5, 2015 to August 4, 2016, 3,867 suspected cases were

reported, of which 879 were laboratory confirmed, causing

a total of 369 deaths, of which 119 were reported among

confirmed cases[2]. Another significant recent outbreak oc-

curred in the Democratic Republic of Congo (DRC) in 2016.

A total of 2,987 suspected cases were reported to the national

surveillance system, with 81 laboratory confirmed cases and

16 deaths [2].

The latest outbreak occurred in Brazil, starting in December

2016. In the period from December 2016 to February 22, 2017,

a YF outbreak has affected Brazil, with 1345 suspected cases,

of which 295 have been confirmed, and 215 deaths reported

to Brazilian Ministry of Health [3].

Although the number of YF outbreaks in Brazil between

the end of 2016 and the beginning of 2017 is higher than

in previous outbreaks, it is important to note that all cases

are wild-type, observed in people who live in rural areas

aGraduate Program in Computational Modeling, Federal University of Juiz de Fora, 36036-900, Juiz de Fora, MG, Brazil

bPresidente Antnio Carlos University - Medical School, Juiz de Fora, MG, Brazil

*Corresponding author. Tel.: +55 32 2102-3481. E-mail address: rezen- [email protected] (C.R.B. Bonin).

or who have had contact with wilderness areas for leisure

or work. Cases were considered as sylvatic transmission

although some cases occurred in cities with previous recent

dengue fever outbreaks, highlighting the imminent risk of YF

reintroduction in urban areas with high infestation of Aedes aegypti. Unfortunately, Brazil, like other major countries in the world, has favorable conditions for this. Urbanization,

population mobility in endemic areas, expansion of these areas

due to climate change, and the resurgence of the Aedes aegypti mosquito increase the likelihood of future YF epidemics in

the absence of effective countermeasures [4]. Furthermore, the

routine YF vaccination is not mandatory in larger centers in

Brazil (such as Rio de Janeiro and So Paulo).

World stocks of YF vaccine may not be prepared to supply

the need for vaccination if a large outbreak occurs. In fact,

this has already occurred in recent outbreaks. In Kinshasa,

capital of the DRC, fractional doses of the YF vaccine were

administered [4]. Some studies that have been used as a basis

for the implementation of this strategy are summarized in

an information published by the World Health Organization

(WHO) [5]. The most recent ones indicate that fractional doses

have the same immunogenicity as standard doses, at least in

the short and medium term [6], [7].

The WHO launched in April 2017 a strategy called Elimi-

nate Yellow fever Epidemics (EYE), which aims to eliminate

YF epidemics in the world by 2026 [4]. Through early

detection and rapid and appropriate response, it is possible

to minimize suffering, damage and propagation [4]. This

strategy has three goals: protect populations at risk, prevent

the international spread of the disease and contain outbreaks

quickly. To achieve these goals, the strategy suggests actions

on different fronts, including research and development for

better tools and practices.

In this scenario, mathematical and computational modeling

presents itself as a tool to assist researches in vaccinology and

public health. Mathematical models have been used for many

years to represent various aspects of the immune system and

related pathologies, but their application to describe the effects

of vaccines has been rather limited [8]. The term computa-

tional vaccinology has been used to refer to computer-aided

vaccine design [9], [10], [11], [12]. In a previous work[13],

we proposed a new use of computational vaccinology, in

the clinical development stage. With the use of mathematical

and computational models, it is possible to experiment, in silico, different scenarios related to vaccination, for answering important questions still open.

This work presents a simplified mathematical-computational

model of the immune response to the YF vaccine. The model

1426

JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, OCTOBER 2017 2

considers the major populations of Human Immune System

(HIS) cells and molecules important in the process of im-

munity acquisition, such as Antigen Presenting Cells (APCs),

B and T lymphocytes, and antibodies, which are considered

the main marker of immunity. The model is then evaluated

using distinct scenarios, and as will be shown, it was able to

qualitatively reproduce some experimental results reported in

the literature.

The remaining of this work is organized as follows. Section

II briefly presents related work. Section III presents the

simplified mathematical model and its computational imple-

mentation. The results of the model are presented in Section

IV. Finally, Section V presents the conclusions and plans for

future works.

II. RELATED WORK

Several computational modeling techniques applied to vac-

cination are analyzed and discussed by Pappalardo et al.[8]. The authors describe what mathematical and computational

modeling are and how they can aid research in vaccination.

