# A Stochastic Switched Epidemic Model with Two Epidemic Diseases

Authors: Amine El Koufi; Abdelkrim Bennar; Noura Yousfi
Journal: Complexity
 (2021)

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## Abstract

In this paper, we study a stochastic epidemic model with double epidemics which includes white noise and telegraph noise modeled by Markovian switching. Sufficient conditions for the extinction and persistence of the diseases are established. In the end, some numerical simulations are presented to demonstrate our analytical results.


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## Body

## 1. Introduction

Compartmental models represent an important mathematical tool to describe the spread of infectious diseases [1–6]. A mathematical model that describes the spread of infectious diseases currently known as the SIR model divides the population into three classes, the sensitive S, infectious I, and recovered R, and was firstly developed by Kermack and Mckendrick [7] in 1927. To take into account the case of recovered individuals who lose their immunity and return to the susceptible compartment, we use SIRS epidemic models investigated by many scientists (see, for example, [8–10]). In classic SIRS epidemic models, the disease caused by one virus but in the real world, there are certain diseases to which the population is infected with two or more viruses (see [11–14]). Zhao et al. [15] established the following SIRS model describing the dynamics of two viruses (see Figure 1):(1)S′t=μ−β1StI1t−β2StI2t−μSt+δRt,I1′t=β1StI1t−μ+γ1I1t,I2′t=β2StI2t−μ+γ2I2t,R′t=γ1I1t+γ2I2t−μ+δRt,where St denotes the number of susceptible individuals at time t, I1t and I2t are, respectively, the numbers of infected individuals with virus A and virus B, Rt denotes the number of removed individuals at time, the parameters β1 and β2 are the disease transmission coefficient with virus A and virus B, respectively, μ represents the birth rate of the population and the natural death rate of the population, γ1 and γ2 are the recovery rates of infectious individuals with virus A and with virus B, and δ is the lost immunity rate.Figure 1
The compartmental diagram for theSIRS model with two epidemic diseases.However, the biological systems are necessarily touched via the environmental noise, which can present an additional degree of realism compared with their corresponding deterministic models. Thus, it is an important component in an ecosystem. Consequently, many works existing in the literature studied the effects of white noise on epidemic models (see, for example, [16–18]). El Fatini et al. [19] studied a stochastic epidemic model. They assumed that environmental perturbations are of the white noise type. They showed the existence and the uniqueness of the solution of their proposed model. They established sufficient conditions for extinction by constructing some suitable Lyapunov function. And they showed sufficient conditions for the persistence of the disease under some conditions on parameters of the model. Liu et al. [20] investigated the dynamical behavior of a higher-order stochastically perturbed SIRI epidemic model with relapse and media coverage. They showed sufficient measures for the existence and uniqueness of an ergodic stationary distribution of their stochastic model by constructing a suitable Lyapunov function. And they obtained sufficient conditions for the elimination of the infectious disease. Tang et al. [21] investigated the threshold of a class of the stochastic SIRS epidemic model with with nonlinear incidence, which determines the extinction and persistence of the disease. In [22], Qi et al. investigated the dynamics of a nonautonomous stochastic SIS epidemic model with nonlinear incidence rate and double epidemic. Rajasekar and Pitchaimani [23] have formulated the following stochastic SIRS model with two viruses:(2)dSt=μ−β1StI1t−β2StI2t−μSt+δRtdt+σ1StdB1t,dI1t=β1StI1t−μ+γ1I1tdt+σ2I1tdB2t,dI2t=β2StI2t−μ+γ2I2tdt+σ3I2tdB3t,dRt=γ1I1t+γ2I2t−μ+δRtdt+σ4RtdB4t,where Bit for i=1,2,3,4 are independent standard Brownian motions which are defined on a complete probability space Ω,ℱ,ℙ with filtration ℱtt≥0 satisfying the usual conditions and σi2 for i=1,2,3,4 denote the intensity of the white noise.On the other hand, the white noise cannot express the phenomena that the population may submit sudden catastrophic collisions in nature. The telegraph noise can be illustrated as a switching between two or more regimes of environment, which differ by factors such as nutrition or socio-cultural factors [24–26]. In fact, switching between environmental regimes is frequently memoryless and the waiting time for the next switching follows the exponential distribution [27]. So, the regime switching can be modeled by a continuous-time Markov chain rtt≥0 taking values in a finite-state space S=1,…,m. Many works have introduced Markov process into their models. For example, in [28], Zhang and Peng addressed a stochastic cholera epidemic model with vaccination. They presented the threshold of a stochastic cholera epidemic model which determines the extinction of the disease and the existence of a unique ergodic stationary distribution. Wang et al. [29] discussed the asymptotic properties of the solution of a stochastic SIRS epidemic model with Beddington–DeAngelis incidence rate and Markovian switching. They proved that the proposed model is stochastically asymptotically stable in the large. In addition, they presented threshold values and conditions which determine disease extinction and persistence. In [30], Li et al. proposed a stochastic SIRS model with vaccination and regime switching. They proved sufficient conditions for the extinction and the existence of the stationary distribution of the population. Motivated by the above facts, in this work, we will initially include telegraph noise into stochastic model (1) to obtain a more realistic model. Then, the following stochastic white and telegraphic noise perturbations on the diseases transmission coefficient are assumed:(3)β1dt⟶β1rtdt+σ1rtdB1t,β2dt⟶β2rtdt+σ2rtdB2t.Then, the stochastic SIRS epidemic model with two viruses (1) can be expressed as the following stochastic system under regime switching:(4)dSt=μrt−β1rtStI1t−β2rtStI2t−μrtSt+δrtRtdt−σ1rtStI1tdB1t−σ2rtStI2tdB2t,dI1t=β1rttI1t−μrt+γ1rtI1tdt+σ1rtStI1tdB1t,dI2t=β2rtStI2t−μrt+γ2rtI2tdt+σ2rtStI2tdB2t,dRt=γ1rtI1t+γ2rtI2t−μrt+δrtRt.This article proposes a stochasticSIRS epidemic model with regime switching and double epidemic hypothesis. Our main purpose is to investigate the effect of the white and telegraph noises on the spread dynamics of the disease in the population. The rest of the paper is as follows. In Section 3, we show the existence and uniqueness of a global positive solution to system (4). In Sections 4 and 5, we study the existence of a stochastic threshold for the extinction and the persistence in mean of the disease. In last section, we present some numerical simulations to demonstrate our main theoretical results.


