# Dynamical Behavior of the SEIS Infectious Disease Model with White Noise Disturbance

Authors: Yuheng Song; Qixing Han
Journal: Journal of Mathematics
 (2022)

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## Abstract

Mathematical model plays an important role in understanding the disease dynamics and designing strategies to control the spread of infectious diseases. In this paper, we consider a deterministic SEIS model with a saturation incidence rate and its stochastic version. To begin with, we propose the deterministic SEIS epidemic model with a saturation incidence rate and obtain a basic reproduction numberR0. Our investigation shows that the deterministic model has two kinds of equilibria points, that is, disease-free equilibrium E0 and endemic equilibrium E∗. The conditions of asymptotic behaviors are determined by the two threshold parameters R0 and R0c. When R0<1, the disease-free equilibrium E0 is locally asymptotically stable, and it is unstable when R0>1. E∗ is locally asymptotically stable when R0c>R0>1. In addition, we show that the stochastic system exists a unique positive global solution. Conditions d>σˇ2/2 and R0s<1 are used to show extinction of the disease in the exponent. Finally, SEIS with a stochastic version has stationary distribution and the ergodicity holds when R0∗>1 by constructing appropriate Lyapunov function. Our theoretical finding is supported by numerical simulations. The aim of our analysis is to assist the policy-maker in prevention and control of disease for maximum effectiveness.


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## Body

## 1. Introduction

Worldwide, human beings have been battling with various diseases and epidemic research is the main means of disease prevention and control in today’s society. Kermack and Mckendrick’s research has laid the foundation for the study of epidemic patterns [1]. Many authors have provided a number of ways for studying the epidemic model. For instance, Li and Ma [2] studied the SIS epidemic model and discussed the conditions about asymptotic behaviors of equilibrium points by defining threshold parameters. Xiao and Ruan [3] studied the stability of equilibrium points for an SIRS model. Alexander and Moghadas [4] used bifurcation analysis to study an SIRS epidemic model. For some diseases, a host will stay in a latent period before becoming infectious. Liu and Yang [5] and Li and Moldowney [6] have investigated the transmission dynamics of disease with a latent period.For plenty of sexually transmitted diseases, infections have no acquired immunity to the disease; therefore, the recovered individuals would become susceptible again. The population in the SEIS epidemic model can be divided into three classes: susceptible (S), exposed (E), and infectious (I), and Nt=St+It+Et. Some mathematical models have been developed in order to understand the disease transmission about the SEIS epidemic model. Naim et al. [7, 8] discussed the two significant equilibrium points and obtained the conditions of asymptotic behavior. Huo et al. [9] obtained the basic reproduction number about the SEIS epidemic model. Fan et al. [10] investigated the SEIS system with bilinear incidence, which can be written as(1)dS=Λ−βSI−dS+γIdt,dE=βSI−d+εE)dt,dI=εE−d+γ+μIdt.The parameters in the model are positive constants and summarized in the following:(2)Λ:the influx or recruitment of the susceptible and the exposed,β:the transmission cofficient betweenSandI,d:natural death rate,μ:the extra disease−related death rate,γ,ε:the transfer rates among the corresponding compartments.They showed that the basic reproductive numberR0=Λβε/dd+εd+μ+γ. If R0≤1, system (1) has a unique disease-free equilibrium P0, which is globally asymptotically stable. If R0>1, the disease-free equilibrium P0 is unstable and there exists a unique endemic equilibrium P∗, which is globally asymptotically stable.Capasso and Serio [11] first proposed the saturated incidence rate gIS=βIS/1+αI to describe the cholera epidemic, where gI tends to a saturation level when I is big enough. βI is used to measure disease infection ability and 1/1+αI is used to measure inhibition effort from the change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals. Under numerous circumstances, the saturation incidence rate is a better reflection of the actual situation. Model (1) with a saturated incidence rate can be described as follows:(3)dS=Λ−βStIt1+αIt−dSt+γItdt,dE=βStIt1+αIt−d+εEtdt,dI=εEt−d+μ+γItdt,where the parameters are the same defined with system (1).But the epidemic models in the real world are inevitably influenced by environment noise like white noise, telegraph noise, Levy noise, and so on, which is of great importance in the epidemic systems. On the one hand, Lahrouz et al. have taken parameter perturbation into epidemic models and have investigated their dynamic behaviors [12, 13]. Gray et al. [14] also showed the SIS epidemic model has a unique global positive solution after parameter perturbation.On the other hand, some authors have considered another noise disturbance, namely, system disturbance, that is, the environmental noise is proportional to the variables, and they have pointed out that this approach is reasonable. For example, Song et al. [15] discussed the stochastic cholera model and obtained stationary distribution. Liu and Wei [16] studied the stochastic SEIS epidemic model and obtained a unique global solution. Liu et al. [17] studied the stochastic S-DI-A models and obtained the persistence and extinction under white noise and telegraph noise. Here, we also consider system disturbance in model (3) and obtain the stochastic system as follows:(4)dS=Λ−βStIt1+αIt−dSt+γItdt+σ1SdB1t,dE=βStIt1+αIt−d+εEtdt+σ2EdB2t,dI=εEt−d+μ+γItdt+σ3IdB3t,where B1t,B2t,B3t are standard one-dimensional independent Wiener processes, σiσi>0,i=1,2,3 on a complete probability space Ω,ℱ,P, with filtration ℱ≥0 satisfying the usual conditions (i.e., it is of increasing and right continuous and ℱ0 contains all P-null sets). We denote by R+n, the positive area in Rn; therefore, we can write(5)R+n=x∈Rn:xi>0for all1≤i≤n.A d-dimensional stochastic differential equation can be written as follows:(6)dXt=fXt,tdt+gXt,tdBtont≥t0,and its initial value is Xt0=x0∈Rd. Define the differential operator L associated with (6) which is given by(7)L=∂∂t+∑k=1dfkx,t∂∂xk+12∑k,j=1dgTx,tgx,tkj∂2∂xk∂xj.Let X(t) be a homogeneous Markov process inEl (El denotes l dimensional Euclidean space); the stochastic equation can be described by(8)dXt=bXdt+∑r=1kgrXdBrt.The diffusion matrix is defined by the following equation:(9)Λx=λijx,λijx=∑r=1kgirxgrjx.