Modeling is defined as human activity involving the repre-

sentation, manipulation, and communication of everyday real

world objects. In this study, two main types of modeling are

considered: Agent-Based Models (ABM) and mathematical

models. The dynamic agents of an ABM can be described

as a function of time, a position and an internal state that

includes the most important properties of the agent, such as

age. Mathematical models are mainly based on differential

equations, whether ordinary or partial, with delayed and/or

stochastic equations. The studies reviewed by the authors[8]

are more related to tumor modeling or to vaccines that use

mechanisms other than live and attenuated virus inoculation.

In this work, we propose a mathematical-computational model

of the immune response to the YF vaccine, which is based

on a live, attenuated viral strain. The model used Ordinary

Differential Equations (ODEs) to model the main cells and

molecules related to the adaptive immune response.

Other works use computational tools to aid the vaccine

design. For example, epitope-mapping algorithms are used

for vaccine design since the 1980s[14]. Since then, new

computational tools have been used for selection of vaccine

targets[15], [16], [17], [18], [19], [20], [21], [22], [23]. Most

of the works focus on using computational science to predict

epitopes[24] or to develop virtual screening approaches (i.e,

the identification of relevant antigens)[25], [26], [27], [28].

This traditional use of computational vaccinology is related

to the pre-clinical development. This work focus in the de-

velopment of mathematical and computational models that

can be used in the clinical development stage, i.e., when the

vaccine is first tested in humans. We argue that it is possible to

carry some experiments in silico, reducing the search space for experiments in vivo or in vitro, and it is possible to eliminate, reinforce or weaken hypotheses and to direct studies, thus

saving time and resources.

A mathematical model, using ODEs, of the human immune

response to vaccination against both YF and smallpox was

presented in a previous work [29]. The aim of the authors were

to evaluate primarily the dynamics of CD8+ T cells and not

the immune response as a whole. The model proposed in this

work differs from those presented by Le et al.[29] because it considers important populations at each stage of the immune

response to YF vaccination, from virus inoculation to APC

antigen presentation and consequent activation of lymphocytes

to the generation of antibodies and memory cells.

III. METHODS

A. Mathematical model

The model proposed in this work was based on a previous

work [30] and consists of a system of 10 ODEs representing

some important populations in the response process generated

by the body after vaccination. The main populations are the

following: viruses, APCs, CD8+ T cells, short-lived and long-

lived plasma cells, B cells and antibodies.

The model can estimate the concentration of all cells and

molecules represented by the equations. In the next section the

values of viremia and antibody concentration will be estimated

and then compared to the values reported in the literature, in

order to qualitatively validate the proposed model.

Equation (1) represents the vaccine virus (V ).

d

dt V = πvV −

cv1V

cv2 + V − kv1V A − kv2V TE (1)

The virus can not proliferate itself, it needs to infect a cell

and use it as a factory for new viruses. This is implicitly

considered in the term πvV , which represents the multipli- cation of the virus in the body, with a production rate πv. The term cv1V

cv2+V denotes a non-specific viral clearance made

by the innate immune system. This function models growth

combined to the saturation phenomenon and is similar to

the Hill family of equations[31]. The term kv1V A denotes specific viral clearance due to antibody signaling, where kv1 is the clearance rate. The term kv2V TE denotes specific viral clearance due to the induction of apoptosis of cells infected

by the YF virus, where kv2 is the clearance rate. APCs are all cells that display antigens complexes on their

surfaces, such as dendritic cells and macrophages. Two stages

of APCs were considered: immature and mature. The first

stage, immature APCs (AP ), is described by Equation (2).

d

dt AP = αAP (AP 0−AP )−βAP AP (kAP 1 +tanh(V −kAP 2))

(2)

The term αAP (AP 0 − AP ) denotes the homeostasis of APCs, where αAP is the homeostasis rate. The term βAP AP (kAP 1 + tanh(V − kAP 2)) denotes the conversion of immature APCs into mature ones. So the same term appears in

Equation (3) with positive sign. βAP represents the convertion rate and (kAP 1 + tanh(V − kAP 2)) is a sigmoidal saturation function in the form of a hyperbolic tangent.

The Equation (3) represents the mature APCs (AP M ).

d

dt AP M = βAP AP (kAP 1 + tanh(V − kAP 2)) − δAP M AP M

(3)

1427

JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, OCTOBER 2017 3

The first term, as just explained, denotes the dynamics of

APCs maturation. The second term, δAP M AP M , denotes the natural decay of the mature APCs, where δAP M is the decay rate.

