# Stability and Asymptotic Behavior of a Regime-Switching SIRS Model with Beddington–DeAngelis Incidence Rate

Authors: Shan Wang; Youhua Peng; Feng Wang
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

A regime-switching SIRS model with Beddington–DeAngelis incidence rate is studied in this paper. First of all, the property that the model we discuss has a unique positive solution is proved and the invariant set is presented. Secondly, by constructing appropriate Lyapunov functionals, global stochastic asymptotic stability of the model under certain conditions is proved. Then, we leave for studying the asymptotic behavior of the model by presenting threshold values and some other conditions for determining disease extinction and persistence. The results show that stochastic noise can inhibit the disease and the behavior will have different phenomena owing to the role of regime-switching. Finally, some examples are given and numerical simulations are presented to confirm our conclusions.


---

## Body

## 1. Introduction

Infectious diseases are one of the greatest enemies of human beings. Whenever they happen, they will bring great disasters to human beings. Therefore, it is of great significance to model and study the infectious mechanism of infectious diseases for disease control. SIR model, usingSt, It, and Rt to express fractions of the susceptible, infected, and removed at time t, as one of the classical infectious disease models, has been studied and extended by many scholars.Owing to the richness and importance of the research content of epidemic models, different scholars study them from different perspectives. Some authors used Lyapunov functions to study the stability of the model [1–3]. The authors in [2] proposed a new technique to study stability of an SIR model with a nonlinear incidence rate by establishing a transformation of variable. More about the stability of stochastic differential equations, we refer to [4, 5]. Some scholars have studied the dynamic behavior of epidemic models and gave the threshold values of disease extinction and persistence so as to give control strategies for disease [6–10]. The authors in [6] have proved that the number R0S can govern the dynamics of the model under intervention strategies by using the Markov semigroup theory. In addition, scholars have studied the ergodicity and stationary distribution of the model by making use of different methods in [11–14]. The authors in [12] generalized the method for analyzing ergodic property of epidemic models, all of which further enrich and improve the theory and application of epidemiology. Markov semigroup approach was used in [14] to obtain the existence of stationary distribution density of the stochastic plant disease system.Parameters involved in models are more or less disturbed by some environmental noise. Mao et al. [15] have proved that the presence of noise can suppress potential population explosion, which shows that environmental noise has a great influence on the behavior of the model. In order to describe this perturbation, stochastic noise driven by continuous Brownian motion is widely studied in epidemic models and other systems with various incidence functions [7–9, 12, 13, 16, 17]. There are several kinds of stochastic noise, one of which is assumed that some parameters in the model are disturbed, such as the contact rate or death rate.Beddington–DeAngelis function is an important incidence rate with the formfS,I=SI/m1+m2S+m3I, which has been studied by some scholars [17–19]. It can be considered as a generalization of many incidence functions, for example,(1)
m1=1,m2=0,m3=0,fS,I=SI [7](2)
m1=1,m2=a,m3=0,fS,I=SI/1+aS [20](3)
m1=1,m2=0,m3=a,fS,I=SI/1+aI [13](4)
m1=0,m2=1,m3=1,fS,I=SI/S+I [21]Hence, a certain SIRS model containing constant population size and stochastic perturbation takes the following form:(1)S.t=μ−μSt−βStItm1+m2St+m3It+θRtdt−σStItm1+m2St+m3ItdBt,I.t=βStItm1+m2St+m3It−μ+γ+dItdt+σStItm1+m2St+m3ItdBt,R.t=γIt−μ+θRtdt,where μ represents the birth and death rate, β denotes the valid contact coefficient, γ means the rate at which the infected is cured and returns to the removed, d is the death rate due to disease, θ expresses the rate of losing immunity and returning to the susceptible, σ represents the intensity of stochastic perturbation, and Bt is the standard Brownian motion.Moreover, the environment in our life often changes, for example, the seasons, temperature and humidity will always change and the mechanism and infectious ability of diseases will change accordingly. Therefore, the parameters in the model will change suddenly and discontinuously, which cannot be depicted by continuous Brownian motion, but can be described by continuous-time Markov chain in finite-state space. Many scholars have studied the epidemic models with Markovian regime-switching, see [8–10, 12]. Due to the rationality and significance of multiple environments in the model, regime-switching is also applied in population model and other fields, see [18, 22, 23]. We refer the readers to [5, 24, 25] for the theory and more knowledge of Markovian switching.As far as we know, although there are a great many number of research studies on the epidemic models with Markovian switching, there is little work on the properties of the regime-switching SIRS model the with Beddington–DeAngelis incidence rate. In this paper, we will discuss the properties of this kind, and its expression is as follows:(2)S.t=μαt−μαtSt−βαtStItm1αt+m2αtSt+m3αtIt+θαtRtdt−σαtStItm1αt+m2αtSt+m3αtItdBt,I.t=βαtStItm1αt+m2αtSt+m3αtIt−μαt+γαt+dαtItdt+σαtStItm1αt+m2αtSt+m3αtItdBt,R.t=γαtIt−μαt+θαtRtdt,where αt is a continuous-time finite-state Markov chain taking values in space Ε=1,2,…,N with transition rate matrix Q=qijN×N, i.e.,(3)ℙαt+Δt=jαt=i=qijΔt+oΔt,ifi≠j,1+qiiΔt+oΔt,ifi=j,for a sufficiently small Δt>0. We assume in this paper that the matrix is conservative and irreducible, which signifies that the unique stationary distribution π=πi for Markov chain exists and satisfies the equations:(4)πQ=0,∑i=1Nπi=1.The outline of this paper is organized as follows. Section2 proves that the model has a unique positive solution and the invariant set is presented. Meanwhile, some important lemmas which will be used later are given. In Section 3, conditions of stochastic asymptotic stability in the large are established by constructing suitable Lyapunov functionals with regime-switching. In Section 4, conditions for disease extinction are discussed and condition of persistence in the mean is also studied by applying some useful inequality techniques. Section 5 presents some examples and their simulations to confirm our theoretical results.


