Periodic solution of a stochastic SIRS epidemic model with seasonal variation

ABSTRACT

In this paper, we consider a stochastic SIRS epidemic model with seasonal variation and saturated incidence. Firstly, we obtain the threshold of stochastic system which determines whether the epidemic occurs or not. Secondly, we prove that there is a non-trivial positive periodic solution if $R_0^s>1$.

KEYWORDS:

• SIRS epidemic model
• periodic solution
• extinction and persistence

2010 MSC:

• 47D07
• 60J60
• 60H10
• 92D30

1. Introduction

Understanding the periodic behaviour of epidemic dynamical system is of paramount importance in applications. This is because many childhood diseases such as measles, chickenpox, and mumps, have been found to be endemic and to exhibit regular oscillatory levels of incidence in large populations [1,3,4,10].

In order to predict sustained oscillatory, many authors take the effect of seasonal variation and stochasticity into account, see, for example [2,6]. However, in Ref. [2,6] the authors prove the the existence of positive periodic solution for a stochastic epidemic model only by numerical methods, but not by theoretical methods. Recently there are several literatures, in which the existence of stochastic periodic solution can be showed analytically (e.g. [8,9,12]). In Ref. [12] the authors considered a periodic stochastic SIR epidemic model with pulse vaccination. Under some conditions a unique positive periodic disease-free solution was obtained. But the existence of non-trivial positive periodic solution could not be obtained by Wang et al. [12]. Lin et al. [8] later filled this gap. They provided the sufficient conditions for the existence of non-trivial periodic solution. Further, Liu et al. [9] considered positive periodic solution for a stochastic non-autonomous SIR epidemic model with logistic growth.

In this paper we continue to do some work in this direction. We consider a stochastic SIRS epidemic model with seasonal variation and saturated incidence, which takes the form of (1) $\begin{array}{rl}\mathrm{d}S\left(t\right)& =\left(\mathrm{\Lambda }\right(t)-\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-\mu (t\left)S\right(t)+\delta (t\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_1\left(t\right)S\left(t\right)\mathrm{d}{B}_1\left(t\right),\\ \mathrm{d}I\left(t\right)& =(\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\left)I\right(t\left)\right)\mathrm{d}t+{\sigma }_2\left(t\right)I\left(t\right)\mathrm{d}{B}_2\left(t\right),\\ \mathrm{d}R\left(t\right)& =\left(\gamma \right(t\left)I\right(t)-(\delta \left(t\right)+\mu \left(t\right)\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_3\left(t\right)R\left(t\right)\mathrm{d}{B}_3\left(t\right),\end{array}$(1) where $S\left(t\right)$, $I\left(t\right)$ and $R\left(t\right)$ denote the number of susceptible, infected and removed individuals at time t respectively. Parameter functions Λ, β, ϵ, γ, μ, δ, ${\sigma }_i,i=1,2,3$ are positive, non-constant and continuous functions of period ω on $(0,+\mathrm{\infty })$; $B_i\left(t\right),i=1,2,3$ are independent standard Brownian motions, defined on a complete probability space $(\mathrm{\Omega },\mathcal{F},\{{\mathcal{F}}_t{\}}_{t\ge 0},\mathbb{P})$ with a filtration $\{{\mathcal{F}}_t{\}}_{t\ge 0}$ satisfying the usual conditions (see [11]).

In fact, model (1(1) $\begin{array}{rl}\mathrm{d}S\left(t\right)& =\left(\mathrm{\Lambda }\right(t)-\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-\mu (t\left)S\right(t)+\delta (t\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_1\left(t\right)S\left(t\right)\mathrm{d}{B}_1\left(t\right),\\ \mathrm{d}I\left(t\right)& =(\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\left)I\right(t\left)\right)\mathrm{d}t+{\sigma }_2\left(t\right)I\left(t\right)\mathrm{d}{B}_2\left(t\right),\\ \mathrm{d}R\left(t\right)& =\left(\gamma \right(t\left)I\right(t)-(\delta \left(t\right)+\mu \left(t\right)\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_3\left(t\right)R\left(t\right)\mathrm{d}{B}_3\left(t\right),\end{array}$(1) ) with constant coefficients and $\delta =0$ has been considered by Yang et al. [14]. They presented the sufficient conditions for the ergodicity and extinction of (1(1) $\begin{array}{rl}\mathrm{d}S\left(t\right)& =\left(\mathrm{\Lambda }\right(t)-\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-\mu (t\left)S\right(t)+\delta (t\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_1\left(t\right)S\left(t\right)\mathrm{d}{B}_1\left(t\right),\\ \mathrm{d}I\left(t\right)& =(\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\left)I\right(t\left)\right)\mathrm{d}t+{\sigma }_2\left(t\right)I\left(t\right)\mathrm{d}{B}_2\left(t\right),\\ \mathrm{d}R\left(t\right)& =\left(\gamma \right(t\left)I\right(t)-(\delta \left(t\right)+\mu \left(t\right)\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_3\left(t\right)R\left(t\right)\mathrm{d}{B}_3\left(t\right),\end{array}$(1) ).

