# Ergodicity of a Nonlinear Stochastic SIRS Epidemic Model with Regime-Switching Diffusions

Authors: Hongxia Liu; Juan Li; Mengnan Chi; Jinlei Liu; Wencai Zhao
Journal: Complexity
 (2020)

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## Abstract

In this paper, taking both white noises and colored noises into consideration, a nonlinear stochastic SIRS epidemic model with regime switching is explored. The threshold parameterRs is found, and we investigate sufficient conditions for the existence of the ergodic stationary distribution of the positive solution. Finally, some numerical simulations are also carried out to demonstrate the analytical results.


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## Body

## 1. Introduction

It is well known that the incidence rate plays a crucial role in studying the dynamics of infectious disease models. In general, bilinear incidenceβSI is considered in most infectious disease models [1, 2]. For example, Li and Ma [3] conducted the qualitative analyses of the SIS epidemic model with vaccination and varying total population size. Nakata and Kuniya [4] introduced the global dynamics of a class of SEIRS epidemic models in a periodic environment. In order to effectively investigate the rapid spread of the disease, it is rewarding to consider the behavioral changes and crowding effect of the infected individuals, as well as choose appropriate parameters to prevent the unbounded contact rate. Capasso and Serio [5] proposed the saturated incidence βSI/1+αI, which is more reasonable than the bilinear incidence. For the detailed introduction of the saturated incidence, see [5].The classical SIRS epidemic model with the saturated incidence rate is in the following form [6, 7]:(1)dStdt=A−βStIt1+αIt−μSt+ωRt,dItdt=βStIt1+αIt−μ+λ+dIt,dRtdt=λIt−μ+ωRt,where St, It, and Rt represent the number of susceptible, infected, and removed individuals at time t, respectively. A denotes an input of new members into the population, β stands for the transmission rate, μ is the natural mortality, d is the death rate relative to the disease, λ is the proportion of the infective class to the recovered class, and ω is the per capita rate of loss of immunity.In nature, it is inevitable for a population to be affected by a variety of random factors [8, 9]. Consequently, it is crucial to consider the randomness which might exist during the transmission of disease [10, 11]. In general, there are two types of random perturbations to be considered in ecosystem modeling: one is white noise which can be described as Brownian motion [12–14], and the other is colored noise (also called telegraph noise) which can be described through a finite-state Markov chain [15–17]. In [15], Liu et al. investigated the threshold behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching. Lin and Jin [16] considered a stochastic SIS epidemic model with regime switching; by verifying a Foster–Lyapunov condition, the threshold condition for the ergodicity is presented. Hu et al. [17] studied a stochastic SIS epidemic model with vaccination and nonlinear incidence under regime switching.Motivated by the above literature, we study a nonlinear stochastic SIRS epidemic model with two kinds of random interference. The model is as follows:(2)dSt=1−qξtAξt−βξtStIt1+αIt−μξtSt+ωξtRtdt+σ1ξtStdB1t,dIt=βξtStIt1+αIt−μξtIt−λξtItdt+σ2ξtItdB2t,dRt=qξtAξt+λξtIt−μξtRt−ωξtRtdt+σ3ξtRtdB3t,where q is a fraction of vaccinated individuals for newborns. The incidence rate α contains the crowding effect of the infected individuals and should not be disturbed by the noises in the environment. Biti=1,2,3 denotes one-dimensional standard Brownian motion, and σii=1,2,3 is the intensity of white noise. ξt is a right-continuous Markov chain taking values in ℳ=1,2,…,m, and the generator matrix of ξt is Γ=γij1≤i,j≤m. The details of the Markov chain are presented in [18], which we omit here.In this paper, the dynamic behaviors of stochastic differential system (2) are discussed. In Section 2, we get the conditions for the extinction and persistence in mean of the infected. In Section 3, we investigate the ergodicity of system (2) by constructing a suitable Lyapunov function. Finally, numerical simulations are given in Section 4.