## 2. Preliminaries

In this section, some background knowledge about differential equations with Markovian switching and several important Lemmas will be proposed, all of which will be used later in the paper.Letℝ+3≔x1,x2,x3xi>0,i=1,2,3. We define Ω,ℱ,ℱt≥0,ℙ as a complete probability space with a filtration ℱt≥0, which satisfies the usual conditions. Consider the SDEs with Markovian switching as follows:(5)dXt=fX,αtdt+gX,αtdBt,where f:ℝn×ℰ⟶ℝn, g:ℝn×ℰ⟶ℝn×m, and Bt=Bt1,…,BtmT is m−dimensional standard Brownian motion. For i∈ℰ and any function VX,i∈C2ℝn×Ε;ℝn, define the operator ℒ by(6)ℒVX,i=VxX,ifX,i+12trgTX,tVxxgX,t+∑j=1NqijVX,j.For a differential system, people are concerned with the existence, uniqueness, form of solutions, and so on. In this paper, we are concerned about whether the model has a unique solution. Can we estimate the range of the solution more precisely? The following lemma is presented to answer these questions.Lemma 1.
For the initial valueS0,I0,R0,α0∈ℝ+3×Ε, there exists a unique positive solution St,It,Rt,αt for model (2). In addition, let Δ≔S,I,R>00≤Nt≔St+It+Rt≤1, then Δ is an invariant set of model (2), i.e., if S0,I0,R0∈Δ, then St,It,Rt∈Δ for all t>0. Moreover, Nt≤maxN0,1.Proof.
We take a piecemeal approach to prove. Let0=τ0<τ1<⋯<τn<⋯ be all jump times of Markov chain αt. When t∈0,τ1, let αt=c0, and we can prove that the model has a positive solution almost surely by constructing appropriate Lyapunov function. This process is common, and we omit here. When jumping to another state αt=c1 for time t∈τ1,τ2, the corresponding parameters in the model will change to another set of numbers and the positive solution can be testified by the same method. Repeat this process on the intervals τ2,τ3, τ3,τ4⋯, and the positive solution can be obtained for t>0.
Then, we prove the range of the solution. Adding three equations in model (2), one has(7)dNt=μαt−μαtNt−dαtItdt≤μαt−μαtNtdt.
For each statei∈ℰ, dNt≤μi−μiNtdt, that is,(8)Nt≤1+N0−1e−μit,where N0=S0+I0+R0. If N0∈Δ, then Nt≤1 holds until the first jump time τ1. When jumping into the next state, we know that the initial value Nτ1≤1 will lead to Nt≤1 for t∈τ1,τ2. Keep repeating the process, Nt≤1 for all t>0. If N0>1, 1+N0−1e−μit will decrease with increase of t for each i∈ℰ, then Nt≤N0. The proof is completed.
Because of this property, we assume that the initial valueS0+I0+R0≤1 below.
Stability is one of the important research topics, which attracts a lot of attention of researchers. In this paper, we will discuss the stochastic stability of the model. In [5, 24], conditions for stochastic asymptotic stability in the large are given.Lemma 2.
Suppose that two functionsμx∈ℝ3;ℝ+ with μP0=0 and Vx,i∈C2ℝ3×ℰ;ℝ+ exist and satisfy(9)ℒVx,i≤−μx,∀x,i∈ℝ3×ℰ,limx⟶∞infi∈ΕVx,i=∞.
Then, the equilibriumP0 of the model is stochastic asymptotically stable in the large.Lemma 3.
LetSt,It,Rt,αt be the solution of model (2); then, it has(10)limsupt⟶∞Rt≤γˇμ^+θ^limsupt⟶∞It,where γˇ=maxi∈ℰri, μ^=mini∈ℰμi, and θ^ is defined in the same way.Proof.
See reference [26] for proof.Lemma 4.
For the solution of model (2), the following formulas hold true:(11)limt⟶∞1t∫0tσαsSsm1αs+m1αsSs+m2αsI1sdBs=0,(12)limt⟶∞1t∫0tσαsSsIsm1αs+m1αsSs+m2αsI1sdBs=0.Proof.
We only prove equation (11). Equation (12) can be testified in the same way. Let Mt≔∫0tσαsSs/m1αs+m1αsSs+m2αsI1sdBs; then, it has(13)limsupt⟶∞M,Mtt≤limsupt⟶∞1t∫0tσˇm1^2dt<∞,a.s.
Thus, we getlimsupt⟶∞Mt/t=0.