## 2. Deterministic Model

### 2.1. The Existence of Equilibria

To study equilibria, we calculate the basic reproduction number first in (3). We rewrite system (3) as follows:(10)dXdt=FX−VX,where F denotes the appearance in new infection individuals and V represents the rate of conversion of individuals in or out of each category. These two matrices in (3) can be shown as follows:(11)F=βSI1+αI00,V=d+εE−εE+d+μ+γI−Λ+βSI1+αI+dS−γI.The regeneration matrix isBC−1, B, C is the 2×2 Jacobian determinant, which is defined by(12)B=∂Fi∂XjE0,C=∂Vi∂XjE0,1≤i,j≤2.System (3) has a disease-free equilibrium E0=Λ/d,0,0. It is easy to get that(13)B=0βΛd00,C=d+ε0−εd+μ+γ.The spectral radius is theρBC−1 =  max1≤i,j≤2λi, where λi represents the eigenvalues of the reproducing matrix [18]. So, we can obtain(14)R0=Λβεdd+εd+μ+γ,where R0 represents the spectral radius of the reproducing matrix. R0 is known as a basic reproduction number in the epidemiology, which represents the average number of the secondary infections created by a single infective individual during its entire infection period in susceptible individuals [19].System (3) has an endemic disease equilibrium E∗=S∗,E∗,I∗, where S∗=Λ+ΛαI∗+γI∗+αγI∗2/d+dαI∗+βI∗, E∗=βI∗Λ+ΛαβI∗2+γβI∗2+αγI∗3β/1+αI∗d+εd+dαI∗+βI∗, and I∗ is the positive solution of equation as follows:(15)aI∗2+bI∗+c=0,where(16)a=αγεβ−αβ+dα2d+εd+μ+γ,b=εΛαβ+γεβ−d+εd+μ+γβ+2dα,c=dd+εd+μ+γR0−1.It is easy to know thata<0 and(17)c<0⟺R0<1,c≥0⟺R0≥1.(15) is a quadratic equation about I∗ since a<0. Solving for Δ=0, we obtain R0=R0c, where Δ is the discriminant of (15) and(18)R0c=b24add+εd+μ+γ+1.We can note the equivalent relation as follows:(19)Δ<0⟺R0>R0c,Δ=0⟺R0=R0c,Δ>0⟺R0<R0c.Based on the above discussion, we can conclude the following result.Theorem 1.
The existence of the endemic equilibrium are as follows:(1)
There is a unique endemic equilibrium in system (3) when 1<R0<R0c;(2)
There is a unique endemic equilibrium in system (3) whenever R0=1; b>0,(3)
There is a unique endemic equilibrium of multiplicity 2 system (3) when R0=R0c and b>0;(4)
There are two endemic equilibrium in system (3) whenever R0<1, b>0;(5)
There is no endemic equilibrium in system (3) if one of the following conditions holds: (i) R0>R0c, (ii) R0≤1 and b<0, and (iii) R0=R0c and b<0.


### 2.2. Stability Analysis of Equilibria

In this part, we will investigate the local stability of equilibria by studying the eigenvalues of the Jacobian matrices of (3) at equilibria and applying Routh–Hurwitz criterion.Theorem 2.
WhenR0<1, the disease-free equilibrium E0 is locally asymptotically stable and is unstable when R0>1.Proof.
The Jacobin matrix atE0 in (3) is as follows:(20)JE0=−d0γ−βΛd0−d+εβΛd0ε−d+μ+γ.
The character equation of system (3) at E0 is(21)λ+dλ2+Pλ+Q=0,where P=2d+μ+γ+ε, Q=d+εd+μ+γ−βΛε/d. Clearly, λ=−d is a roots of (21). All other roots of (21) are determined by the following equation:(22)λ2+Pλ+Q=0.
It has negative roots ifd+εd+μ+γ−βΛε/d>0, which is equivalent to R0<1, then the disease-free equilibrium E0 is locally asymptotically stable. Obviously, E0 is unstable if R0>1.Theorem 3.
System (3) has the unique endemic equilibrium E∗=S∗,E∗,I∗, and it is locally asymptotically stable when R0c>R0>1.Proof.
The Jacobin matrix of (3) at E∗ is(23)JE∗=−d−βI∗1+αI∗0γ−βS∗1+αI∗2βI∗1+αI∗−d+εβS∗1+αI∗20ε−d+μ+γ.
The character equation of system (3) at E∗ can be expressed as(24)λ3+a1λ2+a2λ+a3=0,where(25)a1=d+μ+γ+d+ε+βI∗1+αI∗+d>0,a2=dd+ε+βI∗d+ε1+αI∗+dd+μ+γ+βI∗d+μ+γ1+αI∗+d+εd+μ+γ−βS∗ε1+αI∗2,a3=βI∗d+εd+μ+γ1+αI∗+dd+εd+μ+γ−γβI∗ε1+αI∗−βS∗dε1+αI∗2.
From the equations ofdE=βSI/1+αI−d+ε=0, dI=εE−d+μ+γI=0, we obtain βS∗/1+αI∗=d+εd+μ+γ/ε. Then,(26)a2=dd+ε+βI∗d+ε1+αI∗+dd+μ+γ+βI∗d+μ+γ1+αI∗+d+εd+μ+γ−d+εd+μ+γ1+αI∗>0,a3=βI∗d+εd+μ+γ1+αI∗+dd+εd+μ+γ−γβI∗ε1+αI∗−dd+εd+μ+γ1+αI∗>0.
By calculating, we have thatH1=a1>0, H2=a1a2−a3>0, and H3=a3a1a2−a3>0, and by Routh–Hurwitz criterion, we obtain that the endemic equilibrium E∗ is locally asymptotically stable.


