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MARKOV MODEL IN ANALYSIS OF SPREAD OF COVID -19

ABSTRACT

The world is experiencing a crisis which is a coronavirus or COVID-19. The disease first arose in Wuhan town of China in December 2019. Since the first patient was hospitalized, there has been 3.44M confirmed cases .1.1M patients have recovered and 244,00 have died as of 3rd May 2020. A convenient understanding of the nature of the illness will enable the minimization of the spread of the disease. Prevention and control are a major step that countries worldwide have undertaken to calm down the spread of Coronavirus. COVID -19 has affected the global economy and has done a lot of harm to people's livelihood. The aim of the study was to employ Markov model to predict the spread of the virus and compared it with other models of prediction. This is essential for policy making that aim to reduce the spread of coronavirus. Markov model has been used on Mexico population to predict the spread of the disease.

INTRODUCTION

The symptoms of the disease include fever in the body, coughing, breathing problem, having muscle pain, sore throat, sputum production (Poundel, Bhandari,2020). Most of the reported cases amounted in minor symptoms, others lead to pneumonia and body parts malfunctioning. The rapid increase of the spread of corona the world's continent has led to some countries to institute lockdowns.

The virus is spread via respiratory droplets that come out when coughing or sneezing. It is also transmitted when one touches contaminated surfaces and then touching your face area. The virus lasts on top of surfaces up to3 days. (National Institutes of Health, 2020). The time of contracting to time of showing the symptoms is between 2-15 days and the average being of 5 days. (Centers for Disease Control and Prevention, 2020 Zhou et al, 2020). The reverse transcription-polymerase chain reaction (RRT -PCR) recommended as a form of diagnosis. The nasopharyngeal swab is used. The diagnosis is done by the determination of risk factors, symptoms, and a CT scan for the chest one showing characteristic of pneumonia. (Jin YH et al, 2020).

The best control measure of the infection includes frequent washing of hands, maintaining physical distance from one another, and keeping the hands off the face. (Perlman, 2020). Using sanitized face masks is good for anyone who is suspected to have contracted the disease the virus and important when going to the public. Reducing social events and gatherings is also paramount. (Tang et al, 2020; Li et al, 2020). So far there is no antiviral treatment or vaccine for the COVID-19 crisis. The ones who have recovered its symptoms have been treated, some supportive care, experimental measures, and isolation.

The World Health Organization (WHO) classified COVID-19 outbreak as a Public Health Emergency of International Concern (PHEIC) on 30th Jan 2020 and a pandemic on 11th March 2020. The transmission has been found throughout 6 WHO regions and most nations have announced emergency alerts throughout.

LITERATURE REVIEW

Coronaviruses cause respiratory diseases to animals and human beings (Lira, 2020). This collection of viruses was thought to affect mainly animals until 2002 and 2003 when there came an outbreak of Severe Acute Respiratory Syndrome (SARS) in China. Bronchitis coronavirus is detected annually in 40x109 Gallus domesticus specimen tested globally. Porcupine transmissible gastroenteritis coronavirus causes the death of 90% infected swine (Cavanagh,2005). After SARS, another coronavirus arose in the Middle East it was referred to as the Middle East Respiratory Syndrome Coronavirus (MERS) (Cui, Li, Shi, 2019). Later in 2019, the coronavirus rose from Wuhan, China and it was referred to as Covid-19. This virus has spread worldwide.

COVID-19 is a worldwide health threat. Obtaining the relevant information on (Kriston, 2020) how it is transmitted, severity and the control measure to be undertaken is important. With the appropriate information on the epidemiology of COVID -19 transmission models are developed to enable relevant policy to be introduced and the necessary control measure to be implemented. Daily decision making and public impressions are dependent on news broadcasted and mostly this information is on new corona cases, recoveries, and total deaths. The figures are recommended to be publicized for surveillance by the World Health Organization.

Due to scarce information on the epidemiology of COVID -19 the traditional modeling of related viruses is used. Various assumptions are made for example incubation time for the disease. In countries where the assumption is not clear or reasonable the application of models to predict the transmission remains very uncertain. The results of models pose a problem to be converted to the language of public health practice.