## 3. Stability of Disease-Free Equilibrium

In this section, the global stochastic asymptotic stability under some conditions of disease-free equilibrium is studied by making use of Lyapunov functionals with regime-switching.For simplicity, let us definefS,I=m1+m2St+m3It and fS,I,i=m1i+m2iSt+m3iIt, then m1i≤fS,I,i≤m1i+m2i∨m3i≔Mi. We also define functions as follows:(14)Aix≔−σ2i2M2ix2+βim1ix−μi+γi+di,and obviously,(15)Ai1≔Ξi=−σ2i2M2i+βim1i−μi+γi+di.Theorem 1.
For the initial valueS0,I0,R0,α0∈Δ×ℰ, if for every i∈ℰ, βiM2i≥σi2m1i and ∑i=1NπiΞi<0, i.e.,(16)Δ≔∑i=1Nπiβi/m1i∑i=1Nπiμi+γi+di+σ2i/2M2i<1,are satisfied, then the disease-free equilibrium is stochastically asymptotically stable in the large.Proof.
ForS0,I0,R0∈Δ, we define the Lyapunov functional with regime-switching as follows:(17)VS,I,R,i=12κ11−S2+1κ2+ωiIκ2+12κ3R2,where κ1, κ2, κ3, and ωi are positive constants which will be specifically determined later. Using the generalized Ito formula to calculate directly, we can obtain that(18)dVS,I,R,i=ℒVS,I,R,i−κ1σiS1−SIfS,I,idBt+1+κ2ωiσiSIκ2fS,I,idBt,ℒVS,I,R,i=−κ11−Sμi1−S−βiSIfS,I,i+θiRdt+κ1σ2iS2I22f2S,I,i+κ21κ2+ωiIκ2−1βiSIfS,I,i−μi+γi+diI+κ2κ2−11κ2+ωiσ2iS2Iκ22f2S,I,i+κ3RγiI−μi+θiR+Iκ2∑j=1Nωjqij.
From Lemma1, we know that St,It,Rt>0 and S+I+R≤1; thus,(19)ℒVS,I,R,i≤−κ1μi1−S2+κ1βiSI1−SfS,I,i−κ1θiIR+κ1σ2iS2I22f2S,I,i+1+κ2ωiIκ2βiSfS,I,i−1+κ2ωiμi+γi+diIκ2+κ2−11+κ2ωiσ2iS2Iκ22f2S,I,i+κ3γiIR−κ3μi+θiR2+Iκ2∑j=1Nωjqij.
Forκ2≤1, then I2≤I≤Iκ2. Choose κ3≤κ1mini∈ℰδi/ri; hence,(20)ℒVS,I,R,i≤−κ1μi1−S2+κ1βiIκ2m1i+κ3γi−κ1θiIR+κ1σ2iS2Iκ22m12i+1+κ2ωiIκ2βiSm1i−1+κ2ωiμi+γi+diIκ2+κ2−11+κ2ωiσ2iS2Iκ22f2S,I,i−κ3μi+θiR2+Iκ2∑j=1Nωjqij≤−κ1μi1−S2−κ3μi+θiR2+1+κ2ωiIκ2κ1βi1+κ2ωim1i+κ1σ2iS221+κ2ωim12i+AiS+κ2σ2iS22m12i+11+κ2ωi∑j=1Nωjqij.
According to the assumptionβiM2i≥σi2m1i, AiS is monotonically increasing in interval 0,1; then, AiS≤Ξi holds true.
LetΞ≔Ξ1,Ξ2,…,ΞNT. Considering the Poisson equation,(21)Qω=∑i=1NπiΞi1,1,…,1T−Ξ.
It has a solutionω=ω1,ω2,…,ωNT, which implies that(22)∑j=1Nωjqij+Ξi=∑i=1NπiΞi.
Therefore,(23)ℒVS,I,R,i≤−κ1μi1−S2−κ3μi+θiR2+1+κ2ωiIκ2κ1βi1+κ2ωim1i+κ1σ2i21+κ2ωim12i+κ2σ2i2m12i+∑i=1NπiΞi−κ2ωi1+κ2ωi∑j=1Nωjqij.
Letκ2 be a sufficiently small constant such that 1+κ2ωi>0 and(24)κ2σ2i2m12i+∑i=1NπiΞi−κ2ωi1+κ2ωi∑j=1Nωjqij<0.
Again, we choose a sufficiently small numberκ1 to make the following inequality holds true:(25)κ12βim1i+σ2i21+κ2ωim12i+κ2σ2i2m12i+∑i=1NπiΞi−κ2ωi1+κ2ωi∑j=1Nωjqij<0.
We can see from (32) that the coefficients are all negative constants. Hence, we arrive at the conclusion by taking advantage of Lemma 2.