## 2. Existence and Uniqueness of the Nonnegative Solution

Throughout this paper, we letΩ,ℱ,ℱtt≥0,ℙ be a complete probability space with a filtration ℱtt≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while ℱ0 contains all ℙ-null sets). Let rtt≥0 be a right-continuous Markov chain on the probability space Ω,ℱ,ℱtt≥0,ℙ taking values in a finite-state space S=1,2,…,m with the generator Φ=ϕuv1≤u,v≤N given, for δ>0, by(5)Prt+δ=v|rt=u=ϕuvδ+oδ,ifu≠v,1+ϕuuδ+oδ,ifu=v.Here,ϕuv is the transition rate from u to v and ϕuv≥0 if u≠v, while(6)ϕuu=−∑u≠vϕuv.Suppose that the Markov chainrt is independent of the Brownian motion B⋅ and it is irreducible. Under this condition, the Markov chain has a unique stationary distribution π=π1,…,πN, which can be determined by solving the linear equation πΦ=0, subject to ∑i=1Nπi=1, and πi>0, ∀i∈S. Thereafter, for any vector h=h1,…,hNT, let h^=mini∈Mhi and hˇ=maxi∈Mhi.We consider the following stochastic system:(7)dϑt=ft,ϑt,rtdt+gt,ϑt,rtdBt,∗where Bt is a d-dimensional standard Wiener process defined on a complete probability space Ω,ℱ,ℱtt≥0,ℙ. Denoted by C1,2ℝd×S;ℝ+, the family of all nonnegative functions Q is defined on ℝd×S such that they are continuously twice differentiable in ϑ. The operator ℒ associated with ∗ is defined as follows:(8)LQϑ,i=Qtt,ϑ,i+Qϑt,ϑ,ift,ϑ,i+12gTt,ϑ,iQϑ,ϑt,ϑ,igt,ϑ,i+∑j∈SϕijQt,ϑ,j,where Qϑ and Qϑ,ϑ represent the gradient and Hessian of Q and T is the transpose of a matrix.By Itô’s formula, ifQ defined on ℝd×S, we have(9)dQϑt,i=LQϑt,idt+Qϑϑt,igϑt,idBt.The total population in system (4) verifies the following equation:(10)dSt+I1t+I2t+Rt=μ−μSt+I1t+I2t+Rtdt.Thus,(11)St+I1t+I2t+Rt=1+S0+I10+I20+R0−1e−μrtt,for allt≥0.Since, forX0∈D, we have(12)St+I1t+I2t+Rt=1,for allt≥0a.s.Let(13)D=S+I1+I2+R∈ℝ+4:S+I1+I2+R=1.By using the same method as in [31, 32], we can easily prove the following theorem.Theorem 1.
LetX0=S0,I10,I20,R0∈D, then there is a unique solution St,I1t,I2t,Rt to system (4) on t≥0, and the solution will remain in D with probability 1.
Firstly, we define the following expressions which we use to determine the extinction of epidemics:(14)∑i=1mπiDi=∑i=1mπi−σ12i+β1i−μi+γ1i,∑i=1mπiAi=∑i=1mπi−σ22i+β2i−μi+γ2i.


