# Positive Recurrence and Extinction of Hybrid Stochastic SIRS Model with Nonlinear Incidence Rate

Authors: Liang Chen; Shan Wang
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this paper, the dynamic behavior of a class of hybrid SIRS model with nonlinear incidence is studied. Firstly, we provide the condition under which the positive recurrence exists and then give the thresholdR0S for disease extinction, that is, when R0S<1, the disease will die out. Finally, some examples are constructed to verify the conclusion.


---

## Body

## 1. Introduction

Disease is one of the most dangerous enemies of human beings. COVID-19 has spread throughout the world since the end of 2019 and brought great harm to people’s life and global economy. It is one of the important measures to understand the characteristics of disease transmission and control the disease to establish and study the disease model. Because of its importance and rich research content, different scholars have used various methods to study dynamical behavior and properties of epidemic models, such as persistence and extinction [1–7], stationary distribution and ergodicity [5, 8–12], stability [13, 14], bifurcation [15, 16], and optimal control of disease [17]. In [17], the optimal vaccination strategies were used to minimize the numbers of the susceptible and infected individuals as well as to maximize recovered individuals.Incidence rate plays an important role in the spread of diseases. Liu et al. [18] abandoned the assumption that incidence rate is proportional to the quantity of infected and susceptible ones to study the influence of nonlinear incidence function on SIRS model. This is more reasonable and suitable for practical situations, because the premise of bilinear incidence rate is that the population is homogeneously mixed, and the probability of being infected for each person is the same. Meng et al. [3] discussed SIS model with the form SI/a+I and double epidemic hypothesis. In [6], Cai et al. studied the ratio-dependent transmission rate with the form gS,I=ShI/Sh+αHh in SIRS epidemic model. Recently, Wei-Xue studied the asymptotic stability of SEIR model with incidence rate gIS=SI/1+aIq [19]. In this paper, we concentrate on general incidence rate with this form.Life is full of stochastic disturbances, which often have an important impact on the system and will essentially change the properties of the system. White noise depicted by Brownian motion and telegraph noise characterized by continuous time Markov chain are two important types of noise. Some scholars have studied the effect ofLe´vy noise on epidemic models [20, 21]. Although many scholars have studied the different properties of epidemic models with white noise and Markovian switching [22–26], there are few studies on the positive recurrence and extinction of SIRS models with the abovementioned general incidence rate. In this paper, we will study the abovementioned properties of stochastic hybrid SIRS model with the following form:(1)dSt=Λαt−dαtSt−βαtStIt1+aαtIq+δαtRtdt+σ1αtStdB1t,dIt=βαtStIt1+aαtIq−dαt+γαt+μαtItdt+σ2αtItdB2t,dRt=γαtIt−dαt+δαtRtdt+σ3αtRtdB3t,where 0<q≤1. St, It, and Rt denote the numbers of the susceptible, infected, and recovered. Λ is the recruitment rate, d represents the natural death rate, β means valid contact rate, μ stands for the disease-caused death rate, γ signifies the recovery rate of infected individuals, and δ expresses the rate at which the recovered ones lose their immunity and go back to the susceptible class.Bit, i=1,2,3 are independent Brownian motions, σik, i=1,2,3 are the intensities of the disturbances and Markov chain αtt≥0 will be introduced in detail in Section 2. Moreover, Section 2 will give an important lemma which will be used in this paper. Obviously, Markovian switching and nonlinear incidence rate make the model more applicable, but also increase the difficulty of research. Section 3 is devoted to prove the positive recurrence and stationary distribution of model (1) by constructing suitable Lyapunov functions. Section 4 gives the threshold for disease extinction. Some examples and their simulations will be presented in Section 5 to verify our results.