## 4. Asymptotic Behavior of Disease

In this section, we shall study the asymptotic behavior of disease in model (2).


### 4.1. Extinction of Disease

First of all, we study the conditions for disease extinction. With these conditions, we can take some measures to adjust the parameters in the model to make the disease go extinct in the long run.Theorem 2.
Assume thatSt,It,Rt,αt is the solution to model (2); if one of the two conditions below holds true:(1)
If(26)R0S≔∑i=1Nπiβ2i/2σ2i∑i=1Nπiμi+γi+di<1.(2)
1/m1i+m2i≤βi/σ2i for all i∈ℰ, and(27)R1S≔∑i=1Nπiβi/m1i+m2i∑i=1Nπiμi+γi+di+σi/2m1i+m2i2<1.Then, the disease will be extinct exponentially almost surely, i.e.,(28)limt⟶∞It=0,a.s.Moreover,(29)limt⟶∞Rt=0,a.s.,limt⟶∞St=1,a.s.Proof.
For case (1), applying the Ito formula to functionlnIt, we can obtain that(30)dlnIt=ℒlnItdt+σαtStfS,I,αtdBt,(31)ℒlnIt=βαtStfS,I,αt−μαt+γαt+dαt−σ2αtS22f2S,I,αt.
DefinegS,I,αt=S/fS,I,αt, then 0<gS,I,αt≤1/m1αt+m2αt and(32)ℒlnIt=−σ2αt2gS,I,αt−βαtσ2αt2+β2αt2σ2αt−μαt+γαt+dαt≤β2αt2σ2αt−μαt+γαt+dαt.
Integrating for the both sides of (30) from 0 to t followed by dividing both sides by t yields(33)lnItt≤lnI0t+1t∫0tβ2αs2σ2αt−μαs+γαs+dαsds+1t∫0tσαsSsfS,I,αsdBs.
From ergodic properties of Markov chainsαt and Lemma 4, we can obtain(34)limsupt⟶∞lnItt≤∑i=1Nπiβ2i2σ2i−μi+γi+di.
With the help of (26), we get that limt⟶∞lnIt/t<0, a.s., which means the disease will go to extinction exponentially almost surely.
For case (2), we can obtain from (31) that if 1/m1i+m2i≤βi/σ2i for all i∈ℰ, then(35)ℒlnIt≤βαtm1αt+m2αt−μαt+γαt+dαt−σ2αt2m1αt+m2αt2.
Thus,(36)lnItt≤lnI0t+1t∫0tβαsm1αs+m2αs−μαs+γαs+dαsds−1t∫0tσ2αs2m1αs+m2αs2ds+1t∫0tσαsSsfS,I,αsdBs.
Therefore, conclusion (28) can be obtained by taking advantage of (27).
The equalitylimt⟶∞Rt=0,a.s. Followed by utilizing Lemma 3, next we prove the second conclusion of (29).
Assume thatA=ω∈Ω:limt⟶∞It=0,limt⟶∞Rt=0; then, the above conclusions tell us that ℙA=1. Thus, for any constant ϵ>0 and ω∈A, there exists a positive constant T=Tω,ε such that(37)It,ω≤ε,Rt,ω≤ε,∀t≥T.
Therefore, according to (7), for t≥T, one has(38)dNtdt=μαt−μαtNt−dαtIt≥μαt−μαtNt−dˇε.
No matter how large thet is, (38) belongs to one of the following formulas:(39)dNtdt≥μi−μiNt−dˇε,i∈Ε.
Applying the variation of constants formula to (39) yields(40)Nt≥μi−dˇεμi+NT−μi−dˇεμiexp−μit−T.
For anyi∈ℰ and the arbitrariness of ε, the right side of (40) tends to 1 when t goes to infinity; then, it has(41)liminft⟶∞Nt1,∀ω∈A.which implies that(42)liminft⟶∞Nt≥1,a.s.
Recalling the assertionNt≤1 in Lemma 1 and the fact that Nt=St+It+Rt, the conclusion that limt⟶∞St=1,a.s. can be obtained.Remark 1.
(1)
From the first condition above in Theorem2, we can see that if the intensity of stochastic perturbation is sufficiently large, R0S<1 is sure to work; then, the disease will be extinct, which shows that the stochastic perturbation has an important influence on the model. When the intensity of stochastic perturbation is small, the disease will still be extinct if the second condition is satisfied.(2)
We can see from the expressions of (16) and (27) that R1S≤Δ; thus, if Δ<1, then R1S<1, which means that stochastically asymptotically stability in the large will also make the disease go extinct.