## 3. Extinction

This section is consecrated to give sufficient conditions so that the epidemics die out. According to [33], we have the following lemma.Lemma 1.
Suppose that there exists a functionUu,i∈C1,2ℝ3×S;ℝ+ and ρu∈ℝ3×;ℝ+ disappears only at E0 such that(15)ℒUu,i≤−ρu,for allu,i∈ℝ3×S,limx⟶∞infj∈SUu,i=∞.Then, the equilibriumE0 of system (4) is globally asymptotically stable in probability.By Lemma1, we identify the conditions for stochastic asymptotic stability of the disease-free equilibrium which are presented by the following theorem.Theorem 2.
Ifβ2i≥σ22i and β1i≥σ12i for all i∈S, and(16)∑i=1mπiDi<0,∑i=1mπiAi<0,then the disease-free equilibrium E0 is globally stochastically asymptotically stable on D.Proof.
We consider the following Lyapunov function:(17)ℋS,I,R,i=m11−S2+η+ωiI11/η+θ+ϖiI21/θ+m2R2,for alli∈S,where m1, m2, η, and θ are real positive constants to be determined later. By Itô’s formula, we calculate ℒℋ as follows:(18)ℒℋ=−2m11−Sμ−μS−β1SI1−β2SI2+δR+m1σ12S2I12+m2σ22S2I22+1ηη+ωiI11/ηβ1S−μ+γ1+12η1η−1η+ωiσ12S2I11/η+1θθ+ϖiI21/θβ2S−μ+γ2+12θ1θ−1θ+ωiσ22S2I21/θ+2m2Rγ1I1+γ2I2−μ+δR+∑k≠i,k∈Sϕikωk−ωi+ϖk−ϖi=−2m11−S2+2m1β1SI11−S+2m1β2SI21−S−2m1δ1−SR+m1σ1S2I12+m1σ2S2I22+1ηη+ωiβ1I11/ηS−1ηη+ωiμ+γ1I11/η+12η1η−1η+ωiσ12S2I11/η+1θθ+ϖiβ2I21/θS−1θθ+ϖiμ+γ2I21/θ+12θ1θ−1θ+ωiσ22S2I21/θ+m2γ1RI1+m2γ2RI2−m2μ+δR2+I11/η∑k≠i,k∈Sϕikωk−ωi+I21/θ∑k≠i,k∈Sϕikϖk−ϖi.
Using the fact thatI≤1−S and S, I∈D, we obtain(19)ℒℋ≤−2m11−S2+2m1β1I11/η+2m1β2I21/θ+m1σ12I11/η+m1σ22I21/θ+1ηη+ωiβ1I11/ηS−1ηη+ωiμ+γ1I11/η+12η1η−1η+ωiσ12S2I1/η+1θθ+ϖiβ2I21/θS−1θθ+ϖiμ+γ2I21/θ+12θ1θ−1θ+ωiσ22S2I21/θ+m2γ1−m1δRI1+m2γ2−m1δRI2−m2μ+δR2+I11/η∑k≠i,k∈Sϕikωk−ωi+I21/θ∑k≠i,k∈Sϕikϖk−ϖi,for all η and θ≥1. Since we can choose m2<mini∈Sm1δi/γ1i,m1δi/γ2i, we obtain(20)ℒℋ≤−2m11−S2−m2μ+δR2+η+ωiηI11/ηηm1η+ωiσ12+2β1−μ+γ1+12ησ12+β1S−σ12S2+ηη+ωi∑k≠i,k∈Sϕikωk−ωi+θ+ωiθI21/θθm1θ+ωiσ22+2β2−μ+γ2+12θσ22+β2S−σ22S2+θθ+ϖi∑k≠i,k∈Sϕikϖk−ϖi≕−2m11−S2−m2μ+δR2+η+ωiηI11/ηηm1η+ωiσ12+2β1+12ησ12+fiS+ηη+ωi∑k≠i,k∈Sϕikωk−ωi+θ+ωiθI21/θθm1θ+ωiσ22+2β2+12θσ22+giS+θθ+ϖi∑k≠i,k∈Sϕikϖk−ϖi,where fix=−σ12ix2+β1ix−μi+γ1i and gix=−σ22ix2+β2ix−μi+γ2i. Note that the functions fix and gix are all increasing on 0,1, which means(21)fix≤fi1≕Di,gix≤gi1≕Ai,then(22)ℒℋ≤−2m11−S2−m2μ+δR2+η+ωiηI11/ηηm1η+ωiσ12+2β1+12ησ12+Di+ηη+ωi∑k≠i,k∈Sϕikωk−ωi+θ+ϖiθI21/θθm1θ+ϖiσ22+2β2+12θσ22+Ai+θθ+ϖi∑k≠i,k∈Sϕikϖk−ϖi.