To predict the transmission rate of the coronavirus across the globe there is a need to understand the model to be used. Markov chain model will be used. To get a better understanding of the model there is a need for understanding of the stochastic process (Adigun, Adeleke, Adewesi, Olubiyi, Halid, Babalola, 2019). It is a mathematical model that develops over time in the probabilistic approach. It is usually defined as a group of random variables. There are various stochastic processes they include: Levy processes, Wiener processes, Random walks, Bernoulli processes, Poisson processes, and Markov processes. In this paper Markov processes are key

Markov chain is a system in mathematics where change of states is based on probabilistic laws (Maltby, Pakornrat, Jackson, 2020). The system shows that no matter how the process is reached at its current state the possible future states are fixed. This means that the probability of a certain state is solely dependent on the time that elapsed and the current state. The time is continuous; real numbers, discrete; integers, or an ordered set. Markov chain describes the process obtained at discrete intervals.

Markov chain has been an essential tool in medicine that influences the decision making in this sector. It is named after Andrew. A Markov (1856-1922) who started the theory of stochastic processes. Markov chain has yielded good results as compared to models such as the SIR model and SIAD model. Multistate models based on Markov processes in the determination of rates of transmission of the disease, and evaluation of stages of disease they yield an error sometimes

(Adigun, Adeleke, Adewesi, Olubiyi, Halid, Babalola, 2019).

STATEMENT OF THE PROBLEM

COVID -19 is one of the zoonotic viruses. It is spread from animals to human beings. COVID -19 is not an influenza. It is a new epidemic with its dynamic (Poundel, Bhandari, 2020). The virus has a high rate of transmission, high recorded deaths, economic and social disruption in the countries around the world.

The mortality cuts across people of ages and sex. People suffering from other underlying diseases such as cancer, asthma, kidney disease, lung disease, diabetes, hypertension, Liver disease, heart disease, or any other related disease have a high mortality rate in case they contract the coronavirus.

This study is aimed to use the Markov chain model to predict or forecast the spread of the virus so that countries around the world can take the right precaution to contain the disease. The information from the model also helps in decision making and policymaking by a country. It also helps us to understand the dynamic nature of COVID-19.

EXISTING MODELS AND SOLUTION APPROACHES.

There also exist dynamic models that predict the transmission of an infectious illness. The models are used since the spread of the disease is a stochastic process in nature (Bertsekas, 2005). The methods enable the making of a dynamic decision in many areas that include: medical treatment optimization (Schaefer et al., 2005), the economics of a nation (Van and Dana, 2003), and research operations ( Powell, 2007). The using techniques in predicting disease spread is limited (Ge et al. (2010).This shows the failure in existing models to fulfill the needed dynamic techniques of optimization.

These two characteristics of infectious illness pose a challenge in using of dynamic optimization methods. A) State-space being Large is prohibitive: The state of an infection is dependent on individuals on each compartment. The size of population growth is directly proportional to the state space size. For example, if the population size is N, the state space size of an SIR model is N(N+1)/2. It is a large state space and it means the methods will lose efficiency extremely fast.

B) The state being unobservable: Limited availability of diagnostic test results makes it hard to determine the status of the disease at any given period. For infectious illness with lengthy periods of incubation as compared to latency, the people who are infectious maybe the ones who are asymptomatic and are unlikely to be tested for a variable amount of time. Symptomatic people may be observed but the asymptomatic ones may be difficult to detect. This means that the entire state of the disease determination is not accurate. To use dynamic programming, the model of the spread of the infection is formulated to probability belief may be established on the status of the disease transmission using real-time data.

Due to problems encountered in the use of dynamic programming the proposed model is the use of a mathematical model. It retains the features of a dynamic technique; the state aggregation reduces state space while the accuracy of the model in predicting the spread of the disease is maintained. The model provides an actual status of the epidemic through observable data. The proposed model is the Markov models.

PROPOSED MODEL AND SOLUTION APPROACH.

In this paper the Markov model will be used. The model is divided into two parts: Markov chain of discrete-time and is used to predict the transmission of COVID-19 in time in the population. For this chain estimation data of the population that is infected is given in time. This information is known as susceptible, infected, recovered (SIR) which is a differential equation model for the equation.