Equation (4) represents the population of naive CD8+ T

cells (TN ).

d

dt TN = αTN (TN0 − TN) − πT AP MTN (4)

The term αTN (TN0 − TN) represents the homeostasis of CD8+ T cells, where αTN is the homeostasis rate. The term πT AP MTN denotes the activation of naive the CD8+ T cells, where πT is the activation rate.

Equation (5) represents the effector CD8+ T cell population

(TE).

d

dt TE = πT AP MTN + kTE AP MTE − δTE TE (5)

The term kTE AP MTE represents the proliferation of effec- tor CD8+ T cells. The term δTE TE represents the natural death of these cells, with δTE representing its decay rate.

Equation (6) represents B cells(B), both naive and effector ones. These populations were not considered separately in

order to simplify the model.

d

dt B = αB(B0 − B) + πBAP MB − βSAP MB

−βLAP MB − βBM AP MB (6)

The term αB(B0 − B) represents the B cells homeosta- sis, where αB is the homeostasis rate. The term πBAP M represents the proliferation of the active B cells. The terms

βSAP MB, βLAP MB and βBM AP MB denote the portions of active B cells that differentiate into short-lived plasma cells,

long-lived plasma cells and memory B cells, respectively. The

activation rates are respectively given by βS, βL and βBM . Equation (7) represents the short-lived plasma cells (PS).

d

dt PS = βSAP MB − δSPS (7)

The term δSPS denotes the natural decay of short-lived plasma cells, where δS is the decay rate.

Equation (8) represents the long-lived plasma cells (PL).

d

dt PL = βLAP MB − δLPL + γMBM (8)

The term δLPL denotes the natural decay of long-lived plasma cells, with δL representing the decay rate. The term, γMBM represents the production of these cells by memory B cells, where γM is the production rate.

Equation (9) corresponds to memory B cells (BM ).

d

dt BM = βBM AP MB +kBM1BM(1−

BM kBM2

)−γMBM (9)

The term kBM1BM(1− BMkBM2 ) represents the logistic growth of memory B cells, i.e., there is a limit to this growth. kBM1 represents the growth rate, and kBM2 limits the growth.

Equation (10) represents the antibodies. The terms πASPS and πALPL are the production of the antibodies by the short- lived and long-lived plasma cells, respectively. The production

rates are given by πAS and πAL, respectively. The term δAA denotes the natural decay of these cells, where δA is the decay rate.

d

dt A = πASPS + πALPL − δAA (10)

The model presented in this paper was based on an earlier

study [30], which described a mathematical model to represent

the human immune response to an infection by YF virus. So

the first difference is that this paper focus on modelling the

effects of the YF vaccine administered subcutaneously.

The previous work modeled the immune response to the YF

virus from infection of epithelial cells to secretion of antibod-

ies, considering various populations of cells and molecules,

in different stages and compartments. There were 19 ODEs

divided into two compartments: one representing the tissue

where the virus proliferate and the other the lymph nodes. In

order to consider all the cells and molecules, the model became

complex. Another issue is related to its adjust to reproduce

some behaviors described in the literature: as the number of

equations and parameters increases, so does the amount of

data and information needed to adjust the model.

The second difference between the two models is that this

work reduces the amount of equations from 19 to 10. The

reduced model described in this work considers only the main

populations of cells and molecules involved in the response to

the vaccine, and abstracts some details that are not crucial to

represent the behavior of the immune response. For example,

the distinct compartments are not represented here. Also, some

populations were not considered because no experimental data

is available to validate the simulations, such as the CD4+ T

cells. In future, more cell or molecule can be included in the

model again, if its role is important to explain or represent

some behavior that the reduced model presented in this section

could not represent. In fact, it is important to remember

that a mathematical model is an abstraction of the reality

and therefore simplifications are always necessary. This is

accentuated when the target of the model is the HIS response,

a complex network that involves many tissues, organs and

cells and performs several processes. The level of abstraction

depends on the purpose of the model. HIS can be seen at

various levels, from the level of substances produced by cells,

such as cytokines, to the level of cells and molecules, as in

the case of the current model. It also can reach the level of

population and epidemiological models that are interested in

a pattern of behavior observed in a population sample.

B. Computational model

For the resolution of the system of ODEs, a code was

implemented using the Python programming language, which

includes libraries for easily solving complex mathematical

problems. The library chosen was scipy [32]. This library has

a package called “integrate”. One of the functions available

in this package is called “odeint”, and it is used to solve

1428

JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, OCTOBER 2017 4

numerically a system of ODEs. The choice of the numerical

method to be used is made automatically by the function

based on the characteristics of the equations. The function

uses an adaptive scheme for both the integration step and the

convergence order. The function can solve the ODEs system

using either the BDF (Backward Differentiation Formula) or

the Adams method[33]. BDF is used for stiff equations and

the implicit Adams method is used otherwise.