This paper is organized as follows. In Section 2, we present some auxiliary results concerning the existence of a periodic Markov process. In Section 3, we obtain the threshold for the epidemic to occur. The existence of non-trivial positive periodic solution is obtained in Section 4.

2. Preliminary

To begin with, we introduce some notations. If $f\left(t\right)$ is an integrable function on $[0,\mathrm{\infty })$, define $⟨f{⟩}_t=\frac{1}{t}{\int }_0^{t}f\left(s\right)\mathrm{d}s,t>0;$ If $f\left(t\right)$ is a bounded function on $[0,\mathrm{\infty })$, define $\check{f}=\underset{t\in [0,\mathrm{\infty })}{sup}f\left(t\right),\hat{f}=\underset{t\in [0,\mathrm{\infty })}{inf}f\left(t\right).$

Next we present some auxiliary results concerning the existence of a periodic Markov process which will be used in the proof of our main result. The author may also refer to [7] for details.

Definition 2.1

A stochastic process $\xi \left(t\right)=\xi (t,\omega )(-\mathrm{\infty }<t<+\mathrm{\infty })$ is said to be periodic with period θ if for every finite sequence of numbers $t_1,t_2,\dots ,t_n$ the joint distribution of random variables $\xi (t_1+h),\dots ,\xi (t_n+h)$ is independent of h, where $h=k\theta (k=\pm 1,\pm 2,\dots )$.

Remark 2.1

It is showed in [7] that a Markov process $x\left(t\right)$ is θ-periodic if and only if its transition probability function is θ-periodic and the function $P_0(t,A)=P\left\{X\right(t)\in A\}$ satisfies the equation $P_0(s,A)={\int }_{R_l}{P}_0(s,\mathrm{d}x)P(s,x,s+\theta ,A)\equiv P_0(s+\theta ,A)$.

Consider the following equation (2) $X\left(t\right)=X\left(t_0\right)+{\int }_{t_0}^{t}b(s,X(s\left)\right)\mathrm{d}s+\sum _{r=1}^k{\int }_{t_0}^t{\sigma }_r(s,X(s\left)\right)\mathrm{d}{B}_r\left(s\right),X\in {\mathbb{R}}^l.$(2)

Lemma 1

Suppose that the coefficient of (2(2) $X\left(t\right)=X\left(t_0\right)+{\int }_{t_0}^{t}b(s,X(s\left)\right)\mathrm{d}s+\sum _{r=1}^k{\int }_{t_0}^t{\sigma }_r(s,X(s\left)\right)\mathrm{d}{B}_r\left(s\right),X\in {\mathbb{R}}^l.$(2) ) are θ-periodic in t and satisfy condition: $\begin{array}{rl}|b(s,x)-b(s,y)|+\sum _{r=1}^k|{\sigma }_r(s,x)-{\sigma }_r(s,y)|& \le B|x-y|,\\ |b(s,x)|+\sum _{r=1}^k|{\sigma }_r(s,x)|& \le B(1+|x|),\end{array}$ in every cylinder $I\times U,$ where B is a constant; and suppose further that there exists a function $V(t,x)\in C^2$ in ${\mathbb{R}}^l$ which is θ-periodic in t, and satisfies the following conditions (3) $\underset{|x|>R}{inf}V(t,x)\to \mathrm{\infty }asR\to \mathrm{\infty }$(3) and (4) $\mathcal{L}V(t,x)\le -1\mathrm{outside}\mathrm{some}\mathrm{compact}\mathrm{set},$(4) where the operator $\mathcal{L}$ is given by $\mathcal{L}=\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\sum _{i=1}^l{b}_i(t,x)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}+\frac{1}{2}\sum _{i,j=1}^l{a}_{ij}(t,x)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j},a_{ij}=\sum _{r=1}^k{\sigma }_r^i(t,x){\sigma }_r^j(t,x).$ Then there exists a solution of (2(2) $X\left(t\right)=X\left(t_0\right)+{\int }_{t_0}^{t}b(s,X(s\left)\right)\mathrm{d}s+\sum _{r=1}^k{\int }_{t_0}^t{\sigma }_r(s,X(s\left)\right)\mathrm{d}{B}_r\left(s\right),X\in {\mathbb{R}}^l.$(2) ) which is a θ-periodic Markov process.