## 2.1. The Existence of Equilibria

To study equilibria, we calculate the basic reproduction number first in (3). We rewrite system (3) as follows:(10)dXdt=FX−VX,where F denotes the appearance in new infection individuals and V represents the rate of conversion of individuals in or out of each category. These two matrices in (3) can be shown as follows:(11)F=βSI1+αI00,V=d+εE−εE+d+μ+γI−Λ+βSI1+αI+dS−γI.The regeneration matrix isBC−1, B, C is the 2×2 Jacobian determinant, which is defined by(12)B=∂Fi∂XjE0,C=∂Vi∂XjE0,1≤i,j≤2.System (3) has a disease-free equilibrium E0=Λ/d,0,0. It is easy to get that(13)B=0βΛd00,C=d+ε0−εd+μ+γ.The spectral radius is theρBC−1 =  max1≤i,j≤2λi, where λi represents the eigenvalues of the reproducing matrix [18]. So, we can obtain(14)R0=Λβεdd+εd+μ+γ,where R0 represents the spectral radius of the reproducing matrix. R0 is known as a basic reproduction number in the epidemiology, which represents the average number of the secondary infections created by a single infective individual during its entire infection period in susceptible individuals [19].System (3) has an endemic disease equilibrium E∗=S∗,E∗,I∗, where S∗=Λ+ΛαI∗+γI∗+αγI∗2/d+dαI∗+βI∗, E∗=βI∗Λ+ΛαβI∗2+γβI∗2+αγI∗3β/1+αI∗d+εd+dαI∗+βI∗, and I∗ is the positive solution of equation as follows:(15)aI∗2+bI∗+c=0,where(16)a=αγεβ−αβ+dα2d+εd+μ+γ,b=εΛαβ+γεβ−d+εd+μ+γβ+2dα,c=dd+εd+μ+γR0−1.It is easy to know thata<0 and(17)c<0⟺R0<1,c≥0⟺R0≥1.(15) is a quadratic equation about I∗ since a<0. Solving for Δ=0, we obtain R0=R0c, where Δ is the discriminant of (15) and(18)R0c=b24add+εd+μ+γ+1.We can note the equivalent relation as follows:(19)Δ<0⟺R0>R0c,Δ=0⟺R0=R0c,Δ>0⟺R0<R0c.Based on the above discussion, we can conclude the following result.Theorem 1.
The existence of the endemic equilibrium are as follows:(1)
There is a unique endemic equilibrium in system (3) when 1<R0<R0c;(2)
There is a unique endemic equilibrium in system (3) whenever R0=1; b>0,(3)
There is a unique endemic equilibrium of multiplicity 2 system (3) when R0=R0c and b>0;(4)
There are two endemic equilibrium in system (3) whenever R0<1, b>0;(5)
There is no endemic equilibrium in system (3) if one of the following conditions holds: (i) R0>R0c, (ii) R0≤1 and b<0, and (iii) R0=R0c and b<0.


## 2.2. Stability Analysis of Equilibria

In this part, we will investigate the local stability of equilibria by studying the eigenvalues of the Jacobian matrices of (3) at equilibria and applying Routh–Hurwitz criterion.Theorem 2.
WhenR0<1, the disease-free equilibrium E0 is locally asymptotically stable and is unstable when R0>1.Proof.
The Jacobin matrix atE0 in (3) is as follows:(20)JE0=−d0γ−βΛd0−d+εβΛd0ε−d+μ+γ.
The character equation of system (3) at E0 is(21)λ+dλ2+Pλ+Q=0,where P=2d+μ+γ+ε, Q=d+εd+μ+γ−βΛε/d. Clearly, λ=−d is a roots of (21). All other roots of (21) are determined by the following equation:(22)λ2+Pλ+Q=0.
It has negative roots ifd+εd+μ+γ−βΛε/d>0, which is equivalent to R0<1, then the disease-free equilibrium E0 is locally asymptotically stable. Obviously, E0 is unstable if R0>1.Theorem 3.
System (3) has the unique endemic equilibrium E∗=S∗,E∗,I∗, and it is locally asymptotically stable when R0c>R0>1.Proof.
The Jacobin matrix of (3) at E∗ is(23)JE∗=−d−βI∗1+αI∗0γ−βS∗1+αI∗2βI∗1+αI∗−d+εβS∗1+αI∗20ε−d+μ+γ.
The character equation of system (3) at E∗ can be expressed as(24)λ3+a1λ2+a2λ+a3=0,where(25)a1=d+μ+γ+d+ε+βI∗1+αI∗+d>0,a2=dd+ε+βI∗d+ε1+αI∗+dd+μ+γ+βI∗d+μ+γ1+αI∗+d+εd+μ+γ−βS∗ε1+αI∗2,a3=βI∗d+εd+μ+γ1+αI∗+dd+εd+μ+γ−γβI∗ε1+αI∗−βS∗dε1+αI∗2.
From the equations ofdE=βSI/1+αI−d+ε=0, dI=εE−d+μ+γI=0, we obtain βS∗/1+αI∗=d+εd+μ+γ/ε. Then,(26)a2=dd+ε+βI∗d+ε1+αI∗+dd+μ+γ+βI∗d+μ+γ1+αI∗+d+εd+μ+γ−d+εd+μ+γ1+αI∗>0,a3=βI∗d+εd+μ+γ1+αI∗+dd+εd+μ+γ−γβI∗ε1+αI∗−dd+εd+μ+γ1+αI∗>0.
By calculating, we have thatH1=a1>0, H2=a1a2−a3>0, and H3=a3a1a2−a3>0, and by Routh–Hurwitz criterion, we obtain that the endemic equilibrium E∗ is locally asymptotically stable.


## 3. Stochastic Model

### 3.1. Existence and Uniqueness of the Global Positive Solution of (4)

In order to prove the dynamical behavior, we first show the unique global positive solution of (4).Theorem 4.
There is a unique positive solutionSt,Et,It of system (4) on t≥0 for any given initial value S0,E0,I0∈R+3, and the solution will remain in R+3 with probability 1; that is, St,Et,It∈R+3 for all t⩾0 almost surely.Proof.
We first consider the equation as follows:(27)dS=Λ−βSev1+αev−dS+γevdt+σ1SdB1t,dE=βSev1+αev−d+εEdt+σ2EdB2t,dv=εEev−d+μ+γ+σ322dt+σ3dB3t,for any initial value S0,E0,I0∈R+3, letting I=ev, the coefficients of system (27) are locally Lipschitz continuous. It exists a unique local solution St,Et,It on t∈0,τe, where τe is the explosion time. In order to prove the unique local solution of system (4) is global and positive, we need to demonstrate that τe=∞ a.s. Let h0>0 be large enough for S0∈1/h0,h0, E0∈1/h0,h0, I0∈1/h0,h0. For any integer h>h0, define the stopping time as(28)τh=inft∈0,τe:St∉1h,horEt∉1h,horIt∉1h,h.
Setinf∅=∞ throughout this paper (as usual ∅ denotes the empty set). Distinctly, τh is increasing as h⟶∞. Set τ∞ =  limh⟶∞τh. Clearly, τ∞⩽τe a.s., we only demonstrate τ∞=∞ a.s. If this statement is false, it exists a pair of constants T>0 and ε∈0,1 such that(29)Pτ∞≤T>ε.
As a result, there is an integerh1≥h0 such that(30)Pτh≤T≥ε∀h≥h1.
Define aC2-function V: R+3⟶R+ by(31)VS,E,I=S−1−logS+E−1−logE+I−1−logI,where VS,E,I is nonnegative function because of x−1−logx≥0, for all x>0. Let h≥h0 and T>0 be arbitrary.
We obtain the result as follows according to Ito^’s formula:(32)dVS,E,I=1−1SΛ−βSI1+αI−dS+γIdt+σ122dt+1−1Sσ1SdB1t+1−1EβSI1+αI−d+εEdt+σ222dt+1−1Eσ2IdB2t+1−1IεE−d+μ+γIdt+σ322dt+1−1Iσ3IdB3t=LVS,E,Idt+1−1Sσ1SdB1t+1−1Eσ2IdB2t+1−1Iσ3IdB3t,where(33)LV=Λ−dS−βSI1+αI+γI−ΛS+d+βI1+αI−γIS+βSI1+αI−d+εE−βSIE1+αI+d+ε+εE−d+μ+γI−εEI+d+μ+γ+σ122+σ222+σ322=Λ−dS+d+βI1+αI−ΛS−dE−βSIE1+αI+d+ε−d+μI−εEI−γIS+d+γ+μ+σ122+σ222+σ322≤Λ+d+βI1+αI+d+ε+d+γ+μ+σ122+σ222+σ322≤Λ+βα+3d+ε+γ+μ+σ122+σ222+σ322≕K.
The rest part of the proof is based on Mao et al. [20].