The experiments were performed using Python version

2.7.10 using the Spyder integrated development environment

(IDE). The execution environment was composed by an Intel

Core i5 1.6 GHz processor, with 8 GB of RAM. The system

runs macOS Sierra version 10.12.5.

IV. COMPUTATIONAL RESULTS

In order to qualitatively validate our model, two experiments

were carried out. The first one simulates a scenario where an

individual was vaccinated against YF for the first time. The

standard dose of the vaccine was used in this scenario. The

results of the simulation were then compared to experimental

data obtained in the literature.

The second scenario is based on an experimental study[6] in

which smaller doses of the YF vaccine were tested. Compared

to the standard dose, the experimental study reported that,

to some extent, the reduction did not affect significantly the

percentage of serum-conversion. In this scenario, the compu-

tational experiments is executed many times, using distinct

values for the vaccine doses. For comparison purposes, the

computational experiments were carried out using the same

values of the experimental study[6].

All the initial values used for the variables as well as the

model parameters are presented in the Appendix. The tuning

of the model parameters was done manually, except for δA, whose value was extracted from the literature.

Usually the literature reports two distinct experimental data.

The first one is the viremia, i.e., the amount of virus present in

the bloodstream. The literature reports the viremia along time.

The second data reported in the literature is the antibody levels

along time. The values obtained by Equations (1) and (10)

are compared to the experimental values from the literature.

A. First vaccination

This section presents the computational results of a simu-

lation in which an individual was first vaccinated against YF.

Although the amount of virus particles varies depending on the

vaccine lot number, ranging from 2.3 to 12 times the minimum

value required by the WHO in the case of 17DD-YFV[6], in

this computational experiment we used a value equal to 27,476

IU. The 17DD-YFV is the YF vaccine developed by Bio-

Manguinhos / Fiocruz, one of the three producers prequalified

by WHO to supply vaccines to international agencies.

Figures 1 and 2 show the comparison of the antibody curve

generated as a result of 100 and 5,000 simulation days, respec-

tively, with the experimental results from the literature[34].

The result of the simulation is presented in separate figures

in order to better observe the increase of the antibody level

in the first days after vaccination. The levels of antibodies

Fig. 1. Antibody curve obtained by the model (line) and experimental data extracted from the literature[34] (dot).

Fig. 2. Antibody curve obtained by the model (line) and experimental data extracted from the literature[34] (dots).

obtained from the literature[34] are in GMT (Geometric Mean

Titers) and refer to time intervals after vaccination. The time

values used in the graph were obtained by averaging the times

of each interval. For example, the first point was the 30-45

day post-vaccination interval, the value used was 37 days,

the corresponding antibody level was 8762.8 IU / mL. From a qualitative point of view, one can observe that the values

obtained from the computational experiments are very close

to the experimental results. Also, the literature reports that the

antibody concentration in the bloodstream peaks at about two

weeks after vaccination[35], a value close to the one obtained

in the computational experiments.

Figure 3 shows the viremia curve obtained by the simula-

tion of the model in comparison with the experimental data

obtained from the literature [6]. As one can observe, in the

simulation the peak viremia value occurs in the fifth day, which

is consistent with the literature, which reports that it occurs be-

tween four and six days after vaccination[36], as well as with

the experimental results[6]. Also, the literature reports that ten

days after vaccination, viremia is undetectable[36], which is

consistent with the computational results. For some patients,

1429

JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, OCTOBER 2017 5

Fig. 3. Viremia curve obtained by model (line) and experimental data obtained from the literature[6](dots). Each dot in time scale represents a distinct patient.

however, the viremia can be detectable, as the experimental

results shows[6].

B. Dose-Response

An experimental work[6] has reported that “doses from

27,476 IU to 587 IU induced similar seroconversion rates

and neutralizing antibodies geometric mean titers (GMTs)”.

Based on this study, a second scenario studies the results of

our model when different dose values are administered. The

values used in the simulation are the same to those used by

the experimental work[6]: 31 IU, 158 IU, 587 IU, 3,013 IU,

20,447 IU and 27,476 IU.

Figure 4 and 5 show the viremia curves obtained by the

model for distinct vaccine doses. Figure 5 uses a smaller

scale to allow the visualization of the simulated viremia

curve obtained after administration of the dose with 587 IU

(represented by diamonds).