## 2. Preliminaries

Some background knowledge about differential equations and an important lemma of the paper will be introduced in this section.Throughout this paper, denote byΩ,ℱ,ℱtt≥0,ℙ the complete probability space with a filtration which satisfies the usual conditions. Furthermore, let ℝ+n=x∈ℝn:xi>0,1≤i≤n and assume that Bit, i=1,2,3 are independent Brownian motions on the probability space.Assume thatαtt≥0 is a right continuous Markov chain in a finite state space N=1,2,…,N and its generator is Γ=γijN×N, satisfying(2)ℙαt+Δ=j|αt=i=γijΔ+oΔ,ifi≠j,1+γiiΔ+oΔ,ifi=j,for Δ↓0. Here, γij>0 if i≠j and ∑j=1Nγij=0, for all i∈N. In this paper, we assume that the Markov chain is irreducible; thus, it has a unique stationary distribution π=π1,π1,…,πN which can be obtained by solving the equation(3)πΓ=0,∑j=1Nπj=1.Considering then-dimensional SDEs with Markovian switching,(4)dXt=fXt,αtdt+gXt,αtdBt,with the initial value Xt0=x0∈ℝn and αt0=i0, where f⋅,⋅:ℝn×N⟶ℝn, g⋅,⋅:ℝn×N⟶ℝn×n, and σx,k=gx,kgTx,k=σijx,k. Let C2,1ℝn×N;ℝ+ be the set of nonnegative functions VX,i defined on ℝn×N and the functions are continuously twice differential in X. The differential operator ℒ on the function Vx,i is defined by(5)ℒVx,i=∑j=1nfjx,i∂Vx,i∂xj+12∑j=1n∑k=1nσkjx,i∂2Vx,i∂xj∂xk+∑j=1nγijVx,j.Lemma 1.
(see [27]). Supposing that the following three conditions hold true,(A1)
γij>0 if i≠j;(A2)
For everyi∈N and the positive constant λ, σx,i satisfies σx,iξ,ξ≥λξ2;(A3)
There exists a bounded setU with compact closure and a nonnegative function Vx,i such that ℒV<−κ for any x∈Uc and i∈N where κ is a positive constant, and then Xt,αt is the positive recurrent. That is to say, there exists a unique stationary distribution Π⋅,⋅ such that(6)ℙlimt⟶∞1t∫0thXt,αtdt=∑i=1N∫ℝnhx,iΠdx,i=1,for any integrable function hx,i with respect to Π.For the model we discuss, our first concern is whether it has a unique positive solution. The following theorem will answer this question.Theorem 1.
For the initial valueS0,I0,R0,i0∈ℝ+3×N, model (1) has a unique solution St,It,Rt and the solution is still in R+3 with probability one.The proof is common; we omit it here.