### 3.2. Extinction

In this part, we will study the condition for the extinction of the disease. As a matter of convenience, we definext =  1/t∫0txudu, R0s  =  4βΛ/d4d+σ∗2, σˇ2=maxσ12,σ22,σ32. Let us present a lemma to prove the extinction of the disease.Lemma 1.
Assume thatd>σˇ2/2. Let St,Et,It be any solution of model (4) with the initial value S0,E0,I0∈R+3. Then,(34)limt⟶+∞Ntt=0,limt⟶+∞Stt=0,limt⟶+∞Itt=0,limt⟶+∞Ett=0.
Moreover,(35)limt⟶+∞lnNtt=0,limt⟶+∞lnStt=0,limt⟶+∞lnItt=0,limt⟶+∞lnEtt=0,a.s,limt⟶+∞1t∫0tSudB1u=0,limt⟶+∞1t∫0tIudB2u=0,limt⟶+∞1t∫0tEudB3u=0a.s,The proof of Lemma1 is similar to Lemma 3.1 of [21]; we are not going to prove that anymore.Theorem 5.
Assume thatd>σˇ2/2. Let St,Et,It be any solution of model (4) with the initial value S0,E0,I0∈R+3. If R0s<1, then(36)limt⟶+∞<Et>=0,limt⟶+∞<It>=0,limt⟶+∞<St>=Λda.s..
Therefore, the disease dies out.Proof.
From (4), we obtain(37)St−S0t+Et−E0t+d+εεIt−I0t=Λ−d<St>+γ<It>−d+γ+μd+εε<It>+1t∫0tσ1SudB1u+1t∫0tσ2EudB2u+1t∫0tσ3IudB3u.
Applying Lemma1 gives(38)limt⟶+∞<St>=Λd+γd−d+γ+μd+εεdlimt⟶+∞<It>.
Define the stochastic process as follows:Zt=Et+It. Using It o^’s formula, we have(39)dlogZt=βStIt1+αItEt+It−d+εEtEt+It+εEtEt+It−d+γ+μItEt+It−σ22Et2+σ32It22Et+It2+σ2EtEt+ItdB2t+σ3ItEt+ItdB3t<βSt−dEtEt+It)−dItEt+It−σ22Et2+σ32It22Et+It2+σ2EtEt+ItdB2t+σ3ItEt+ItdB3t.
Integrating both sides of (39) yields(40)logZtt<β<St>−d−1t∫0tσ22Es2+σ32Is22Es2+Is2+Es2+Is2ds+1t∫0tσ2EuEu+IudB2u+1t∫0tσ3IuEu+IudB3u+logZ0t.
Lettingt⟶∞ gives(41)limsupt⟶+∞logZtt<βΛd+βγd−βd+γ+μd+εεdlimt⟶+∞<It>−d−σ∗24+limsupt⟶+∞logZ0t+limsupt⟶+∞1t∫0tσ2EuEu+IudB2u+limsupt⟶+∞1t∫0tσ3IuEu+IudB3u=R0s−1+limt⟶+∞M1tt+limt⟶+∞M2tt,where σ∗2=minσ22,σ32, M1t=∫0tσ2Eu/Eu+IudB2u, M2t=∫0tσ3Iu/Eu+IudB3u is a real-valued continuous local martingale, M10=0, M20=0 and(42)limsupt⟶+∞M1,M1tt=limsupt⟶+∞∫0tσ22Eu2/Eu+Iu2dut<σ22<∞,limsupt⟶+∞M2,M2tt=limsupt⟶+∞∫0tσ32Eu2/Eu+Iu2dut<σ32<∞.
Therefore, by strong law of large numbers, we must have(43)limt⟶+∞M1tt+limt⟶+∞M2tt=0,and(44)limsupt⟶+∞logZtt<R0s−1a.s.
This means(45)limt⟶+∞Et=0,limt⟶+∞It=0a.s.
It follows from (38)(46)limt⟶+∞<St>=Λda.s.
The proof is completed.