Since the generator matrixΦ is irreducible, then for D=D1,…,DmT, there exists Ξ=ω1,…,ωmT solution of the Poisson system:(23)ΦΞ=−D+∑i=1mπiDiE⟶.And for A=A1,…,AmT, there exists Θ=ϖ1,…,ϖmT solution of the Poisson system:(24)ΦΘ=−A+∑i=1mπiAiE⟶,where E⟶ denotes the column vector with all its entries equal to 1.
Substituting (23) and (24) in (22), we get(25)ℒℋ≤−2m11−S2−m2μ+δR2+η+ωiηI11/ηηm1η+ωiσ12+2β1+12ησ12+ηη+ωiDi−∑i=1mπiDi+∑i=1mπiDi+θ+ϖiθI21/θθm1θ+ϖiσ22+2β2+12θσ22+θθ+ϖiAi−∑i=1mπiAi+∑i=1mπiAi.
Choosingη0 and θ0, such that(26)C1≕maxi∈Sσ122η0+η0η0+ωiDi−∑i=1mπiDi+∑i=1mπiDi<0,C2≔maxi∈Sσ222θ0+θ0θ0+ϖiAi−∑i=1mπiAi+∑i=1mπiAi<0.
Since we chooseη>max−ωi,η0,1 and θ>max−ϖi,θ0,1 such that(27)m1<min−η+minωiηmaxσ12+2maxβ1×C1,−θ+minϖiθmaxσ22+2maxβ2×C2.
According to Lemma1, the proof is completed.Theorem 3.
LetSt,I1t,I2t,Rt be the solution of system (4), then for any given initial value S0,I10,I20,R0∈D, model (4) has the following property:(28)limsupt⟶⟶∞1tln1−S+I1+I2+R≤∑i=1mπiβ12i−μiσ12i2σ12i+β22i−μiσ22i2σ22i,a.s.
If(29)∑i=1mπiβ12i−μiσ12i2σ12i+β22i−μiσ22i2σ22i<0holds, then the disease-free E0 is almost surely exponentially stable in D. Namely, the diseases die out with probability one.Proof.
Consider the following function:(30)GS,I1,I2,R,i=ln1−S+I1+I2+R.
By Itô’s formula, we obtain(31)dG=11−S+I1+I2+R−μr1−S−μrI1−μrI2−μr+2δrrR+2β1rSI1+2β2rSI2dt−2σ12rSI11−S+I1+I2+R2dt−2σ22rSI21−S+I1+I2+R2dt+2σ1rSI11−S+I1+I2+RdB1+2σ2rSI21−S+I1+I2+RdB2.
Note that(32)11−S+I1+I2+R−μr1−S−μrI1−μI2−μr+2δrR≤−μr.
Thus, we have(33)dG≤−2σ1rSI1−S+I1+I2+R−β1r2σ12r+β12r2σ12r−μr2dt+−2σ2rSI1−S+I1+I2+R−β2r2σ22r+β22r2σ22r−μr2dt+2σ1rSI11−S+I1+I2+RdB1+2σ2rSI21−S+I1+I2+RdB2.≤β12r−μrσ12r2σ12r+β22r−μrσ22r2σ22rdt+2σ1rSI11−S+I1+I2+RdB1+2σ2rSI21−S+I1+I2+RdB2.
Integrating both sides of (33) from 0 to t and dividing by t, we get(34)Gtt≤G0t+1t∫0tβ12ru−μruσ12ru2σ12ru+β22ru−μruσ22ru2σ22rudu+1t∫0t2σ1ruSuI1u1−Su+I1u+I2u+RudB1u+1t∫0t2σ2ruSuI2u1−Su+I1u+I2u+RudB2u.
LetFit=∫0t2σirSuIiu/1−Su+I1u+I2u+RudBiui=1,2, then(35)Fit,Fit=∫0t2σiruSuIiu1−Su+I1u+I2u+Ru2du≤4σˇi,t<∞.
Then, by strong law of large numbers, we obtain(36)limt⟶∞Fitt=0,a.s.
By the ergodic theory of the Markov chain, we can obtain(37)limsupt⟶⟶∞1t∫0tβ12ru−μruσ12ru2σ12ru+β22ru−μruσ22ru2σ22rudu≤∑i=1mπiβ12i−μiσ12i2σ12i+β22i−μiσ22i2σ22i,a.s.
Hence, if∑i=1mπiβ12i−μiσ12i/2σ12i+β22i−μiσ22i/2σ22i<0, then the disease-free E0 is almost surely exponentially stable in D.