As one can observe, all doses greater than 3,013 IU produce

high levels of viremia. Although the viremia increases with

the use of doses with higher concentrations, the antibody

response presents a very subtle difference, as one can observe

in Figure 6. The 587 IU dose, which presented a much smaller,

unremarkable viremia in Figure 4 compared to the doses with

higher concentrations, was also able to induce an antibody

response similar to those induced by formulations with higher

concentrations.

Figure 7 presents the antibody curves obtained by simu-

lating 4,000 days after vaccination. The results suggest that

the duration of immunity does not appear to be affected by

the use of vaccine formulations with distinct concentrations:

all doses above 587 IU present similar results. Although

the reference paper has studied the duration of immunity

for a smaller period of time, approximately 10 months after

vaccination[6], its conclusions were similar to the one obtained

by the computational experiments: “GMTs of each group

were not statistically different from the reference vaccine.”.

The computational results are also in agreement with other

studies. The first one[37] concluded: “there was no correlation

between the level and duration of detectable 17D viremia

Fig. 4. Viremia curves obtained by the model when distinct initial values of V (vaccine virus) are used. The values represent distinct vaccine doses. For doses equal to 31 IU and 158 IU, the viremia was equal to zero.

Fig. 5. Viremia curves obtained by the model when distinct initial values of V (vaccine virus) are used. The values represent distinct vaccine doses. The scale was changed in order to better visualize the viremia induced after administration of a dose with 587 IU. For doses equal to 31 IU and 158 IU, the viremia was equal to zero.

and the postvaccination nAb level”. Another paper presents a

similar conclusion[38]: ”it was also found that the serological

response was not related to virus dose as the titres obtained

with high or low doses of virus was at the same level”.

It is noteworthy that doses using 31 IU and 158 IU did

not produce viremia nor antibody titers, so the curves are

superimposed on the x-axis.

C. Discussion

In this work we consider that the vaccine does not cause

adverse events, such as Yellow fever vaccine-associated vis-

cerotropic disease (YEL-AVD) and Yellow fever vaccine-

associated neurotropic disease (YEL-AND), because they are

rare.

Although YF has been used in this study, it must be

stressed that the concept presented in the mathematical model

is generic enough to represent the action of other diseases

or vaccines in the human immune system. However, the

1430

JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, OCTOBER 2017 6

Fig. 6. Antibody curves generated by the computational model. The model simulates the antibody concentrations during a period of 50 days for different doses of the vaccine. For doses equal to 31 IU and 158 IU, the antibody curves were equal to zero.

Fig. 7. Antibody curves generated by the computational model. The model simulates the antibody concentrations during a period of 4,000 days for different doses of the vaccine. For doses equal to 31 IU and 158 IU, the antibody curves were equal to zero.

initial conditions and parameters were adjusted to describe the

Human Immune System (HIS) response to the YF vaccine. The

immune response to a distinct vaccine would probably require

changes in both initial conditions and parameters values.

The obtainment of experimental data to adjust and validate

our model was not a trivial task. Studies on the duration of

immunity are difficult to interpret because different groups

use distinct methods to evaluate seroprotection. There is no

well-established serological value of protection in humans and

cellular immunity data is very scarce. Also, the values reported

for viremia use distinct units, which cannot be converted into

distinct ones due to the different methods used to obtain data.

These factors made it difficult to obtain experimental data

compatible with the standards and units used in the model

presented in this work, and consequently to use more studies

available in the literature to adjust and validate our model.

Since the model parameters represent some biological pro-

cesses, such as activation, neutralization and decay, after the

model is validated it will be possible to predict which effect

a given value change in a parameter will cause in the model

as a whole.

V. CONCLUSION

This work presented a reduced mathematical-computational

model to represent the immune response to the YF vaccine.

In other to validate the model, two distinct scenarios were

simulated. The first one simulates the immune response to the

administration of the standard dose of the 17DD-YFV. The

second one simulates the immune response to distinct doses

of vaccine. Two key values, viremia and antibody level, were

collected and compared to the values reported in the literature.

From a qualitative point of view, the results obtained by the

computational model reproduced the clinical results that can

be found in the literature.

As future works, we would like to improve the qualitative

results obtained from our model. Additional computational

experiments, such as the effects of a) a booster dose and b) a

reduction in the population of TCD8+ naive. Also, a sensitivity

analysis will be performed to identify sensitive parameters and

to identify connections between change in parameters values

and computational results.