### 3.3. Stationary Distribution

In this part, we will prove the stationary distribution of model (4).Lemma 2 (see [22]).
The Markov processXt has a unique ergodic stationary distribution μ⋅ if there exists a bounded domain U∈El with a regular boundary Γ and if A1: there is a positive number M such that ∑i,j=1lλijxξiξj≥Mξ2,x∈U,ξ∈Rl, A2: there exists a non-negative C2− function V such that LV is negative for any El/U. Then(47)PxlimT⟶∞1T∫0TfXtdt=∫Elfxμdx=1,for all x∈El, where f⋅ is a function integrable with respect to the measure μ.Theorem 6.
IfR0∗:=Λβε/d+σ12/2d+ε+σ22/2d+γ+μ+σ32/2>1, for any initial value S0,E0,I0∈R+3, system (4) has a stationary distribution μ⋅ and the ergodicity holds.Proof.
In order to show Theorem6, we first prove the condition A1. Selecting(48)M=minσ12S2,σ22E2,σ32I2,S,E,I∈W¯>0,we obtain that(49)σ12S2ξ12+σ22E2ξ22+σ32I2ξ32>Mξ2,where ξ=ξ1,ξ2,ξ3. Therefore, A1 in Lemma 2 holds.
Now, we prove the conditionA2. Define a C2− function Q:R+3⟶R+ as follows:(50)QS,E,I=ζ−logS−l1logE−l2logI+S+E+Iθ+1−logS−logE−logI⩴ζV1+V2+V3+V4+V5,where(51)l1=d+σ12/2d+ε+σ22/2,l2=d+σ12/2d+γ+μ+σ32/2.
ζ is sufficiently large, θ is a nonnegative constant, and they satisfy the following condition:(52)−ζD+K≤−2,where K is a constant defined in the proof and D=3d+σ12/22/3Λβϵ/d+ϵ+σ22/2d+γ+μ+σ32/21/3−d+σ12/21/3>0 according to R0∗>1. Define(53)G=supS+E+I∈0,∞Λθ+1S+E+Iθ−12θ+1BS+E+Iθ+1<∞,where(54)B≔d−12θσˇ2>0.
It is easy to know thatQSt,Et,It is a continuous function. Therefore, there exists a minimum point S0,E0,I0 in QSt,Et,It in the interior of R+3. We define a C2-and nonnegative function V:R+3⟶R+ as follows:(55)VSt,Et,It=QSt,Et,It−QS0,E0,I0.
According to Ito^’s formula, we have(56)LV1=−ΛS+l1βSI1+αIE+l2εEI+βI1+αI+d−γIS+l1d+ε+l2d+γ+μ+σ12+l1σ22+l2σ322.
Using inequalitya+b+c≥3abc3,a,b,c>0, one gets(57)LV1≤−3l1l2Λβε1+αI1/3+βI1+αI+d−γIS+l1d+ε+l2d+γ+μ+σ12+l1σ22+l2σ322=−3Λβεd+σ12/22d+ε+σ22/2d+γ+μ+σ32/21+αI1/3+3d+σ122+βI1+αI−γIS=−3d+σ1222/3Λβεd+ε+σ22/2d+γ+μ+σ32/21+αI1/3−d+σ1221/3+βI1+αI−γIS.
Similarly,(58)LV2=θ+1S+E+IθΛ−dS−dE−dI−μI+12θθ+1S+E+Iθ−1σ12S2+σ22E2+σ32I2<θ+1S+E+IθΛ−dS+E+I+12θθ+1S+E+Iθ+1σˇ2=θ+1S+E+IθΛ−θ+1S+E+Iθ+1d−12θσˇ2≤G−12θ+1BS+E+Iθ+1<G−12θ+1BSθ+1+Eθ+1+Iθ+1,where G and B are defined in (53) and (54), respectively.(59)LV3=−ΛS+βI1+αI+d−γIS+σ122,LV4=−βSI1+αIE+d+ε+σ222,LV4=−βSI1+αIE+d+ε+σ222,LV5=−εEI+d+γ+μ+σ322.
Hence,(60)LV<−3ζd+σ1222/3Λβεd+ε+σ22/2d+γ+μ+σ32/21+αI1/3−d+σ1221/3+ζ+1βI1+αI−βSI1+αIE−12θ+1BSθ+1+Eθ+1+Iθ+1−ΛS−εEI+G+d+σ122+d+ε+σ222+d+γ+μ+σ322−ζ+1γIS.
Define a bounded subset(61)W=m≤S≤1m,m≤I≤1m,m3≤E≤1m3,where m is a sufficiently small number and satisfies the conditions as follows:(62)−ζD^+ζ+1βm+K<−1,(63)−Λm+H<−1,(64)−β1+αmm+H<−1,(65)−θ+1B2mθ+1+H<−1,(66)−θ+1B2m3θ+1+H<−1,where(67)D^=3d+σ1222/3Λβεd+ε+σ22/2d+γ+μ+σ32/21+αm1/3−d+σ1221/3,K=G+3d+ε+γ+μ+σ122+σ222+σ322,H=ζ+1βα+3ζd+σ122+G+3d+ε+γ+μ+σ122+σ222+σ322.
Sincem is sufficiently small, condition (62) holds because of (52). Let(68)R+3\W=W1∪W2∪⋯∪W6,where(69)W1=S,E,I∈R+3,0<S<m,W2=S,E,I∈R+3,0<I<m,W3=S,E,I∈R+3,S≥m,I≥m,0<E<m3,,W4=S,E,I∈R+3,S>1m,W5=S,E,I∈R+3,I>1m,W6=S,E,I∈R+3,E>1m3.
Now, consider six cases as follows:Case 1.
IfS,E,I∈W1,(70)LV<−ΛS+ζ+1βα+3ζd+σ122+G+3d+ε+γ+μ+σ122+σ222+σ322<−Λm+H.
By (63), we have(71)LV<−1.Case 2.
IfS,E,I∈W2,(72)LV<−3ζd+σ1222/3Λβεd+ε+σ22/2d+γ+μ+σ32/21+αI1/3−d+σ1221/3+ζ+1βI+G+3d+ε+γ+μ+σ122+σ222+σ322<−ζD^+ζ+1βm+K.
It follows from (62) that(73)LV<−1.Case 3.
IfS,E,I∈W3, we can see that(74)LV<−βSI1+αIE+ζ+1βα+3ζd+σ122+G+3d+ε+γ+μ+σ122+σ222+σ322<−β1+αmm+H<−1.
The last inequality holds according to (64).Case 4.
IfS,E,I∈W4, concluding from condition (65) that(75)LV<−θ+1B2Sθ+1+ζ+1βα+3ζd+σ122+G+3d+ε+γ+μ+σ122+σ222+σ322<−θ+1B2mθ+1+H<−1.Case 5.
IfS,E,I∈W5,(76)LV<−θ+1B2Iθ+1+ζ+1βα+3ζd+σ122+G+3d+ε+γ+μ+σ122+σ222+σ322<−θ+1B2mθ+1+H.
Making use of (65), we obtain that(77)LV<−1.Case 6.
IfS,E,I∈W6,(78)LV<−θ+1B2Eθ+1+ζ+1βα+3ζd+σ122+G+3d+ϵ+γ+μ+σ122+σ222+σ322<−θ+1B2m3θ+1+H.
Applying (66) yields(79)LV<−1.
Finally, we have(80)LV<−1,S,E,I∈R+3\W.Thus, conditionA1 of Lemma 2 holds, and condition A2 of Lemma 2 has also established. Finally, we could draw a conclusion that system (4) has an ergodic property.