## 4. Persistence

In this section, we explain sufficient conditions for the persistence of two epidemics.We defineht=1/t∫0thudu if h is an integrable function on 0,+∞.Definition 1.
System (4) is said to be persistent in the mean, if(38)liminft⟶∞1t∫0tIudu>0,a.s.
Let(39)R0s=∑i=0mπi2μi+γ1i+γ2i+σ1i2+σ2i2−k1iμi.Theorem 4.
LetSt,I1t,I2t,Rt be the solution of model (4). If R0s>0, then(40)liminft⟶∞I1+I2t≥I∗,liminft⟶∞Rt≥minγ^1,γ^2μˇ+δˇI∗,liminft⟶∞1−St≥μˇ+δˇ+minγ^1,γ^2μˇ+δˇI∗,where I∗=R0s/D¯ with D¯ is positive constant.Proof.
By system (4), one can obtain that(41)ℒ−lnI1=−β1iS+μi+γ1i+σ12i2S2,ℒ−lnI2=−β2iS+μi+γ2i+σ22i2S2.
Moreover,(42)ℒS+I1+I2=μi−μiS−μi+γ1iI1−μi+γ2iI2+δiR,ℒR=γ1iI1+γ2iI2−μi+δiR.
So, we consider the Lyapunov functionT defined on D by(43)TS,I1,I2,i=−lnI1−lnI2−k1iS+I1+I2−k2iR−υi,where k1i,k2iT is the unique positive solution of the following system:(44)β1i+β2i−k1iμi+∑l=1mϕilk1l=0,k1iδi−k2iμi+δi+∑l=1mϕilk2l=0.
Using Itô’s formula, one can get that(45)dT=LTdt−σ1iS+k1iSI1dB1t−σ2iS+k2iSI2dB2t,where(46)LT=−β1S+μi+γ1i+σ12i2S2−β2iS+μi+γ2i+σ22i2S2−k1iμi−μiS−μi+γ1iI1−μi+γ2iI2+δiR−k2iγ1iI1+γ2iI2−μi+δiR−∑l=1mϕilk1lS+I1+I2−∑l=1mϕilk2lR−∑l=1mϕilυl.
Using the fact thatS∈0,1, we deduce that(47)LT≤2μi+γ1i+γ2i+σ12i2+σ22i2−k1iμi−∑l=1mϕilυl−β1i+β2i−k1iμi+∑l=1mϕilk1lS−k1iδi−k2iμi+δi+∑l=1mϕilk2lR+k1iμi+γ1i−k2iγ1i−∑l=1mϕilk1lI1+k1iμi+γ2i−k2iγ2i+∑l=1mϕilk1lI2.
From (44), we get that(48)LT≤2μi+γ1i+γ2i+σ12i2+σ22i2−k1iμi−∑l=1mϕilυl+β1i+β2i+k1iγ1iI1+β1i+β2i+k1iγ2iI2.
In addition, we also have(49)LT≤2μi+γ1i+γ2i+σ12i2+σ22i2−k1iμi−∑l=1mϕilυl+βˇ1+βˇ2+kˇ1γˇ1I1+βˇ1+βˇ2+kˇ1γˇ2I2≤2μi+γ1i+γ2i+σ12i2+σ22i2−k1iμi−∑l=1mϕilυl+D¯I1+I2≔−R0i−∑l=1mϕilυl+D¯I1+I2,where(50)D¯=maxβˇ1+βˇ2+kˇ1γˇ1,βˇ1+βˇ2+kˇ1γˇ2.
Since the generator matrixΦ is irreducible, then for R0=R01,…,R0mT, there exists Υ=υ1,…,υmT solution of the Poisson system:(51)ΦΥ=−R0+∑i=1mπiR0i11⋅⋅1,which implies that(52)LT≤−R0s+D¯I1+I2.
Integrating (45) from 0 to t and dividing by t on both sides, we obtain(53)Ttt≤T0t−R0s+D¯I1+I2t−M1tt−M2tt,where Mjt=σjiSdBjt (for j=1,2) is a continuous martingale with M0=0 and its quadratic variation is as follows:(54)Mjt,Mjt=∫0tσjiSu2du≤σˇj2t.
Thus, the strong law of large number for local martingales [27] implies that(55)limt⟶∞Mjtt=0,a.s.
SinceS,I1,I1,R∈D, we can derive(56)TS,I1,I2,i=−lnI1−lnI2−k1iS+I1+I2−k2iR−υi≥−kˇ1−kˇ2−υˇ≔C.
Taking the superior limit on both sides of (45) and combining with (52), (55), and (56), we deduce that(57)liminft⟶∞I1+I2t≥R0sD¯≔I∗,a.s.
Integrating the third equation of (4) and dividing both sides by t, we have(58)Rt≥minγ^1,γ^2μˇ+δˇI1+I2t−Rt−R0μ^+δ^t.
Sincelimt⟶∞Rt−R0/t=0 and using (57), we get(59)liminft⟶∞Rt≥minγ^1,γ^2μˇ+δˇI∗.
Furthermore, it follows by1−S=I+R that(60)liminft⟶∞1−St≥μˇ+δˇ+minγ^1,γ^2μˇ+δˇI∗.
This completes the proof.