APPENDIX A

INITIAL CONDITION AND PARAMETERS TABLES

TABLE I MODEL PARAMETERS

Parameter Value

πV 4.0 (day −1)

cv1 2 × 103 (virus titer × day−1) cv2 3 × 101 (virus titer) kv1 4.875 × 10−4 (day−1 × A−1) kv2 1.5694 × 10−3 (day−1 × T−1E ) αAP 2.5 × 10−3 (day−1) βAP 3.0 × 10−1 (day−1) kAP 1 1.0 (dimensionless) kAP 2 2 × 102 (virus titer) δAP M 5.38 × 10−1 (day−1) αTN 2.17 × 10−4 (day−1) πT 1 × 10−2 (day−1) kTE 1 × 10−5 (day−1) δTE 1 × 10−1 (day−1) αB 6.0 (day

−1) πB 1.77 × 10−3 (day−1) βS 6.72 × 10−1 (day−1) βL 8.05 × 10−3 (day−1) βBM 1 × 10−3 (day−1) δS 2.0 (day

−1) δL 2.22 × 10−4 (day−1) γM 1.95 × 10−6 (day−1) kBM1 1 × 10−5 (day−1) kBM2 10.0 (BM ) πAS 5 × 10−1 (day−1) πAL 1.7 × 10−1 (day−1) δA 4 × 10−2 (day−1)*

*Value extracted from [39] and [40] apud [41]

1431

JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, OCTOBER 2017 7

TABLE II MODEL VARIABLES AND INITIAL VALUES

Variable Description Initial value V Vaccine virus 27476 AP Immature APCs 10

3* AP M Mature APCs 0 TN Naive CD8+ T cells 10

3* TE Effectors CD8+ T cells 0 B B cells 103* PS Short-lived plasma cells 0 PL Long-lived plasma cells 0 BM Memory B cells 0 A Antibodies 0

*Values based on [41]

ACKNOWLEDGMENT

The authors would like to express their thanks to CAPES,

CNPq, FAPEMIG and UFJF for funding this work.

REFERENCES

[1] M. Theiler and H. H. Smith, “The use of yellow fever virus modified by in vitro cultivation for human immunization,” J. Exp. Med., vol. 65, no. 6, pp. 787–800, May 1937.

[2] WHO, “Situation report - yellow fever,” October 2016. [Online]. Available: http://apps.who.int/iris/bitstream/10665/249520/1/yellowfeversitrep12Aug16- eng.pdf

[3] L. Z. Goldani, “Yellow fever outbreak in brazil, 2017,” Brazilian Journal of Infectious Diseases, vol. 21, no. 2, pp. 123–124, 2017.

[4] WHO, “Weekly epidemiological record,” April 2017. [Online]. Available: http://apps.who.int/iris/bitstream/10665/255040/1/WER9216.pdf?ua=1

[5] ——, “Fractional dose yellow fever vaccine as a dose- sparing option for outbreak response,” July 2016. [Online]. Available: http://apps.who.int/iris/bitstream/10665/246236/1/WHO-YF- SAGE-16.1-eng.pdf?ua=1&ua=1

[6] R. M. Martins, M. d. L. S. Maia, R. H. G. Farias, L. A. B. Camacho, M. S. Freire, R. Galler, A. M. Y. Yamamura, L. F. C. Almeida, S. M. B. Lima, R. M. R. Nogueira et al., “17dd yellow fever vaccine: a double blind, randomized clinical trial of immunogenicity and safety on a dose- response study,” Human vaccines & immunotherapeutics, vol. 9, no. 4, pp. 879–888, 2013.

[7] A. C. Campi-Azevedo, P. de Almeida Estevam, J. G. Coelho-dos Reis, V. Peruhype-Magalhães, G. Villela-Rezende, P. F. Quaresma, M. d. L. S. Maia, R. H. G. Farias, L. A. B. Camacho, M. da Silva Freire et al., “Subdoses of 17dd yellow fever vaccine elicit equivalent virolog- ical/immunological kinetics timeline,” BMC infectious diseases, vol. 14, no. 1, p. 391, 2014.

[8] F. Pappalardo, D. Flower, G. Russo, M. Pennisi, and S. Motta, “Com- putational modelling approaches to vaccinology,” Pharmacological re- search, vol. 92, pp. 40–45, 2015.

[9] I. A. Doytchinova and D. R. Flower, “Quantitative approaches to computational vaccinology,” Immunology and Cell Biology, vol. 80, no. 3, p. 270, 2002.