### 3.4. Numerical Simulation

In this part, we will give numerical simulation to illustrate the theoretical result. Discretizing system (4), we obtain the equations as follows:(81)Sk+1=Sk+Λ−βSkIk1+αIk−dSk+γIkΔt+σ1ς1,k△tSk+σ122ς1,k2△t−△tSk,Ek+1=Ek+βSkIk1+αIk−d+εEk△t+σ2ς2,k△tEk+σ222ς2,k2△t−△tEk,Ik+1=Ik+εEk−d+μ+γIk△t+σ3ς3,k△tIk+σ322ς3,k2△t−△tIk,where ς1,k, ς2,k, ς3,k are N0,1-distributed independent Gaussian random variables. We select the parameters as follows: Λ = 2, ε = 0.3, β = 0.5, μ = 0.3, d = 0.4, γ = 0.1, α=1, σ1=0.1, σ2=0.1, and σ3=0.1 with initial S0,E0,I0=0.5,0.3,0.2. We calculate (82)R0∗=Λβεd+σ12/2d+ε+σ22/2d+γ+μ+σ32/2≈1.305>1,which satisfies Theorem 6; system (4) has stationary distribution. Figure 1 conforms this.Figure 1
(a) The red line describes the solution about the stochastic system, and the blue line describes the solution about the deterministic system. (b) The density function of the solution.
(a)(b)Next, the noise intensity is changed and the other parameters remain unchanged. Lettingσ1=σ2=σ3=0.03, we get(83)R0∗=Λβεd+σ12/2d+ε+σ22/2d+γ+μ+σ32/2≈1.336>1,and the picture is depicted as shown in Figure 2.Figure 2
(a) The red line describes the solution about the stochastic system, and the blue line describes the solution about the deterministic system. (b) The density function of the solution.
(a)(b)


## 3.1. Existence and Uniqueness of the Global Positive Solution of (4)

In order to prove the dynamical behavior, we first show the unique global positive solution of (4).Theorem 4.
There is a unique positive solutionSt,Et,It of system (4) on t≥0 for any given initial value S0,E0,I0∈R+3, and the solution will remain in R+3 with probability 1; that is, St,Et,It∈R+3 for all t⩾0 almost surely.Proof.
We first consider the equation as follows:(27)dS=Λ−βSev1+αev−dS+γevdt+σ1SdB1t,dE=βSev1+αev−d+εEdt+σ2EdB2t,dv=εEev−d+μ+γ+σ322dt+σ3dB3t,for any initial value S0,E0,I0∈R+3, letting I=ev, the coefficients of system (27) are locally Lipschitz continuous. It exists a unique local solution St,Et,It on t∈0,τe, where τe is the explosion time. In order to prove the unique local solution of system (4) is global and positive, we need to demonstrate that τe=∞ a.s. Let h0>0 be large enough for S0∈1/h0,h0, E0∈1/h0,h0, I0∈1/h0,h0. For any integer h>h0, define the stopping time as(28)τh=inft∈0,τe:St∉1h,horEt∉1h,horIt∉1h,h.
Setinf∅=∞ throughout this paper (as usual ∅ denotes the empty set). Distinctly, τh is increasing as h⟶∞. Set τ∞ =  limh⟶∞τh. Clearly, τ∞⩽τe a.s., we only demonstrate τ∞=∞ a.s. If this statement is false, it exists a pair of constants T>0 and ε∈0,1 such that(29)Pτ∞≤T>ε.
As a result, there is an integerh1≥h0 such that(30)Pτh≤T≥ε∀h≥h1.
Define aC2-function V: R+3⟶R+ by(31)VS,E,I=S−1−logS+E−1−logE+I−1−logI,where VS,E,I is nonnegative function because of x−1−logx≥0, for all x>0. Let h≥h0 and T>0 be arbitrary.
We obtain the result as follows according to Ito^’s formula:(32)dVS,E,I=1−1SΛ−βSI1+αI−dS+γIdt+σ122dt+1−1Sσ1SdB1t+1−1EβSI1+αI−d+εEdt+σ222dt+1−1Eσ2IdB2t+1−1IεE−d+μ+γIdt+σ322dt+1−1Iσ3IdB3t=LVS,E,Idt+1−1Sσ1SdB1t+1−1Eσ2IdB2t+1−1Iσ3IdB3t,where(33)LV=Λ−dS−βSI1+αI+γI−ΛS+d+βI1+αI−γIS+βSI1+αI−d+εE−βSIE1+αI+d+ε+εE−d+μ+γI−εEI+d+μ+γ+σ122+σ222+σ322=Λ−dS+d+βI1+αI−ΛS−dE−βSIE1+αI+d+ε−d+μI−εEI−γIS+d+γ+μ+σ122+σ222+σ322≤Λ+d+βI1+αI+d+ε+d+γ+μ+σ122+σ222+σ322≤Λ+βα+3d+ε+γ+μ+σ122+σ222+σ322≕K.
The rest part of the proof is based on Mao et al. [20].