## 3. Positive Recurrence of Model (1)

In this section, positive recurrence of model (1) will be discussed by utilizing the theory put forward by Khasminskii under some conditions.Theorem 2.
For any initial valueSt,It,Rt,α0∈ℝ+3×N, if(7)R1S=∑k=1NπkΛkβk1/33∑k=1Nπkdk+σ12k/2∑k=1Nπkdk+γk+μk+σ22k/2>1,then the solution St,It,Rt,αt of model (1) has a unique stationary distribution, which means that the disease will persist.Proof.
In order to prove Theorem2, we only need to check that the conditions in Lemma 1 are satisfied. First of all, we give some definitions of notations. Let aˇ≔maxi∈Nai, a^≔mini∈Nai, and a∨b=maxa,b. The first condition (A1) holds obviously from the assumption of Markov chain.
Let(8)Uε=ε,1ε×ε,1ε×ε,1ε,where ϵ is a sufficiently small number. Thus, Uϵ¯⊂ℝ+3. From the expression of model (1), we can see that σS,I,R,k=diagσ12kS2,σ22kI2,σ32kR2, k∈N, which implies that σS,I,R,k is positive definite and thus(9)λk≔minS,I,R∈Uϵ¯σ12kS2,σ22kI2,σ32kR2>0.
Hence, for anyξ∈U, it has σS,I,R,kξ,ξ≥λ^ξ2.
We verify that condition (A2) holds. Now, we move to prove condition (A3). Some functions are defined as follows:(10)V1=−c1lnS−c2lnI+aˇμ^qIq,V2=−lnS−lnI,V3=1p+1S+I+Rp+1,and the function V˜=MV1+ωk+V2+V3, where the constants c1, c2, p, M, and ωk will be specified later. Using the continuity of the function V˜, we can take the minimum value of the function V˜S∗,I∗,R∗,ω∗ and let(11)VS,I,R,ω≔V˜S,I,R,ω−V˜S∗,I∗,R∗,ω∗.In this way, the nonnegativity of the selected function V can be guaranteed.
By making use of theIto^ formula, we get that(12)ℒV1=−c1SΛk−dkS−βkSI1+akIq+δkR+c1σ12k2−c2IβkSI1+akIq−dk+γk+μkI+c2σ22k2+aˇμ^Iq−1βkSI1+akIq−dk+γk+μkI+aˇq−1σ22kIq2μ^=−c1ΛkS+c1dk+c1βkI1+akIq−c1δkRS+c1σ12k2−c2βkS1+akIq+c2dk+γk+μk+c2σ22k2+1+aˇβkIqSμ^1+akIq−aˇμ^Iqdk+γk+μk+aˇq−12μ^σ22kIq−1≤−c1ΛkS−c2βkS1+aˇIq−aˇIq−1+c1dk+σ12k2+c1βkI1−qak+c2dk+γk+μk+σ22k2+aˇβkSμ^ak+1≤−3c1c2Λkβk1/3+c1dk+σ12k2+1+c2dk+γk+μk+σ22k2+c1βˇa^I1−q+aˇβˇμ^a^S.
LetAk=−3c1c2Λkβk1/3+c1dk+σ12k/2+c2dk+γk+μk+σ22k/2, and select(13)c11/3=∑k=1NπkΛkβk1/3∑k=1Nπkdk+σ12k/22/3∑k=1Nπkdk+γk+μk+σ22k/21/3,as well as(14)c21/3=∑k=1NπkΛkβk1/3∑k=1Nπkdk+σ12k/21/3∑k=1Nπkdk+γk+μk+σ22k/22/3.
Consider the system(15)Γω=∑k=1NπkAk1,1,…,1T−A,where A=A1,A2,…,ANT. Hence, it has the solution ω=ω1,ω2,…,ωNT such that ∑j=1Nγkjωj+Ak=∑k=1NπkAk. Therefore, we obtain that(16)ℒV1+ωk≤Ak+1+c1βˇa^I1−q+aˇβˇμ^a^S+∑j=1Nγkjωj≤−λ+c1βˇa^I1−q+aˇβˇμ^a^S.
Here,λ≔R1S−1 and R1S is defined in (7).