[10] V. Brusic and N. Petrovsky, “Bioinformatics for characterisation of aller- gens, allergenicity and allergic crossreactivity,” Trends in immunology, vol. 24, no. 5, pp. 225–228, 2003.

[11] P. D. Taylor and D. R. Flower, Immunoinformatics and Computational Vaccinology: A Brief Introduction. Boston, MA: Springer US, 2007, pp. 23–46.

[12] D. R. Flower, Bioinformatics for vaccinology. John Wiley & Sons, 2008.

[13] C. R. B. Bonin, G. C. Fernandes, R. W. dos Santos, and M. Lobosco, “Mathematical modeling based on ordinary differential equations: A promising approach to vaccinology,” Human vaccines & immunother- apeutics, vol. 13, no. 2, pp. 484–489, 2017.

[14] C. DeLisi and J. A. Berzofsky, “T-cell antigenic sites tend to be am- phipathic structures,” Proceedings of the National Academy of Sciences, vol. 82, no. 20, pp. 7048–7052, 1985.

[15] N. Kumar, B. S. Hendriks, K. A. Janes, D. de Graaf, and D. A. Lauffenburger, “Applying computational modeling to drug discovery and development,” Drug Discov. Today, vol. 11, no. 17-18, pp. 806–811, Sep 2006.

[16] A. S. De Groot, L. Moise, J. A. McMurry, and W. Martin, Epitope- Based Immunome-Derived Vaccines: A Strategy for Improved Design and Safety. New York, NY: Springer US, 2009, pp. 39–69.

[17] F. M. Oliveira, I. E. Coelho, M. D. Lopes, A. G. Taranto, M. C. Junior, L. L. Santos, J. A. Villar, C. T. Fonseca, and D. D. Lopes, “The Use of Reverse Vaccinology and Molecular Modeling Associated with Cell Proliferation Stimulation Approach to Select Promiscuous Epitopes from Schistosoma mansoni,” Appl. Biochem. Biotechnol., vol. 179, no. 6, pp. 1023–1040, Jul 2016.

[18] R. Rappuoli, M. J. Bottomley, U. D’Oro, O. Finco, and E. De Gregorio, “Reverse vaccinology 2.0: Human immunology instructs vaccine antigen design,” J. Exp. Med., vol. 213, no. 4, pp. 469–481, Apr 2016.

[19] M. Michalik, B. Djahanshiri, J. C. Leo, and D. Linke, “Reverse Vaccinology: The Pathway from Genomes and Epitope Predictions to Tailored Recombinant Vaccines,” Methods Mol. Biol., vol. 1403, pp. 87–106, 2016.

[20] F. Andreoni, G. Amagliani, and M. Magnani, “Selection of Vaccine Candidates for Fish Pasteurellosis Using Reverse Vaccinology and an In Vitro Screening Approach,” Methods Mol. Biol., vol. 1404, pp. 181– 192, 2016.

[21] Y. T. Yang, Y. H. Chow, K. N. Hsiao, K. C. Hu, J. R. Chiang, S. C. Wu, P. Chong, and C. C. Liu, “Development of a full-length cDNA-derived enterovirus A71 vaccine candidate using reverse genetics technology,” Antiviral Res., vol. 132, pp. 225–232, Aug 2016.

[22] M. Meunier, M. Guyard-Nicodeme, E. Hirchaud, A. Parra, M. Chemaly, and D. Dory, “Identification of Novel Vaccine Candidates against Campylobacter through Reverse Vaccinology,” J Immunol Res, vol. 2016, p. 5715790, 2016.

[23] A. S. De Groot, A. Bosma, N. Chinai, J. Frost, B. M. Jesdale, M. A. Gonzalez, W. Martin, and C. Saint-Aubin, “From genome to vaccine: in silico predictions, ex vivo verification,” Vaccine, vol. 19, no. 31, pp. 4385–4395, Aug 2001.

[24] E. M. Lafuente and P. A. Reche, “Prediction of MHC-peptide binding: a systematic and comprehensive overview,” Curr. Pharm. Des., vol. 15, no. 28, pp. 3209–3220, 2009.

[25] R. Gomez-Bombarelli, J. Aguilera-Iparraguirre, T. D. Hirzel, D. Duve- naud, D. Maclaurin, M. A. Blood-Forsythe, H. S. Chae, M. Einzinger, D. G. Ha, T. Wu, G. Markopoulos, S. Jeon, H. Kang, H. Miyazaki, M. Numata, S. Kim, W. Huang, S. I. Hong, M. Baldo, R. P. Adams, and A. Aspuru-Guzik, “Design of efficient molecular organic light- emitting diodes by a high-throughput virtual screening and experimental approach,” Nat Mater, Aug 2016.