## 3.2. Extinction

In this part, we will study the condition for the extinction of the disease. As a matter of convenience, we definext =  1/t∫0txudu, R0s  =  4βΛ/d4d+σ∗2, σˇ2=maxσ12,σ22,σ32. Let us present a lemma to prove the extinction of the disease.Lemma 1.
Assume thatd>σˇ2/2. Let St,Et,It be any solution of model (4) with the initial value S0,E0,I0∈R+3. Then,(34)limt⟶+∞Ntt=0,limt⟶+∞Stt=0,limt⟶+∞Itt=0,limt⟶+∞Ett=0.
Moreover,(35)limt⟶+∞lnNtt=0,limt⟶+∞lnStt=0,limt⟶+∞lnItt=0,limt⟶+∞lnEtt=0,a.s,limt⟶+∞1t∫0tSudB1u=0,limt⟶+∞1t∫0tIudB2u=0,limt⟶+∞1t∫0tEudB3u=0a.s,The proof of Lemma1 is similar to Lemma 3.1 of [21]; we are not going to prove that anymore.Theorem 5.
Assume thatd>σˇ2/2. Let St,Et,It be any solution of model (4) with the initial value S0,E0,I0∈R+3. If R0s<1, then(36)limt⟶+∞<Et>=0,limt⟶+∞<It>=0,limt⟶+∞<St>=Λda.s..
Therefore, the disease dies out.Proof.
From (4), we obtain(37)St−S0t+Et−E0t+d+εεIt−I0t=Λ−d<St>+γ<It>−d+γ+μd+εε<It>+1t∫0tσ1SudB1u+1t∫0tσ2EudB2u+1t∫0tσ3IudB3u.
Applying Lemma1 gives(38)limt⟶+∞<St>=Λd+γd−d+γ+μd+εεdlimt⟶+∞<It>.
Define the stochastic process as follows:Zt=Et+It. Using It o^’s formula, we have(39)dlogZt=βStIt1+αItEt+It−d+εEtEt+It+εEtEt+It−d+γ+μItEt+It−σ22Et2+σ32It22Et+It2+σ2EtEt+ItdB2t+σ3ItEt+ItdB3t<βSt−dEtEt+It)−dItEt+It−σ22Et2+σ32It22Et+It2+σ2EtEt+ItdB2t+σ3ItEt+ItdB3t.
Integrating both sides of (39) yields(40)logZtt<β<St>−d−1t∫0tσ22Es2+σ32Is22Es2+Is2+Es2+Is2ds+1t∫0tσ2EuEu+IudB2u+1t∫0tσ3IuEu+IudB3u+logZ0t.
Lettingt⟶∞ gives(41)limsupt⟶+∞logZtt<βΛd+βγd−βd+γ+μd+εεdlimt⟶+∞<It>−d−σ∗24+limsupt⟶+∞logZ0t+limsupt⟶+∞1t∫0tσ2EuEu+IudB2u+limsupt⟶+∞1t∫0tσ3IuEu+IudB3u=R0s−1+limt⟶+∞M1tt+limt⟶+∞M2tt,where σ∗2=minσ22,σ32, M1t=∫0tσ2Eu/Eu+IudB2u, M2t=∫0tσ3Iu/Eu+IudB3u is a real-valued continuous local martingale, M10=0, M20=0 and(42)limsupt⟶+∞M1,M1tt=limsupt⟶+∞∫0tσ22Eu2/Eu+Iu2dut<σ22<∞,limsupt⟶+∞M2,M2tt=limsupt⟶+∞∫0tσ32Eu2/Eu+Iu2dut<σ32<∞.
Therefore, by strong law of large numbers, we must have(43)limt⟶+∞M1tt+limt⟶+∞M2tt=0,and(44)limsupt⟶+∞logZtt<R0s−1a.s.
This means(45)limt⟶+∞Et=0,limt⟶+∞It=0a.s.
It follows from (38)(46)limt⟶+∞<St>=Λda.s.
The proof is completed.


## 3.3. Stationary Distribution

In this part, we will prove the stationary distribution of model (4).Lemma 2 (see [22]).
The Markov processXt has a unique ergodic stationary distribution μ⋅ if there exists a bounded domain U∈El with a regular boundary Γ and if A1: there is a positive number M such that ∑i,j=1lλijxξiξj≥Mξ2,x∈U,ξ∈Rl, A2: there exists a non-negative C2− function V such that LV is negative for any El/U. Then(47)PxlimT⟶∞1T∫0TfXtdt=∫Elfxμdx=1,for all x∈El, where f⋅ is a function integrable with respect to the measure μ.Theorem 6.
IfR0∗:=Λβε/d+σ12/2d+ε+σ22/2d+γ+μ+σ32/2>1, for any initial value S0,E0,I0∈R+3, system (4) has a stationary distribution μ⋅ and the ergodicity holds.Proof.
In order to show Theorem6, we first prove the condition A1. Selecting(48)M=minσ12S2,σ22E2,σ32I2,S,E,I∈W¯>0,we obtain that(49)σ12S2ξ12+σ22E2ξ22+σ32I2ξ32>Mξ2,where ξ=ξ1,ξ2,ξ3. Therefore, A1 in Lemma 2 holds.
Now, we prove the conditionA2. Define a C2− function Q:R+3⟶R+ as follows:(50)QS,E,I=ζ−logS−l1logE−l2logI+S+E+Iθ+1−logS−logE−logI⩴ζV1+V2+V3+V4+V5,where(51)l1=d+σ12/2d+ε+σ22/2,l2=d+σ12/2d+γ+μ+σ32/2.
ζ is sufficiently large, θ is a nonnegative constant, and they satisfy the following condition:(52)−ζD+K≤−2,where K is a constant defined in the proof and D=3d+σ12/22/3Λβϵ/d+ϵ+σ22/2d+γ+μ+σ32/21/3−d+σ12/21/3>0 according to R0∗>1. Define(53)G=supS+E+I∈0,∞Λθ+1S+E+Iθ−12θ+1BS+E+Iθ+1<∞,where(54)B≔d−12θσˇ2>0.
It is easy to know thatQSt,Et,It is a continuous function. Therefore, there exists a minimum point S0,E0,I0 in QSt,Et,It in the interior of R+3. We define a C2-and nonnegative function V:R+3⟶R+ as follows:(55)VSt,Et,It=QSt,Et,It−QS0,E0,I0.
According to Ito^’s formula, we have(56)LV1=−ΛS+l1βSI1+αIE+l2εEI+βI1+αI+d−γIS+l1d+ε+l2d+γ+μ+σ12+l1σ22+l2σ322.
Using inequalitya+b+c≥3abc3,a,b,c>0, one gets(57)LV1≤−3l1l2Λβε1+αI1/3+βI1+αI+d−γIS+l1d+ε+l2d+γ+μ+σ12+l1σ22+l2σ322=−3Λβεd+σ12/22d+ε+σ22/2d+γ+μ+σ32/21+αI1/3+3d+σ122+βI1+αI−γIS=−3d+σ1222/3Λβεd+ε+σ22/2d+γ+μ+σ32/21+αI1/3−d+σ1221/3+βI1+αI−γIS.
Similarly,(58)LV2=θ+1S+E+IθΛ−dS−dE−dI−μI+12θθ+1S+E+Iθ−1σ12S2+σ22E2+σ32I2<θ+1S+E+IθΛ−dS+E+I+12θθ+1S+E+Iθ+1σˇ2=θ+1S+E+IθΛ−θ+1S+E+Iθ+1d−12θσˇ2≤G−12θ+1BS+E+Iθ+1<G−12θ+1BSθ+1+Eθ+1+Iθ+1,where G and B are defined in (53) and (54), respectively.(59)LV3=−ΛS+βI1+αI+d−γIS+σ122,LV4=−βSI1+αIE+d+ε+σ222,LV4=−βSI1+αIE+d+ε+σ222,LV5=−εEI+d+γ+μ+σ322.
Hence,(60)LV<−3ζd+σ1222/3Λβεd+ε+σ22/2d+γ+μ+σ32/21+αI1/3−d+σ1221/3+ζ+1βI1+αI−βSI1+αIE−12θ+1BSθ+1+Eθ+1+Iθ+1−ΛS−εEI+G+d+σ122+d+ε+σ222+d+γ+μ+σ322−ζ+1γIS.
Define a bounded subset(61)W=m≤S≤1m,m≤I≤1m,m3≤E≤1m3,where m is a sufficiently small number and satisfies the conditions as follows:(62)−ζD^+ζ+1βm+K<−1,(63)−Λm+H<−1,(64)−β1+αmm+H<−1,(65)−θ+1B2mθ+1+H<−1,(66)−θ+1B2m3θ+1+H<−1,where(67)D^=3d+σ1222/3Λβεd+ε+σ22/2d+γ+μ+σ32/21+αm1/3−d+σ1221/3,K=G+3d+ε+γ+μ+σ122+σ222+σ322,H=ζ+1βα+3ζd+σ122+G+3d+ε+γ+μ+σ122+σ222+σ322.
Sincem is sufficiently small, condition (62) holds because of (52). Let(68)R+3\W=W1∪W2∪⋯∪W6,where(69)W1=S,E,I∈R+3,0<S<m,W2=S,E,I∈R+3,0<I<m,W3=S,E,I∈R+3,S≥m,I≥m,0<E<m3,,W4=S,E,I∈R+3,S>1m,W5=S,E,I∈R+3,I>1m,W6=S,E,I∈R+3,E>1m3.
Now, consider six cases as follows:Case 1.
IfS,E,I∈W1,(70)LV<−ΛS+ζ+1βα+3ζd+σ122+G+3d+ε+γ+μ+σ122+σ222+σ322<−Λm+H.
By (63), we have(71)LV<−1.Case 2.
IfS,E,I∈W2,(72)LV<−3ζd+σ1222/3Λβεd+ε+σ22/2d+γ+μ+σ32/21+αI1/3−d+σ1221/3+ζ+1βI+G+3d+ε+γ+μ+σ122+σ222+σ322<−ζD^+ζ+1βm+K.
It follows from (62) that(73)LV<−1.Case 3.
IfS,E,I∈W3, we can see that(74)LV<−βSI1+αIE+ζ+1βα+3ζd+σ122+G+3d+ε+γ+μ+σ122+σ222+σ322<−β1+αmm+H<−1.
The last inequality holds according to (64).Case 4.
IfS,E,I∈W4, concluding from condition (65) that(75)LV<−θ+1B2Sθ+1+ζ+1βα+3ζd+σ122+G+3d+ε+γ+μ+σ122+σ222+σ322<−θ+1B2mθ+1+H<−1.Case 5.
IfS,E,I∈W5,(76)LV<−θ+1B2Iθ+1+ζ+1βα+3ζd+σ122+G+3d+ε+γ+μ+σ122+σ222+σ322<−θ+1B2mθ+1+H.
Making use of (65), we obtain that(77)LV<−1.Case 6.
IfS,E,I∈W6,(78)LV<−θ+1B2Eθ+1+ζ+1βα+3ζd+σ122+G+3d+ϵ+γ+μ+σ122+σ222+σ322<−θ+1B2m3θ+1+H.
Applying (66) yields(79)LV<−1.
Finally, we have(80)LV<−1,S,E,I∈R+3\W.Thus, conditionA1 of Lemma 2 holds, and condition A2 of Lemma 2 has also established. Finally, we could draw a conclusion that system (4) has an ergodic property.