Again, we apply theIto^ formula to −lnS, −lnI, and V3 to obtain(17)ℒ−lnS=−ΛkS+dk+βkI1+akIq−δkRS+σ12k2,ℒ−lnR=−γkIR+dk+δk+σ32k2,ℒV3=S+I+RpΛk−dkS+I+R−μkI+pS+I+Rp−1σ12k2S2+σ22k2I2+σ12k2R2,≤ΛˇS+I+Rp−d^−p2σˇ12+σˇ22+σˇ32S+I+Rp+1≤supS,I,R∈ℝ+3ΛˇS+I+Rp−d^−p/2σˇ12+σˇ22+σˇ322S+I+Rp+1−d^−p/2σˇ12+σˇ22+σˇ322S+I+Rp+1≕E−d^−p/2σˇ12+σˇ22+σˇ322S+I+Rp+1.
Here, we choose sufficiently small constantp such that(18)d^−p2σˇ12+σˇ22+σˇ32>0.
Thus,E<∞ and(19)ℒV3≤E−d^−p/2σˇ12+σˇ22+σˇ322Sp+1+Ip+1+Rp+1.
Consequently,(20)ℒV≤−Mλ+Mc1+1βˇa^I1−q+Maˇβˇμ^a^S−Λ^S−δ^RS−γ^IR+2dˇ+δˇ+E+σˇ12+σˇ322−d^−p/2σˇ12+σˇ22+σˇ322Sp+1+Ip+1+Rp+1.
Select sufficiently largeM and sufficiently small ϱ such that the following formulas hold:(21)−Λ^ϱ+F≤−1,(22)−Mλ+Mc1+1βˇa^ϱ1−q+G≤−1,(23)−γ^ϱ+H≤−1,(24)−Mλ−d^−p/2σˇ12+σˇ22+σˇ3241ϱp+1+K≤−1,where the constants F,G,H,andK are all finite and their representations can been seen in (27)–(29) and (31).
For the constantϱ selected above, define a set(25)Dϱ=S,I,R∈ℝ+3:ϱ≤S≤1ϱ,ϱ≤I≤1ϱ,ϱ2≤R≤1ϱ2.
Hence, we can divide the setℝ+3∖Dϱ into six regions(26)D1=S,I,R∈ℝ+3:0<S<ϱ,D2=S,I,R∈ℝ+3:0<I<ϱ,D3=S,I,R∈ℝ+3:0<R<ϱ2,I≥ϱ,D4=S,I,R∈ℝ+3:S>1ϱ,D5=S,I,R∈ℝ+3:I>1ϱ,D6=S,I,R∈ℝ+3:R>1ϱ2.
In what follows, we will prove thatℒV≤−1 in ℝ+3∖Dϱ, that is, ℒV≤−1 in the regions Di, i=1,…,6.Case 1: whenS,I,R∈D1, it has(27)ℒV≤−Λ^ϱ+F,whereF=supS,I,R∈ℝ+3−Mλ+Mc1+1βˇ/a^I1−q+Maˇβˇ/μ^a^S+2dˇ+δˇ+E+σ12ˇ+σ32ˇ/2−d^−p/2σˇ12+σˇ22+σˇ32/2Sp+1+Ip+1+Rp+1<∞.From the result of (21), one has ℒV≤−1.Case 2: whenS,I,R∈D2, it has(28)ℒV≤−Mλ+Mc1+1βˇa^ϱ1−q+G,whereG=supS,I,R∈ℝ+3Maˇβˇ/μ^a^S+2dˇ+δˇ+E+σˇ12+σˇ32/2−d^−p/2σˇ12+σˇ22+σˇ32/2Sp+1+Ip+1+Rp+1<∞.ℒV≤−1 can be obtained by (22).Case 3: whenS,I,R∈D3, we have(29)ℒV≤−γ^IR+H≤−γ^ϱ+H.Here,H=supS,I,R∈ℝ+3−Mλ+Mc1+1βˇ/a^I1−q+Maˇβˇ/μ^a^S+2dˇ+δˇ+E+σˇ12+σˇ32/2−d^−p/2σˇ12+σˇ22+σˇ32/2Sp+1+Ip+1+Rp+1<∞.ℒV≤−1 can be obtained from the result of (23).Case 4: whenS,I,R∈D4, we have(30)ℒV≤−Mλ−d^−p/2σˇ12+σˇ22+σˇ324Sp+1+K,≤−Mλ−d^−p/2σˇ12+σˇ22+σˇ3241ϱp+1+K,where(31)K=supS,I,R∈ℝ+3Mc1+1βˇa^I1−q+Maˇβˇμ^a^S+2dˇ+δˇ+E+σˇ12+σˇ322−d^−p/2σˇ12+σˇ22+σˇ324Sp+1+Ip+1+Rp+1<∞.
Thus, we can get thatℒV≤−1 with the help of (24).
Similarly,S,I,R∈D5 and S,I,R∈D6 can be proved in a similar way. So far, we have proved that ℒV≤−1 in region ℝ+3∖Dϱ. By virtue of Lemma 1, the positive recurrence of model (1) can be obtained.Remark 1.
We can see from the proof that in order to obtain the positive recurrence of the model, different Lyapunov functions are constructed to reach the goal.