## 2. The Extinction and Persistence of the Disease

In system (2), let Nt=St+It+Rt; then, we have(3)dNt=Aξt−μξtNtdt+σ1ξtStdB1t+σ2ξtItB2t+σ3ξtRtdB3t.From Lemma 2.1 and Lemma 2.2 of [19], we have the following.Lemma 1.
For any initial valueS0,I0,R0,r0∈R+3×ℳ, the solution St,It,Rt of system (2) has the following properties:(1)
limt⟶∞St+It+Rt/t=0 a.s.(2)
limt⟶∞∫0tSudB1u/t=0, limt⟶∞∫0tIudB2u/t=0, limt⟶∞∫0tRudB3u/t=0 a.s.Definition 1.
(1)
Iflimt⟶∞It=0, then the disease tends to be extinct(2)
Iflimt⟶∞inf1/tE∫0tIzdz>0, then the disease tends to be persistent in mean
Define a matrix(4)M=μ1−γ11⋯−γ1m0⋯0⋮⋮⋮⋮−γm1⋯μm−γmm0⋯0−ω1⋯0μ1+ω1−γ11⋯−γ1m⋮⋮⋮⋮0⋯−ωm−γm1⋯μm+ωm−γmm.
Following Lemma1 in [20], we obtain that M is a nonsingular M-matrix. By Theorem 2.10 (see Mao and Yuan [21]), for vector β=β1,β2,…,βm,0,0,…,0T∈R2m, the equation Mη=β has a unique solution:(5)η=d11,d12,…,d1m,d21,d22,…,d2mT∈R+2m.
That is,(6)βk−d1kμk+∑l∈ℳγkld1l=0,k=1,2,…,m,d1kωk−d2kμk+ωk+∑l∈ℳγkld2l=0,k=1,2,…,m.
Then, we define a parameter(7)Rs=Σk∈ℳπkd1k1−qkAk+d2kqkAk−μk+λk+σ22k2.Theorem 1.
For any initial valueS0,I0,R0,r0∈R+3×ℳ, assume that α+d^1−d⌣2>0; then,(1)
The diseaseIt of system (2) is extinct when Rs<0(2)
The diseaseIt of system (2) is persistent in mean when Rs>0Proof.
Define the function(8)USt,It,Rt,k=U1St,It,Rt,k+U2It,where(9)U1St,It,Rt,k=d1kSt+It+d2kRt,U2It=lnIt+αIt.
Then, applying Itô’s formula, we have(10)LU1=d1k1−qkAk−μkSt−μk+λkIt+ωkRt+d2kqkAk+λkIt−μk+ωkRt+St+It∑l=1mγkld1l+Rt∑l=1mγkld2l=d1k1−qkAk+d2kqkAk+−d1kμk+∑l=1mγkld1lSt−μk+λkd1k−λkd2k−∑l=1mγkld1lIt+d1kωk−d2kμk+ωk+∑l=1mγkld2lRt=d1k1−qkAk+d2kqkAk−βkSt−βk+λkd1k−λkd2kIt,(11)LU2=βkSt1+αIt−μk+λk+σ22k2+αβkStIt1+αIt−αμk+λkIt=βkSt−αμk+λkIt−μk+λk+σ22k2.
By (10) and (11), one can obtain that(12)LU=d1k1−qkAk+d2kqkAk−μk+λk+σ22k2−βk+λkd1k−λkd2k+αμk+αλkIt=d1k1−qkAk+d2kqkAk−μk+λk+σ22k2−βk+λkα+d1k−d2k+αμkIt.
An application of generalized Itô’s formula yields(13)USt,It,Rt,ξt=US0,I0,R0,ξ0+∫0tLUSz,Iz,Rz,ξzdz+∫0td1ξzσ1ξzSzdB1z+∫0t1+α+d1ξzIzσ2ξzdB2z+∫0td2ξzσ3ξzRzdB3z+∫0t∫RUSz,Iz,Rz,i0+ℓξz,l−USz,Iz,Rz,ξzμdz,dl,where μdz,dl=γdz,dl−μdldz is a martingale measure. By using Lemma 1.9 of [21], we can get(14)EUSt,It,Rt,ξt=EUS0,I0,R0,ξ0+E∫0tLUSz,Iz,Rz,ξzdz.
Then,(15)EUSt,It,Rt,ξtt=EUS0,I0,R0,ξ0t+E1t∫0td1ξz1−qξzAξz+d2ξzqξzAξz−μξz+λξz+σ22ξz2dz−E1t∫0tβξz+λξzα+d1ξz−d2ξz+αμξzIzdz.
Hence,(16)Rsβ⌣+λ⌣α+d⌣1−d^2+αμ⌣≤limt⟶∞inf1tE∫0tIzdz≤limt⟶∞sup1tE∫0tIzdz≤Rsβ^+λ^α+d^1−d⌣2+αμ^.
In conclusion, based on the conditionα+d^1−d⌣2>0, we have the following:(1)
IfRs<0, then limt⟶∞It=0, a.s.(2)
IfRs>0, then limt⟶∞inf1/tE∫0tIzdz≥Rs/β⌣+λ⌣α+d⌣1−d^2+αμ⌣>0, and the disease It of system (2) is persistent in meanRemark 1.
According to Theorem1, if the intensity of white noise is large enough that the condition Rs<0 holds, then the disease dies out with probability 1. Conversely, if Rs>0, the disease of system (2) is persistent in mean. This means that the presence of environmental noise is conducive to disease control.