## 3.4. Numerical Simulation

In this part, we will give numerical simulation to illustrate the theoretical result. Discretizing system (4), we obtain the equations as follows:(81)Sk+1=Sk+Λ−βSkIk1+αIk−dSk+γIkΔt+σ1ς1,k△tSk+σ122ς1,k2△t−△tSk,Ek+1=Ek+βSkIk1+αIk−d+εEk△t+σ2ς2,k△tEk+σ222ς2,k2△t−△tEk,Ik+1=Ik+εEk−d+μ+γIk△t+σ3ς3,k△tIk+σ322ς3,k2△t−△tIk,where ς1,k, ς2,k, ς3,k are N0,1-distributed independent Gaussian random variables. We select the parameters as follows: Λ = 2, ε = 0.3, β = 0.5, μ = 0.3, d = 0.4, γ = 0.1, α=1, σ1=0.1, σ2=0.1, and σ3=0.1 with initial S0,E0,I0=0.5,0.3,0.2. We calculate (82)R0∗=Λβεd+σ12/2d+ε+σ22/2d+γ+μ+σ32/2≈1.305>1,which satisfies Theorem 6; system (4) has stationary distribution. Figure 1 conforms this.Figure 1
(a) The red line describes the solution about the stochastic system, and the blue line describes the solution about the deterministic system. (b) The density function of the solution.
(a)(b)Next, the noise intensity is changed and the other parameters remain unchanged. Lettingσ1=σ2=σ3=0.03, we get(83)R0∗=Λβεd+σ12/2d+ε+σ22/2d+γ+μ+σ32/2≈1.336>1,and the picture is depicted as shown in Figure 2.Figure 2
(a) The red line describes the solution about the stochastic system, and the blue line describes the solution about the deterministic system. (b) The density function of the solution.
(a)(b)


## 4. Conclusion

A SEIS epidemic model is a simpler epidemic model than others, but it still plays an important role in the control and treatment of epidemic diseases today. In this article, we derive the basic reproduction numberR0 for the deterministic SEIS system, which plays an important role in epidemiology. We carry out local asymptotic stability analysis for the disease-free equilibrium point, which is locally asymptotically stable if R0<1. The endemic equilibrium point E∗ is locally asymptotically stable if R0>1, which means that the disease is chronic. With the help of the stochastic Lyapunov analysis method, we obtain the persistence and extinction of the disease for the stochastic system, that is, the disease will die out if R0s<1, and the system has ergodic stationary distribution if R0∗>1. This shows that stationary distribution does not depend on interior equilibrium in the deterministic system (3). In addition, it is obvious that R0∗<R0, this means that noise can accelerate the prevalence of the disease. If in a laboratory, the people can provide noise to control development about the scope and the time of the epidemic and gain accurate data, we can obtain interesting fruits. The aim of this research is to identify the parameters of interest for the control of the disease, we hope to assist the policy-maker in the disease prevention and control.



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*Source: 2747320-2022-08-22.xml*