## 4. Extinction of Disease

In this section, we will discuss the conditions for extinction of disease in model (1). Let us start with a lemma as follows.Lemma 2.
Assume that there exists some constantp>1 such that d^−p−1/2σˇ12∨σˇ22∨σˇ32>0 and St,It,Rt,αt is the solution of model (1) with the initial value St,It,Rt,α0, then(32)limt⟶∞St+It+Rtt=0,a.s.By the nonnegative property ofSt,It,Rt, it has(33)limt⟶∞Stt=0,limt⟶∞Itt=0,limt⟶∞Rtt=0,a.s.In addition,(34)limt⟶∞∫0tσ1αsSsdB1st=0,limt⟶∞∫0tσ2αsIsdB2st=0,limt⟶∞∫0tσ3αsRsdB3st=0,a.s.The proof is similar to that in [28]; we omit it here.Theorem 3.
Letm≔maxi∈Nδˇ/di+δi and c≔maxi∈Nβi/di. If di+γi+μi−mγi≥0 for each i∈N, then(35)limsupt⟶∞lnItt≤∑i=1NπicΛi−di+γi+μi+σ22i2,a.s.Assume further that(36)R0S≔c∑i=1NπiΛi∑i=1Nπidi+γi+μi+σ22i/2<1.Then, the disease will extinct exponentially almost surelyProof.
ApplyingIto^ formula to lnIt yields(37)dlnIt=βαtS1+aαtIq−dαt+γαt+μαt+σ22αt2dt+σ2αtdB2t,≤βαtS−dαt+γαt+μαt+σ22αt2dt+σ2αtdB2t.
Add the first two equations in (1) to get(38)dS+I=Λαt−dαtS+δαtR−dαt+γαt+μαtIdt+σ1αtSdB1t+σ2αtIdB2t≤Λαt−dαtS+δˇR−dαt+γαt+μαtIdt+σ1αtSdB1t+σ2αtIdB2t.
This, along with the third equation in (1), yields(39)dS+I+mR≤Λαt−dαtS+δˇ−mdαt+δαtR−dαt+γαt+μαt−mγαtIdt+σ1αtStdB1t+σ2αtItdB2t+mσ3αtRtdB3t≤Λαt−dαtSdt+σ1αtStdB1t+σ2αtItdB2t+mσ3αtRtdB3t.
Integrating (24) and cdS+I+mR from 0 to t and dividing by t for both sides of formula, one has(40)lnItt≤lnI0t+1t∫0tcΛαsds+1t∫0tβαs−cdαsSsds−1t∫0tdαs+γαs+μαs+σ22αs2ds−cSt+It+mRtt+cS0+I0+mR0t+1t∫0tcσ1αsSsdB1s+1t∫0tcσ2αsIsdB2s+1t∫0tmcσ3αsRsdB3s+1t∫0tσ2αsdB2s.
By virtue of Lemma2 and ergodicity of Markov chain, (35) can be obtained. If R0S<1, then lnIt/t<0, which means that the disease will extinct exponentially. The proof is completed.


## 5. Examples and Simulations

In this section, some examples are constructed and their simulations are presented to verify the abovementioned results.Letαtt≥0 be the Markov chain with state space N=1,2 and its generator is Q1=−mmn−n. Thus, its stationary distribution is π=π1,π2=n/m+n,m/m+n.Example 1.
We will give the values of the abovementioned parameters in the two states in the form of vectors, that is,a=3,1 means a1=3, a2=1 in two states. To verify the result of Theorem 2, we assume that m=1, n=2, q=0.5, a=2,1, Λ=0.4,0.35, d=0.1,0.08, β=0.5,0.4, μ=0.1,0.05, γ=0.5,0.7, δ=0.1,0.3, σ1=0.2,0.4, σ2=0.4,0.1, and σ1=0.3,0.2 such that the threshold in Theorem 2, R1S=1.6761>1. According to the result of the Theorem 2, the model has unique stationary distribution and the disease will persist, see Figure 1.Figure 1
The trajectory of SIRS model 1 with values of parameters in example1 and initial value S0=0.1, I0=0.5, and R0=0.2.Example 2.
To verify the result of Theorem3, let m=1, n=2, q=0.5, a=2,1, Λ=0.2,0.3, d=0.1,0.08, β=0.2,0.3, μ=0.1,0.05, γ=0.5,0.35, δ=0.2,0.4, σ1=0.5,0.7, σ2=1,0.3, and σ1=0.4,1 such that the threshold in Theorem 3, R0S=0.8974<1. According to the result of Theorem 3, the disease will die out, see Figure 2.Figure 2
The trajectory of SIRS model 1 with values of parameters in example2 and initial value S0=0.1, I0=0.5, and R0=0.2.


## 6. Conclusions

In this paper, the dynamic behavior of a class of hybrid SIRS model with nonlinear incidence is studied. Firstly, the condition under which the positive recurrence exists is provided, and then we give the thresholdR0S for disease extinction, that is, when R0S<1, the disease will die out. Finally, some examples are constructed to verify the conclusion.Some other issues deserve further study. For example, the incidence function can be more generalized, orLe´vy noise can be added to make the model more realistic. We can also investigate other properties of the model, such as stability and so on. These are left to study in the future.



---

*Source: 6699024-2021-03-03.xml*