Remark 2.2

According to the proof of Lemma 1, linear growth condition is only used to guarantee the existence and uniqueness of the solution of (2(2) $X\left(t\right)=X\left(t_0\right)+{\int }_{t_0}^{t}b(s,X(s\left)\right)\mathrm{d}s+\sum _{r=1}^k{\int }_{t_0}^t{\sigma }_r(s,X(s\left)\right)\mathrm{d}{B}_r\left(s\right),X\in {\mathbb{R}}^l.$(2) ).

3. Extinction and persistence of the disease

According to the similar arguments in [5], we know that system (1(1) $\begin{array}{rl}\mathrm{d}S\left(t\right)& =\left(\mathrm{\Lambda }\right(t)-\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-\mu (t\left)S\right(t)+\delta (t\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_1\left(t\right)S\left(t\right)\mathrm{d}{B}_1\left(t\right),\\ \mathrm{d}I\left(t\right)& =(\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\left)I\right(t\left)\right)\mathrm{d}t+{\sigma }_2\left(t\right)I\left(t\right)\mathrm{d}{B}_2\left(t\right),\\ \mathrm{d}R\left(t\right)& =\left(\gamma \right(t\left)I\right(t)-(\delta \left(t\right)+\mu \left(t\right)\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_3\left(t\right)R\left(t\right)\mathrm{d}{B}_3\left(t\right),\end{array}$(1) ) has a unique global positive solution for any $\left(S\right(0),I(0),R(0\left)\right)\in {\mathbb{R}}_{+}^3$. In the following result we determine the threshold for the disease to occur.

Theorem 3.1

Assume $\left(S\right(0),I(0),R(0\left)\right)\in {\mathbb{R}}_{+}^3$. When $⟨R_0{⟩}_{\omega }>0$ and ${\int }_0^{\omega }(\mu \left(t\right)-({\sigma }_1^2\left(t\right)\vee {\sigma }_2^2\left(t\right)\vee {\sigma }_3^2\left(t\right)\left)/2\right)\mathrm{d}t>0,$ the disease I will persist in the sense that $\frac{⟨R_0{⟩}_{\omega }}{\check{h}+\check{\alpha }\check{k}}\le \underset{t\to \mathrm{\infty }}{lim inf}⟨I{⟩}_t\le \underset{t\to \mathrm{\infty }}{lim sup}⟨I{⟩}_t\le \frac{⟨R_0{⟩}_{\omega }}{\hat{h}+\hat{\alpha }\hat{k}},$ where $R_0\left(t\right)=\mathrm{\Lambda }\left(t\right)v\left(t\right)-\left(\mu \right(t)+\epsilon (t)+\gamma (t)+{\sigma }_2^2\left(t\right)/2),$ $h\left(t\right):=v\left(t\right)\left(\epsilon \right(t)+\gamma (t\left)\right)+\beta \left(t\right)-\gamma \left(t\right)u\left(t\right),$ $k\left(t\right):=\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right),$ $v\left(t\right)$ and $u\left(t\right)$ are the unique positive ω-periodic solution of the equation $v^{\prime }\left(t\right)=\mu \left(t\right)v\left(t\right)-\beta \left(t\right)$ and $u^{\prime }\left(t\right)=\left(\delta \right(t)+\mu (t\left)\right)u\left(t\right)-\delta \left(t\right)v\left(t\right),$ respectively.