## 3. Ergodic Stationary Distribution of System (2)

The study of the ergodicity and stationary distribution has been widely concerned by many scholars [22, 23]. In this section, in order to investigate the ergodic property of system (2), we establish a suitable Lyapunov function with Markov conversion.Theorem 2.
For any initial valueS0,I0,R0,ξ0∈R+3×ℳ, if Rs>0, stochastic process St,It,Rt,ξt of system (2) is ergodic and has a unique stationary distribution in R+3×ℳ.Proof.
Letxt=lnSt, yt=lnIt, and zt=lnRt. System (2) can be transformed into(17)dxt=1−qξtAξtext−βξteyt1+αeyt−μξt+ωξteztext−12σ12ξtdt+σ1ξtdB1t,dyt=βξtext1+αeyt−μξt−λξt−12σ22ξtdt+σ2ξtdB2t,dzt=qξtAξtezt+λξteytezt−μξt−ωξt−12σ32ξtdt+σ3ξtdB3t.
The ergodic property of system (2) is the same as that of system (17). We need to prove that system (17) satisfies the three conditions of Lemma 2.2 in [18]. Obviously, conditions (1) and (2) are satisfied. Hence, we only need to prove condition (3).
Define(18)Hx,y,z,k=1ϑ+1ex+ey+ezϑ+1−ad1kex+ey+d2kez+y+αey+τk−x−z,where ϑ∈0,1,σ⌣2=max1≤i≤3,k∈ℳσ⌣i2k, and a>0 satisfy μ^−ϑ/2σ⌣2>0, −aRs+2μ⌣+ω⌣+1/2σ⌣12+1/2σ⌣32≤−2. τk will be determined later. There exists a unique point x0,y0,z0,k which is the minimum value of Hx,y,z,k.
Set(19)Vx,y,z,k=Hx,y,z,k−Hx0,y0,z0,k.
Denote(20)V1=1ϑ+1ex+ey+ezϑ+1,V2=−d1kex+ey+d2kez+y+αey+τk,V3=−x,V4=−z.
An application of generalized Itô’s formula yields(21)LV1=ex+ey+ezϑAk−μkex−μkey−μkez+ϑ2ex+ey+ezϑ−1σ12ke2x+σ22ke2y+σ32ke2z≤3ϑAkeϑx+eϑy+eϑz−μkeϑ+1x−μkeϑ+1y−μkeϑ+1z+ϑ2σ12keϑ+1x+ϑ2σ22keϑ+1y+ϑ2σ32keϑ+1z=−μk−ϑ2σ12keϑ+1x−μk−ϑ2σ22keϑ+1y−μk−ϑ2σ32keϑ+1z+3ϑAkeϑx+eϑy+eϑz≤−μ^−ϑ2σ⌣2eϑ+1x−μ^−ϑ2σ⌣2eϑ+1y−μ^−ϑ2σ⌣2eϑ+1z+3ϑA⌣eϑx+eϑy+eϑz,(22)LV2=−d1k1−qkAk−d2kqkAk+μk+λk+σ22k2−∑l=1mγklτl+exd1kμk−βk−∑l∈ℳγkld1l+eyd1k+αμk+λk−d2kλk−∑l∈ℳγkld1l+ez−d1kωk+d2kμk+ωk−∑l∈ℳγkld2l=−G0k−∑l=1mγklτl+eyβk+d1k−d2kλk+αμk+λk,where we used equation (4) and (23)G0k=d1k1−qkAk+d2kqkAk−μk−λk−σ22k2.
Letτ=τ1,…,τmT be the solution of the following Poisson system:(24)Γτ=∑k=1mπkG0k1⟶−G01⋮G0m,where G=G01,G02,…,G0mT.
Substituting this equality into (22), we obtain(25)LV2=−∑k=1mπkG0k+eyβk+d1k−d2kλk+αμk+λk,(26)LV3=−1−qkAkex−βkey1+αey+ωkezex−μk−12σ12k,(27)LV4=−qkAkez−λkeyez+μk+ωk+12σ32k.
Combining (21) and (25)–(27), we have (28)LVx,y,z,k=LV1+aLV2+LV3+LV4≤−μ^−ϑ2σ⌣2eϑ+1x+3ϑA⌣eϑx−1−q⌣A^ex−μ^−ϑ2σ⌣2eϑ+1y+3ϑA⌣eϑy+aeyβ⌣+λ⌣d⌣1−d^2+αμ⌣+λ⌣+β⌣ey−μ^−ϑ2σ⌣2eϑ+1z+3ϑA⌣eϑz−q^A^ez−aRs+2μ⌣+ω⌣+12σ⌣12+12σ⌣32.
Consequently, takeρ sufficiently large, and let U=−ρ,ρ×−ρ,ρ×−ρ,ρ; then, we have LVx,y,z,k≤−1 for any x,y,z,k∈Uc×ℳ. According to Lemma 2.2 in [18], we know that xt,yt,zt,ξt is ergodic and positive recurrent.
This proof is completed.


