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The COVID-19 pandemic has become a great challenge to scientific, biological and medical research as well as to economic and social sciences. Hence, the objective of infectious disease modeling-based data analysis is to recover these dynamics of infectious disease spread and to estimate parameters that govern these dynamics. The random aspect of epidemics leads to the development of stochastic epidemiological models. We establish a stochastic combined model using numerical scheme Euler, Markov chain and Susceptible-Exposed-Infected-Recovery (SEIR) model. The combined SEIR model was used to predict how epidemics will develop and then to act accordingly. These COVID-19 data were analyzed from several countries such as Italy, Russia, USA and Iran.

The epidemic outbreak of COVID-19 virus is of great interest to researches because it is fast spreading and contagious. Mathematical analysis and modeling as with Kingman, 1982 [

Epidemiology models predict how N individuals among a population move among four groups: susceptible S, exposed E, infective I and removed R. In this context “removed” means individuals who are either recovered from the disease and immune to further infection, or dead as in Tang, 2019 [

In epidemiology, the dynamics of infectious disease are often characterized by the nonlinear and the quick variation in sizes of different population groups (states). One of the most widely used models is the Susceptible-Infectious-Removed (SIR) model Bailey, 1975 [

I t = I 0 + ∫ 0 t b ( u , I u ) d u + ∫ 0 t σ ( u , I u ) d B u (1)

where b ( u , I u ) ∈ ℝ d × ℝ d → ℝ d and σ ( u , I u ) ∈ ℝ d × ℝ d → ℝ d × d denote respectively the drift of the disease transmission (infection rate) and the Brownian motion diffusion ( B t ) , ∀ t ∈ [ 0, T ] .

In order to approximate the number of affected individuals I n = ( I t n ; t ∈ [ 0, T ] ) , we apply the Euler scheme. The principle of this numerical scheme is based on the following points:

1) divide the interval [ 0, T ] : 0 = t 0 < t 1 < ⋯ < t n − 1 < t n = T and we denote

Π = sup i = 0 , ⋯ , n − 1 t i + 1 − t i .

2) approximate integrals in Equation (1) and we get

∫ t i t i + 1 b ( s , I s ) d s ≃ b ( t i , I t i ) ( t i + 1 − t i ) ,

∫ t i t i + 1 σ ( s , I s ) d B s ≃ σ ( t i , I t i ) ( B t i + 1 − B t i ) .

3) For i = 0 , ⋯ , n − 1 , calculate I t i + 1 n in terms of I t i n as following

I t i + 1 n = I t i n + b ( t i , I t i n ) ( t i + 1 − t i ) + σ ( t i , I t i n ) ( B t i + 1 − B t i ) .

The sequence of random variables ( B t i + 1 − B t i ) is that of independent random variables normally distributed.

The mostly widely studied class of epidemic models, and the one on which we focus in this paper, is the class of susceptible/infected-infectious/removed or SIR models. Under SIR (Susceptible-Infectious-Removed) model, the population consists of three groups/compartments categories.

S , I − 1, R + 1 ← ︸ γ I S , I , R → ︸ β S I S − 1, I + 1, R

In each group, the population transitions from the current state ( S , I , R ) to ( S − 1, I + 1, R ) with rate β S I and to ( S , I − 1, R + 1 ) with rate γ I .

We use a Combined Susceptible-Exposed-Infected-Recovery (CSEIR) model to illustrate stochastic model representation. CSEIR model is an extension of the SIR model. The transition events between different states of the CSEIR model are depicted in

The current state is ( S , E , I , R ) . The infection events appear in transitions to state ( S − 1, E + 1, I , R ) and ( S − 1, E + 1, I , R ) with respectively rates δ β S and β S I . We suppose that Exposed compartment is proportional to infected group by the factor δ . Infected individual becomes infectious is within the transition to state ( S , E − 1, I + 1, R ) with rate α S E . Last transition to state ( S , E , I − 1, R + 1 ) explains a removal event with rate γ I . Next section focuses on the transition probabilities based on the following stochastic differential equations

{ d S d t = − β S N ( I + δ E ) d I d t = β S I N + α E − γ I d E d t = δ β S E N − α E d R d t = γ I .

The challenge is to calculate transition probability (see Gillespie, 1977 and 2000 [

In order to assess the random aspect of epidemics models, we emphasize on the probabilistic version of the CSEIR model. As mentioned our model divides population into four compartments. Each compartment is modeled as a continuous time Markov chain over discrete state space as in

p ( s , e , i ) ; ( s ′ , e ′ , i ′ ) = { ℙ [ ( S n + 1 , E n + 1 , I n + 1 ) = ( s ′ , e ′ , i ′ ) | ( S n , E n , I n ) = ( s , e , i ) ] ℙ [ ( S n + 1 , E n + 1 , I n + 1 ) = ( s ′ = s + k , e ′ = e + l , i ′ = i + j ) | ( S n , E n , I n ) = ( s , e , i ) ] .

Since the total population size should remain constant, we assume that k ∈ { − 1,0 } and l , j ∈ { − 1,0,1 } . Under the assumption that no other instantaneous transitions are allowed, we get:

p ( s , e , i ) ; ( s ′ , e ′ , i ′ ) = { β s i N Δ t ( k , l , j ) = ( − 1 , 0 , 1 ) ; δ β s e Δ t ( k , l , j ) = ( − 1 , 1 , 0 ) ; α e Δ t ( k , l , j ) = ( 0 , − 1 , 1 ) ; γ i Δ t ( k , l , j ) = ( 0 , 0 , − 1 ) 1 − ( β s i N + δ β s e + α e + γ i ) Δ t ( k , l , j ) = ( 0 , 0 , 0 ) ; 0 otherwise .

In this section we have taken up the simulation study of the COVID 19 virus. To assess the performance of our combined SEIR model we have considered the number of infected and recovered cases of COVID 19 epidemic registered up during March and April 2020 in four countries Italy, Russia, USA and Iran. The results below of the simulation study based on the CSEIR model are incorporated graphically. Statistical data for the spread of COVID 19 in Italy, Russia, USA and Iran are collected from the website of World Health Organization provided by the Department of the Ministry of Health in each country [^{th} April 2020 and 33.000 infected Iranian by the first week of April 2020 as in ^{th} 2020, the situation in USA and Russia is less critical. Simulations using the CSEIR model predict around one million infected in USA and more than 150.000 infected in Russia by the end of April 2020 as in

Simulation of recovered individuals using CSEIR is shown in Figures 7-9 and ^{th}, 2020 during the initial outbreak of COVID-19 in both Italy and Iran; the second stage is from March 20^{th} to April 30^{th}. Our model predicted around 30.000 recovered Italian as in ^{th} 2020. The decrease of the recovered curve depends on the strength of the healthcare system and the steps taken up to stop the spreading of this virus in each country.

In this paper, we established a combined stochastic model through analyzing the existing data of an epidemic situation. Here, we studied the spread of COVID-19 (Infected individuals) to see whether the situation is still under control or not. Basically, if we could down the spread of the virus enough, we will be able to keep infections at a manageable level so that hospitals won’t be overcrowded. Finally, Statisticians and data experts around the globe need a constant flow of new and disaggregated data to monitor the spread of the virus. Stepping up this challenge is saving lives.

The author declares no conflicts of interest regarding the publication of this paper.

Elhiwi, M. (2021) Stochastic Model for the Spread of the COVID-19 Virus. Applied Mathematics, 12, 24-31. https://doi.org/10.4236/am.2021.121003