Remark 3.1

In fact, we have that $\begin{array}{rl}v\left(t\right)& =\frac{{\int }_t^{t+\omega }\exp \left\{{\int }_s^t\mu \right(\tau )\mathrm{d}\tau \left\}\beta \right(s)\mathrm{d}s}{1-\exp \{-{\int }_0^{\omega }\mu (\tau )\mathrm{d}\tau \}},t\ge 0,\\ u\left(t\right)& =\frac{{\int }_t^{t+\omega }\exp \left\{{\int }_s^t\right(\mu \left(\tau \right)+\delta \left(\tau \right))\mathrm{d}\tau \left\}\delta \right(s\left)v\right(s)\mathrm{d}s}{1-\exp \{-{\int }_0^{\omega }(\mu \left(\tau \right)+\delta \left(\tau \right))\mathrm{d}\tau \}},t\ge 0.\end{array}$ From the equations which $v\left(t\right)$ and $u\left(t\right)$ satisfy, it follows that $(u-v{)}^{\prime }(t)=(\delta \left(t\right)+\mu \left(t\right)\left)\right(u\left(t\right)-v\left(t\right))+\beta (t)$. This means $u\left(t\right)<v\left(t\right),t\in [0,\omega ]$. Hence, $h\left(t\right)>0$ on $[0,\omega ]$.

Proof.