### 4.2. Permanence of Disease

Next, we move forward to analyze the conditions of disease persistence. First of all, a definition about persistence is given.Definition 1.
The disease in model (2) is called persistent in the mean if there exists a constant η>0 such that(43)liminft⟶∞1t∫0tIudu≥η,a.s.Theorem 3.
Assume thatSt,It,Rt,αt is the solution to model (2) with initial value S0,I0,R0,α0 and 0<S0+I0+R0≤1. If(44)R2S≔∑i=1Nπiμiλ∑i=1Nπiμi+di+ri+σ2i/2m1i+m2i2>1,where λ=maxi∈ℰμim1i+m2i/βi, then the disease will be persistent in the mean.Proof.
We prove this theorem in two steps. First, let us prove that the following inequality holds true for a certain positive constantC:(45)HS,I,i≔λ−Iβim1i+m2iS+m3iI−μiS+CI≥0.where I,S∈0,1,i∈ℰ.
It is clear thatHS,0,i=λβi/m1i+m2iS−μiS≥λβi/m1i+m2i−μiS≥0, and(46)∂H∂I=−βiSm1i+m2iS+λm3im1i+m2iS+m3iI2+C≥C−maxi∈Εβim1i+m2i+λm3im12i.
We can see easily that, for sufficiently large constantC, ∂H/∂I>0; then, from the monotone increasing property of HS,I,i, there exists a constant ϵ, HS,I,i>0 for S∈0,1, I∈0,ε, and all i∈ℰ. Next, we prove that HS,I,i>0 for S∈0,1, I∈ε,1, and all i∈ℰ. According to the expression of HS,I,i,(47)HS,I,i≥−βiSIm1i+m2iS+m3iI−μiS+CI≥Cϵ−maxi∈ℰβim1i+μi.
For sufficiently large constantC, HS,I,i>0 for I∈ε,1. Now, inequality (45) has been proved.
In what follows, we will demonstrate our conclusion in the theorem. From the first equation of model (2), we know that(48)dSt≥μαt−μαtSt−βαtStItfS,I,αtdt−σαtStItfS,I,αtdBt≥μαt−λβαtStfS,I,αt−CItdt−σαtStItfS,I,αtdBt.
According to the result of (31), we obtain that(49)ℒλlnItdt=λβαtStfS,I,αt−λμαt+γαt+dαt−λσ2αtS22f2S,I,αt≥μαt−λμαt+γαt+dαt+σ2αt2m1αt+m2αt2dt−CItdt−dSt−σαtStItfS,I,αtdBt.
Making use of the result of (49), integrating for the both sides of dλlnIt=ℒλlnItdt+λσαtStIt/fS,I,αtdBt from 0 to t and dividing by t, one has(50)C1t∫0tIsds≥1t∫0tμαs−λμαs+γαs+dαsds−1t∫0tλσ2αs2m1αs+m2αs2−λlnIt−lnI0t−St−S0t+1t∫0tλ−1σαsSsIsfS,I,αsdBs.
Using the boundedness ofSt, It, and Lemma 4 along with the ergodic property of Markov chain αt, we take the limit inferior for both sides; then,(51)liminft⟶∞1t∫0tIsds≥1C∑i=1Nπiμi−λμi+γi+di+σ2i2m1i+m2i2.
Therefore, if (44) is satisfied, then liminft⟶∞1/t∫0tIsds>0, which means the disease will be persistent in the mean.
The proof is completed.Corollary 1.
From the expressions ofR1S in (27) and R2S in (44), we have R1S≥R2S. R1S=R2S holds if and only if μi, βi, m1i, and m2i in the formulas are fixed constants.Proof.
(52)R1SR2S=λ∑i=1Nπiβi/m1i+m2i∑i=1Nπiμi=maxμim1i+m2i/βi∑i=1Nβi/μim1i+m2iπiμi∑i=1Nπiμi≥∑i=1Nπiμi∑i=1Nπiμi=1.
As can be seen from the above proof, if and only ifmaxi∈ℰμim1i+m2i/βi=μim1i+m2i/βi for every i∈ℰ, R1S=R2S holds true.Remark 2.
(1)
R1S≥R2S means that Theorems 2 and 3 are not in conflict. If R1S<1, then R2S<1, and the disease will die out. If R2S>1, then R1S>1, and the disease will be persistent.(2)
If there is no regime-switching in model (2), i.e., there is only one environment, then(53)R1S=R2S≔R=β/m1+m2μ+γ+d+σ2/2m1+m22,can be considered as the threshold of disease persistence and extinction in the model. We can see from formula (53) that R will increase with the increase of contact coefficient β. When R>1, the disease will be persistent in the long run. Therefore, it will be one of the important ways to control infectious diseases to reduce the value of transmission coefficient β by isolating the infected and limiting the people’s going out, which is widely used when SARS virus and new coronavirus pneumonia spread in China.
Due to the existence of regime-switching, the behavior of disease will have different phenomena. Examples2 and 3 will reveal some interesting things.