[26] A. R. Geanes, H. P. Cho, K. D. Nance, K. M. McGowan, P. J. Conn, C. K. Jones, J. Meiler, and C. W. Lindsley, “Ligand-based virtual screen for the discovery of novel M5 inhibitor chemotypes,” Bioorg. Med. Chem. Lett., Jul 2016.

[27] Y. Xu, L. Yue, Y. Wang, J. Xing, Z. Chen, Z. Shi, R. Liu, Y. C. Liu, X. Luo, H. Jiang, K. Chen, C. Luo, and M. Zheng, “Discovery of Novel Inhibitors Targeting Menin-Mixed Lineage Leukemia (MLL) Interface Using Pharmacophore- and Docking-Based Virtual Screening,” J Chem Inf Model, Aug 2016.

[28] S. Khalili, H. Mohammadpour, M. Shokrollahi Barough, and P. Kokhaei, “ILP-2 modeling and virtual screening of an FDA-approved library:a possible anticancer therapy,” Turk J Med Sci, vol. 46, no. 4, pp. 1135– 1143, 2016.

[29] D. Le, J. D. Miller, and V. V. Ganusov, “Mathematical modeling provides kinetic details of the human immune response to vaccination,” Frontiers in Cellular and Infection Microbiology, vol. 4, no. 177, 2015.

[30] C. Bonin, R. W. dos Santos, G. Fernandes, and M. Lobosco, “Compu- tational modeling of the immune response to yellow fever,” Journal of Computational and Applied Mathematics, vol. 295, pp. 127–138, 2016.

[31] S. Goutelle, M. Maurin, F. Rougier, X. Barbaut, L. Bourguignon, M. Ducher, and P. Maire, “The Hill equation: a review of its capabilities in pharmacological modelling,” Fundam Clin Pharmacol, vol. 22, no. 6, pp. 633–648, Dec 2008.

[32] Odeint, “Odeint’s homepage,” September 2014. [Online]. Available: http://docs.scipy.org

[33] R. J. LeVeque, Finite difference methods for ordinary and partial differ- ential equations - steady-state and time-dependent problems. SIAM, 2007.

[34] Collaborative Group for Studies on Yellow Fever Vaccines, “Duration of post-vaccination immunity against yellow fever in adults,” Vaccine, vol. 32, no. 39, pp. 4977–4984, 2014.

1432

JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, OCTOBER 2017 8

[35] T. P. Monath, M. Gershman, J. E. Staples, and A. D. Barrett, “38 - yellow fever vaccine,” in Vaccines (Sixth Edition), 6th ed., S. A. Plotkin, W. A. Orenstein, and P. A. Offit, Eds. London: W.B. Saunders, 2013, pp. 870 – 968.

[36] T. P. Monath, “Yellow fever vaccine,” Expert Review of Vaccines, vol. 4, no. 4, pp. 553–574, 2005.

[37] B. Reinhardt, R. Jaspert, M. Niedrig, C. Kostner, and J. L’age-Stehr, “Development of viremia and humoral and cellular parameters of immune activation after vaccination with yellow fever virus strain 17d: a model of human flavivirus infection,” Journal of medical virology, vol. 56, no. 2, pp. 159–167, 1998.

[38] O. de Souza Lopes, S. S. D. de Almeida Guimarães, and R. de Carvalho, “Studies on yellow fever vaccine iiidose response in volunteers,” Journal of biological standardization, vol. 16, no. 2, pp. 77–82, 1988.

[39] P. Vieira and K. Rajewsky, “The bulk of endogenously produced igg2a is eliminated from the serum of adult c57bl/6 mice with a half-life of 6– 8 days,” European journal of immunology, vol. 16, no. 7, pp. 871–874, 1986.

[40] ——, “The half-lives of serum immunoglobulins in adult mice,” Euro- pean journal of immunology, vol. 18, no. 2, pp. 313–316, 1988.

[41] H. Y. Lee, D. J. Topham, S. Y. Park, J. Hollenbaugh, J. Treanor, T. R. Mosmann, X. Jin, B. M. Ward, H. Miao, J. Holden-Wiltse, A. S. Perelson, M. Zand, and H. Wu, “Simulation and prediction of the adaptive immune response to influenza A virus infection,” J. Virol., vol. 83, no. 14, pp. 7151–7165, Jul 2009.