Markov Chain Application

This model has been used in most of the countries to do the prediction of the spread of the infection. In this case will use Mexico to understand how the model is applied.32 states are considered. Therefore, E= {𝑒𝑖: 𝑒 ∈ 𝑀, 𝑖 ∈ ℕ} M represent Mexican territory covered, the set maps to

S = {𝑠1, 𝑠2, …, 𝑠32} of discrete states f: E → S in a bijective way. Let U be this process. It means that T ⊂ U (Lira, 2020). T stands for parameters that are temporal. T = {𝑡𝑗 |𝑗 ∈ ℕ} where [j]=[day]. It is the corresponding unit of the temporal parament obtained that day. Next is establishing conditions to be met by the probability. ∀𝑠𝑖𝑅𝑠𝑘∃𝑝𝑖𝑘 ∈ Ω, for every existence of the relationship between the i-th state //and the k-th state there is always ik which is the associated probability for the change of i-th towards k-th state. The probability is the balance for all existing possible interactions between i-th state and any k-th state, for instance if i-th state interacts with n different states, it means that the associates' probability for every siRsk=pik=1/n. The interaction of states with each other means that there exists a sharing of boundary territories. The stochastic matrix used to describe the process is:

Where

The vector in the initial process displays COVID-19 characteristic is Mexico as per 11 th March 2020 by the Ministry of Health. The cases we as follow: Mexico City: five, Coahuila: one, Chiapas: one, Sinaloa: one(recovered), Mexico state: two, Queretaro: one. The total amount of reported cases can be presented as a set of cases Where |C|=11. The probabilities of initial vector 𝜋0 for any state 𝜋=#cases/|C|. Vector can be formulated for any discrete time ti ∈ T,

Through computation of powers of the matrix, probability distributions are evaluated when,n=2,4,8,20,160,320 and it remind us that the [t]=[day].With this information computation of probability of distribution of COVID 19 in the coming days is determined.

Figure 1:A graph comparing probability distribution of COVID-19 in Mexico for period 11/03/2030-21/01/2021(Lira, 2020).

When nth power is computed we realize stationary distribution of set variables. There is a decrease in inflection points n=320 and the tendency as you go to stabilization where the peak of infection is reduced, and the spread of the disease is evident. From the data a probability distribution is established.

From Markov chain there is a slow infective rate of COVID-19and the tendency to spread through the whole Mexico town when n=320, the c≥1 case per state in case no preventive measure is undertaken.

Figure 2:States' probability density when n=320 (Lira, 2020).

b) SIR Model

The model enables the evaluation of susceptible, infected, and recovered people in the population. This is how the model is established (Lira, 2020):

S represents a susceptible population, I am the infected population, and R is the population that has recovered. By use of chemical kinetics a set of reaction is created:

s0 is the initial susceptible population and s0-x to represent susceptible people that may be lost over time by COVID-19,

Therefore.

The differential equation is.

Then integrate.

These pieces of information can be obtained:

To compute for the number of the infected:

I=x-y

R=y

Then,

This is a non -homogeneous linear differential equation:

Introducing the integrating factor:

Integrating:

Substituting the limits:

Recalling that:

Therefore, the initial susceptible population is:

S0=146.54x106

The constants are:

Table 1:No.of estimated: Infected(I),susceptible(S),recovered(R) throughout the period.(Cases pb: probable cases ( Lira,2020).

CONCLUSION AND FUTURE RESEARCH

During this pandemic with a high number of reported cases daily there is a need to nowcast and forecast transmission rate. This information is important for public health planning and control both internationally and locally. Without intervention from public health the prediction made by the model will not be of value. The spread will be immense.

Over the years there has been a substantial effort in the development of ways to enable real-time decision making in case of an epidemic outbreak. For long dynamic programming has been employed to do prediction emerging infectious diseases. To solve the problem while a large population is in place discrete-time Markov chain models. These models have several advantages. A) The model can use dynamic optimization to enable the selection of optimal dynamic health policies. b) A little state space size can be used to predict the spread of an infectious disease in a large population with a high level of accuracy and computation time. The current stochastic models have been limited to a moderate-sized population. The computer simulation has been used to obtain the stochastic dynamics of diseases that are highly infections and in large populations (Ele laboratory, 2020).

Markov models provide a mathematical framework to obtain all the stochastic dynamics of illnesses that are infectious. It allows the modeler to include their ways in the representation of the known history of the infection and the spreading characteristics (Yaesoubi, Cohen, 2011).

The model has some disadvantages that should not be overlooked. The framework needs to utilize grids as a way of reducing the Markov model state size for modeling the spread of disease for a large population. The techniques used to optimize the size of the grid to minimize the error of approximation and the development of alternative methods to predict the spread of infectious diseases are gaps to be researched in the future.

APPENDIX A: Markov Process Definition

The transition rates involved in the system (Zhang, You, Cai, Sun, Hu, 2020):

It determines the Markov process continuous time of the model.

(i) Infection:

(ii) Quarantine:

(iii) Symptoms Production:

(iv) Hospitalization:

(v) Symptom relief:

(vi) Recovery:

(vii) Death

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