## 4.1. Extinction of Disease

First of all, we study the conditions for disease extinction. With these conditions, we can take some measures to adjust the parameters in the model to make the disease go extinct in the long run.Theorem 2.
Assume thatSt,It,Rt,αt is the solution to model (2); if one of the two conditions below holds true:(1)
If(26)R0S≔∑i=1Nπiβ2i/2σ2i∑i=1Nπiμi+γi+di<1.(2)
1/m1i+m2i≤βi/σ2i for all i∈ℰ, and(27)R1S≔∑i=1Nπiβi/m1i+m2i∑i=1Nπiμi+γi+di+σi/2m1i+m2i2<1.Then, the disease will be extinct exponentially almost surely, i.e.,(28)limt⟶∞It=0,a.s.Moreover,(29)limt⟶∞Rt=0,a.s.,limt⟶∞St=1,a.s.Proof.
For case (1), applying the Ito formula to functionlnIt, we can obtain that(30)dlnIt=ℒlnItdt+σαtStfS,I,αtdBt,(31)ℒlnIt=βαtStfS,I,αt−μαt+γαt+dαt−σ2αtS22f2S,I,αt.
DefinegS,I,αt=S/fS,I,αt, then 0<gS,I,αt≤1/m1αt+m2αt and(32)ℒlnIt=−σ2αt2gS,I,αt−βαtσ2αt2+β2αt2σ2αt−μαt+γαt+dαt≤β2αt2σ2αt−μαt+γαt+dαt.
Integrating for the both sides of (30) from 0 to t followed by dividing both sides by t yields(33)lnItt≤lnI0t+1t∫0tβ2αs2σ2αt−μαs+γαs+dαsds+1t∫0tσαsSsfS,I,αsdBs.
From ergodic properties of Markov chainsαt and Lemma 4, we can obtain(34)limsupt⟶∞lnItt≤∑i=1Nπiβ2i2σ2i−μi+γi+di.
With the help of (26), we get that limt⟶∞lnIt/t<0, a.s., which means the disease will go to extinction exponentially almost surely.
For case (2), we can obtain from (31) that if 1/m1i+m2i≤βi/σ2i for all i∈ℰ, then(35)ℒlnIt≤βαtm1αt+m2αt−μαt+γαt+dαt−σ2αt2m1αt+m2αt2.
Thus,(36)lnItt≤lnI0t+1t∫0tβαsm1αs+m2αs−μαs+γαs+dαsds−1t∫0tσ2αs2m1αs+m2αs2ds+1t∫0tσαsSsfS,I,αsdBs.
Therefore, conclusion (28) can be obtained by taking advantage of (27).
The equalitylimt⟶∞Rt=0,a.s. Followed by utilizing Lemma 3, next we prove the second conclusion of (29).
Assume thatA=ω∈Ω:limt⟶∞It=0,limt⟶∞Rt=0; then, the above conclusions tell us that ℙA=1. Thus, for any constant ϵ>0 and ω∈A, there exists a positive constant T=Tω,ε such that(37)It,ω≤ε,Rt,ω≤ε,∀t≥T.
Therefore, according to (7), for t≥T, one has(38)dNtdt=μαt−μαtNt−dαtIt≥μαt−μαtNt−dˇε.
No matter how large thet is, (38) belongs to one of the following formulas:(39)dNtdt≥μi−μiNt−dˇε,i∈Ε.
Applying the variation of constants formula to (39) yields(40)Nt≥μi−dˇεμi+NT−μi−dˇεμiexp−μit−T.
For anyi∈ℰ and the arbitrariness of ε, the right side of (40) tends to 1 when t goes to infinity; then, it has(41)liminft⟶∞Nt1,∀ω∈A.which implies that(42)liminft⟶∞Nt≥1,a.s.
Recalling the assertionNt≤1 in Lemma 1 and the fact that Nt=St+It+Rt, the conclusion that limt⟶∞St=1,a.s. can be obtained.Remark 1.
(1)
From the first condition above in Theorem2, we can see that if the intensity of stochastic perturbation is sufficiently large, R0S<1 is sure to work; then, the disease will be extinct, which shows that the stochastic perturbation has an important influence on the model. When the intensity of stochastic perturbation is small, the disease will still be extinct if the second condition is satisfied.(2)
We can see from the expressions of (16) and (27) that R1S≤Δ; thus, if Δ<1, then R1S<1, which means that stochastically asymptotically stability in the large will also make the disease go extinct.