## 5. Simulation

In the following, we give numerical simulations to illustrate the main theoretical results.Example 1.
We consider the Markov chainrtt≥0 taking values in S=1,2, with generator defined by the following:(61)Φ=−112−2.
The parameter values of (4) are presented in Table 1.
Exact computations supply us∑i=12πiDj=−0.0352 and ∑i=12πiAj=−0.0467. Consequently, by Theorem 2, the disease-free equilibrium E0 is globally stochastically asymptotically stable on D. Figure 2 confirms this result.Table 1
Parameter values used in numerical simulations.
StatesCoefficientsμβ1β2δγ1γ2σ1σ210.250.50.60.50.40.30.20.120.30.60.40.40.20.40.120.2Figure 2
Sample paths ofSt in blue, I1t in red, I2t in black, Rt in green, and rt in brown in Example 1.Example 2.
In this example, we use the parameter values given in Table2 and the Markov chain used in Example 1. By simply computing, we obtain R0s>0. Theorem 4 shows that the two epidemics persist in the population. Figure 3 confirms this result.Table 2
Parameter values used in numerical simulations.
StatesCoefficientsμβ1β2δγ1γ2σ1σ210.010.90.80.40.20.30.60.820.010.80.80.40.30.20.70.7Figure 3
Sample paths ofSt in blue, I1t in red, I2t in black, Rt in green, and rt in brown in Example 1.


## 6. Conclusion

This paper investigates a stochastic switched epidemic model with double epidemic diseases and cure rate affected with a composite of environmental perturbations modeled by white and color noises. We established conditions for extinction and persistence of diseases.The theoretical result given in this paper shows that the stationary distribution of Markov chainrt controls the propagation of the epidemic in the population. To know, if the Markov chain rt takes the values which lead to find that ∑i=1mπiDi<0 and ∑i=1mπiAi<0, then the epidemic diseases die out. We have given numerical simulations in order to support and illustrate the main results of this paper.In this perspective, we prepare to include Lévy noises into epidemic models (4) to describe the effect of sudden and dangerous phenomena such as pandemic, tsunami, and volcano.



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