## 4. Conclusions and Numerical Simulations

This paper investigated a nonlinear epidemic disease model with two kinds of noise disturbances. The threshold of extinction and persistence in mean is obtained.(i)
IfRs<0, the infected individuals tend to become extinct(ii)
IfRs>0, the infected individuals are persistent in mean(iii)
IfRs>0, the stochastic process St,It,Rt,ξt of system (2) is ergodic and has a unique stationary distributionTo verify the correctness of the theoretical analysis, numerical simulation is employed in the following example.Example 1.
In system (2), let ξt be a right-continuous Markov chain taking values in ℳ=1,2 with the generator(29)Γ=−331−1.
Then, the unique stationary distribution ofξt is π=π1,π2=1/4,3/4. Let α=0.2, and other coefficients in system (2) are selected as follows.
Fork=ii=1,2, A1,A2=1.3,1.1, q1,q2=0.2,0.3, β1,β2=0.6,0.7, μ1,μ2=0.45,0.5, λ1,λ2=0.2,0.3, ω1,ω2=0.3,0.2, σ11,σ12=0.2,0.2, and σ31,σ32=0.05,0.2.
Firstly, letσ21σ22=0.05,0.1. This gives Rs=0.52874>0. According to Theorem 1, the infectious disease is permanent (Figure 1).
Then, letσ21,σ22=1.2,1.2. This gives Rs=−0.1902<0. Based on Theorem 1, the disease It goes extinct (Figure 2). It is easy to see that the disease tends to become extinct when the intensity of white noise increases.
Lastly, letσ21σ22=0.05,0.1,μ1,μ2=0.2,0.15. This gives Rs=3.8277>0. From Theorem 2, the stochastic process St,It,Rt,ξt of system (2) is ergodic and has a unique stationary distribution (Figure 3). Figure 3(a) shows the Markov chain, and Figure 3(b) shows a stationary distribution of the stochastic SIRS model. From Figures 3(c)–3(e), we obtain the distributions of St,It, and Rt in stochastic system (2), respectively. This means stochastic SIRS model (2) has an ergodic stationary distribution.Figure 1
Simulations of the solutionSt,It,Rt to stochastic model (2) with S0,I0,R0=1,0.7,0.8. (a) k=1,σ21=0.05. (b) k=2,σ22=0.1.
(a)(b)Figure 2
Simulations of the solutionSt,It,Rt to stochastic model (2) with S0,I0,R0=1,0.7,0.8. (a) k=1,σ21=1.2. (b) k=2,σ22=1.2.
(a)(b)Figure 3
(a) Markov chain. (b), (c), (d), and (e) Distributions ofSt,It, and Rt of model (2).
(a)(b)(c)(d)(e)



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