## 4.2. Permanence of Disease

Next, we move forward to analyze the conditions of disease persistence. First of all, a definition about persistence is given.Definition 1.
The disease in model (2) is called persistent in the mean if there exists a constant η>0 such that(43)liminft⟶∞1t∫0tIudu≥η,a.s.Theorem 3.
Assume thatSt,It,Rt,αt is the solution to model (2) with initial value S0,I0,R0,α0 and 0<S0+I0+R0≤1. If(44)R2S≔∑i=1Nπiμiλ∑i=1Nπiμi+di+ri+σ2i/2m1i+m2i2>1,where λ=maxi∈ℰμim1i+m2i/βi, then the disease will be persistent in the mean.Proof.
We prove this theorem in two steps. First, let us prove that the following inequality holds true for a certain positive constantC:(45)HS,I,i≔λ−Iβim1i+m2iS+m3iI−μiS+CI≥0.where I,S∈0,1,i∈ℰ.
It is clear thatHS,0,i=λβi/m1i+m2iS−μiS≥λβi/m1i+m2i−μiS≥0, and(46)∂H∂I=−βiSm1i+m2iS+λm3im1i+m2iS+m3iI2+C≥C−maxi∈Εβim1i+m2i+λm3im12i.
We can see easily that, for sufficiently large constantC, ∂H/∂I>0; then, from the monotone increasing property of HS,I,i, there exists a constant ϵ, HS,I,i>0 for S∈0,1, I∈0,ε, and all i∈ℰ. Next, we prove that HS,I,i>0 for S∈0,1, I∈ε,1, and all i∈ℰ. According to the expression of HS,I,i,(47)HS,I,i≥−βiSIm1i+m2iS+m3iI−μiS+CI≥Cϵ−maxi∈ℰβim1i+μi.
For sufficiently large constantC, HS,I,i>0 for I∈ε,1. Now, inequality (45) has been proved.
In what follows, we will demonstrate our conclusion in the theorem. From the first equation of model (2), we know that(48)dSt≥μαt−μαtSt−βαtStItfS,I,αtdt−σαtStItfS,I,αtdBt≥μαt−λβαtStfS,I,αt−CItdt−σαtStItfS,I,αtdBt.
According to the result of (31), we obtain that(49)ℒλlnItdt=λβαtStfS,I,αt−λμαt+γαt+dαt−λσ2αtS22f2S,I,αt≥μαt−λμαt+γαt+dαt+σ2αt2m1αt+m2αt2dt−CItdt−dSt−σαtStItfS,I,αtdBt.
Making use of the result of (49), integrating for the both sides of dλlnIt=ℒλlnItdt+λσαtStIt/fS,I,αtdBt from 0 to t and dividing by t, one has(50)C1t∫0tIsds≥1t∫0tμαs−λμαs+γαs+dαsds−1t∫0tλσ2αs2m1αs+m2αs2−λlnIt−lnI0t−St−S0t+1t∫0tλ−1σαsSsIsfS,I,αsdBs.
Using the boundedness ofSt, It, and Lemma 4 along with the ergodic property of Markov chain αt, we take the limit inferior for both sides; then,(51)liminft⟶∞1t∫0tIsds≥1C∑i=1Nπiμi−λμi+γi+di+σ2i2m1i+m2i2.
Therefore, if (44) is satisfied, then liminft⟶∞1/t∫0tIsds>0, which means the disease will be persistent in the mean.
The proof is completed.Corollary 1.
From the expressions ofR1S in (27) and R2S in (44), we have R1S≥R2S. R1S=R2S holds if and only if μi, βi, m1i, and m2i in the formulas are fixed constants.Proof.
(52)R1SR2S=λ∑i=1Nπiβi/m1i+m2i∑i=1Nπiμi=maxμim1i+m2i/βi∑i=1Nβi/μim1i+m2iπiμi∑i=1Nπiμi≥∑i=1Nπiμi∑i=1Nπiμi=1.
As can be seen from the above proof, if and only ifmaxi∈ℰμim1i+m2i/βi=μim1i+m2i/βi for every i∈ℰ, R1S=R2S holds true.Remark 2.
(1)
R1S≥R2S means that Theorems 2 and 3 are not in conflict. If R1S<1, then R2S<1, and the disease will die out. If R2S>1, then R1S>1, and the disease will be persistent.(2)
If there is no regime-switching in model (2), i.e., there is only one environment, then(53)R1S=R2S≔R=β/m1+m2μ+γ+d+σ2/2m1+m22,can be considered as the threshold of disease persistence and extinction in the model. We can see from formula (53) that R will increase with the increase of contact coefficient β. When R>1, the disease will be persistent in the long run. Therefore, it will be one of the important ways to control infectious diseases to reduce the value of transmission coefficient β by isolating the infected and limiting the people’s going out, which is widely used when SARS virus and new coronavirus pneumonia spread in China.
Due to the existence of regime-switching, the behavior of disease will have different phenomena. Examples2 and 3 will reveal some interesting things.


## 5. Examples and Simulations

In this section, some examples will be proposed and their numerical simulations are presented to verify our theoretical results above.Letαtt≥0 be a Markov chain taking values in the space ℰ=1,2 with the transition rate matrix:(54)Q=−mmn−n.Hence, there exits two environments, say, Environment 1 and Environment 2, and its stationary distributionπ=π1,π2=n/m+n,m/m+n.Example 1.
First, we verify the results of Lemma1 and Theorem 1. Assume that m=2 and n=1; then, the stationary distribution π=π1,π2=1/2,2/3. Let S0=0.12, I0=0.5, R0=0.3, μ1=0.15, μ2=0.2, β1=0.5, β2=0.3, d1=0.15, d2=0.1, θ1=0.4, θ2=0.25, γ1=0.3, γ2=0.4, σ1=0.8, σ1=1, m11=1, m12=1, m21=1, m22=2, m31=2, and m32=3; thus, S0+I0+R0<1, βiM2i≥σi2mi, and ∑i=1NπiΞi<0 are satisfied for i=1,2.
According to Lemma1, Nt=St+It+Rt≤1 will always hold, see Figure 1(a). In order to verify the stability of the model, we select two different sets of parameters. The parameters in set 1 are as above, and its trajectory is shown Figure 1(b). Parameters in the other set are the same except β1=0.5, β2=0.3, m=1, and n=2. Moreover, the initial values are different; let S0=0.4, I0=0.35, and R0=0.2, and its trajectory can be seen in Figure 1(c).Figure 1
The trajectories with parameters in Example1: (a) the trajectory of Nt; (b and c) the trajectories with different initial values.
(a)(b)(c)Figure 2
(a) The trajectory of Environment 1 withR=1.3483>1 and all parameters in Example 2; (b) the trajectory of the regime-switching model with R1S<1 with parameters in Example 2.
(a)(b)Figure 3
(a) The trajectory of Environment 1 withR=0.9756<1 and parameters in Example 3; (b) the trajectory of the regime-switching model with R1S>1 and parameters in Example 3.
(a)(b)Example 2.
Next, let us test Theorem2. Since the parameters of the first conclusion can be selected from Example 1, we only verify the second conclusion. Assume that m=1 and n=2, and let β1=0.6, β2=0.4, γ1=0.2, γ2=0.4, σ1=0.3, σ1=5, m21=0, m22=2, m31=2, and m32=3; the other parameters are the same as those in Example 1. Then, R1S=0.8313<1 and 1/m1i+m2i≤βi/σ2i are satisfied, although R=1.3483>1 in Environment 1. The data means that the disease will be persistent in Environment 1 (see Figure 2(a)), but it will go to extinction as a result of regime-switching, which shows the important role of Markovian Switching. The simulation can be seen in Figure 2(b).Example 3.
Finally, we verify Theorem3. Let m=3, n=12, μ1=0.2, μ2=0.3, β1=0.9, β2=0.95, d1=0.1, d2=0.08, γ1=0.15, γ2=0.2, σ1=0.3, σ2=0.2, m11=1, m12=1, m21=1, m22=0, m31=2, and m32=1. By calculation, we can get R2S=1.0123>1 while R=0.9756<1 in Environment 1, which means the disease will die out in Environment 1 (see Figure 3(a)), but it will continue as a result of Markovian switching, see Figure 3(b).


## 6. Conclusions

In this paper, we study the regime-switching SIRS model with the Beddington–DeAngelis incidence rate. We first prove that the model we discuss has a unique positive solution. Secondly, we give the conditions of global stochastic asymptotic stability by the Lyapunov method. Then, the thresholds of disease behavior are given by some useful inequality technique. Finally, some examples are given and numerical simulations are presented to confirm our conclusions.In addition, some more topics are worth further studying. More complex models can be considered to better reflect the actual situation, for example, the model with more general incidence rate, more perturbations such as jump noise, or the model with the effect of time delay. We will keep these for our future research.



---

*Source: 7181939-2020-07-15.xml*