By using the similar arguments as in Zhao and Jiang [15], we can get that if ${\int }_0^{\omega }(\mu \left(t\right)-({\sigma }_1^2\left(t\right)\vee {\sigma }_2^2\left(t\right)\vee {\sigma }_3^2\left(t\right)\left)/2\right)\mathrm{d}t>0,$ then (5) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^{t}v\left(s\right){\sigma }_1\left(s\right)S\left(s\right)\mathrm{d}{B}_1\left(s\right)=0,\mathrm{a}.\mathrm{s}.\\ & \underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^{t}v\left(s\right){\sigma }_2\left(s\right)I\left(s\right)\mathrm{d}{B}_2\left(s\right)=0,\mathrm{a}.\mathrm{s}.\\ & \underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^{t}u\left(s\right){\sigma }_3\left(s\right)R\left(s\right)\mathrm{d}{B}_3\left(s\right)=0,\mathrm{a}.\mathrm{s}.\\ & \underset{t\to +\mathrm{\infty }}{lim}\frac{S\left(t\right)}{t}=0,\underset{t\to +\mathrm{\infty }}{lim}\frac{I\left(t\right)}{t}=0,\underset{t\to +\mathrm{\infty }}{lim}\frac{R\left(t\right)}{t}=0,\mathrm{a}.\mathrm{s}.\end{array}$(5) Applying Itô formula, we have $\begin{array}{rl}& \mathrm{d}\left[v\right(t\left)\right(S\left(t\right)+I\left(t\right))+u(t\left)R\right(t\left)\right]\\ & =v\left(t\right)\mathrm{d}\left(S\right(t)+I(t\left)\right)+v^{\prime }\left(t\right)\left(S\right(t)+I(t\left)\right)\mathrm{d}t+u\left(t\right)\mathrm{d}R\left(t\right)+u^{\prime }\left(t\right)R\left(t\right)\mathrm{d}t\\ & =\left(v\right(t\left)\mathrm{\Lambda }\right(t)-\beta (t\left)S\right(t)-[v\left(t\right)\left(\epsilon \right(t)+\gamma (t\left)\right)+\beta \left(t\right)-\gamma \left(t\right)u\left(t\right)\left]I\right(t\left)\right)\mathrm{d}t\\ & +v\left(t\right){\sigma }_1\left(t\right)S\left(t\right)\mathrm{d}{B}_1\left(t\right)+v\left(t\right){\sigma }_2\left(t\right)I\left(t\right)\mathrm{d}{B}_2\left(t\right)+u\left(t\right){\sigma }_3\left(t\right)R\left(t\right)\mathrm{d}{B}_3\left(t\right)\\ & =\left(v\right(t\left)\mathrm{\Lambda }\right(t)-\beta (t\left)S\right(t)-h(t\left)I\right(t\left)\right)\mathrm{d}t\\ & +v\left(t\right){\sigma }_1\left(t\right)S\left(t\right)\mathrm{d}{B}_1\left(t\right)+v\left(t\right){\sigma }_2\left(t\right)I\left(t\right)\mathrm{d}{B}_2\left(t\right)+u\left(t\right){\sigma }_3\left(t\right)R\left(t\right)\mathrm{d}{B}_3\left(t\right).\end{array}$ This together with Equation (5(5) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^{t}v\left(s\right){\sigma }_1\left(s\right)S\left(s\right)\mathrm{d}{B}_1\left(s\right)=0,\mathrm{a}.\mathrm{s}.\\ & \underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^{t}v\left(s\right){\sigma }_2\left(s\right)I\left(s\right)\mathrm{d}{B}_2\left(s\right)=0,\mathrm{a}.\mathrm{s}.\\ & \underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^{t}u\left(s\right){\sigma }_3\left(s\right)R\left(s\right)\mathrm{d}{B}_3\left(s\right)=0,\mathrm{a}.\mathrm{s}.\\ & \underset{t\to +\mathrm{\infty }}{lim}\frac{S\left(t\right)}{t}=0,\underset{t\to +\mathrm{\infty }}{lim}\frac{I\left(t\right)}{t}=0,\underset{t\to +\mathrm{\infty }}{lim}\frac{R\left(t\right)}{t}=0,\mathrm{a}.\mathrm{s}.\end{array}$(5) ) implies (6) $\begin{array}{rl}0& =\underset{t\to \mathrm{\infty }}{lim}\frac{v\left(t\right)\left(S\right(t)+I(t\left)\right)+u\left(t\right)R\left(t\right)}{t}\\ & =\underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^t\mathrm{\Lambda }\left(s\right)v\left(s\right)\mathrm{d}s-\underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^t\beta \left(s\right)S\left(s\right)\mathrm{d}s-\underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^{t}h\left(s\right)I\left(s\right)\mathrm{d}s\\ & =⟨\mathrm{\Lambda }v{⟩}_{\omega }-\underset{t\to \mathrm{\infty }}{lim}⟨\beta S{⟩}_t-\underset{t\to \mathrm{\infty }}{lim}⟨hI{⟩}_t.\end{array}$(6) By Itô formula, we have $\mathrm{d}\log I\left(t\right)=(\frac{\beta \left(t\right)S\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-\mu (t)-\epsilon (t)-\gamma (t)-\frac{{\sigma }_2^2\left(t\right)}{2})\mathrm{d}t+{\sigma }_2\left(s\right)\mathrm{d}{B}_2.$ That is, (7) $\begin{array}{rl}\log I\left(t\right)& =\log I\left(0\right)+{\int }_0^t[\frac{\beta \left(s\right)S\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}-\mu (s)-\epsilon (s)-\gamma (s)-\frac{{\sigma }_2^2\left(s\right)}{2}]\mathrm{d}s+{\int }_0^t{\sigma }_2\left(s\right)\mathrm{d}{B}_2\\ & =\log I\left(0\right)+{\int }_0^t\left[\beta \right(s\left)S\right(s)-\mu (s)-\epsilon (s)-\gamma (s)-\frac{{\sigma }_2^2\left(s\right)}{2}]\mathrm{d}s\\ & -{\int }_0^t\frac{\alpha \left(s\right)\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}\mathrm{d}s+{\int }_0^t{\sigma }_2\left(s\right)\mathrm{d}{B}_2\\ & =⟨R_0{⟩}_{\omega }t-{\int }_0^{t}h\left(s\right)I\left(s\right)\mathrm{d}s-{\int }_0^t\frac{\alpha \left(s\right)\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}\mathrm{d}s+\varphi \left(t\right),\end{array}$(7) where $\begin{array}{rl}\varphi \left(t\right)& =\log I\left(0\right)+{\int }_0^t\beta \left(s\right)S\left(s\right)\mathrm{d}s+{\int }_0^{t}h\left(s\right)I\left(s\right)\mathrm{d}s\\ & -⟨R_0{⟩}_{\omega }t-{\int }_0^t\left(\mu \right(s)+\epsilon (s)+\gamma (s)+\frac{{\sigma }_2^2\left(s\right)}{2})\mathrm{d}s+{\int }_0^t{\sigma }_2\left(s\right)\mathrm{d}{B}_2.\end{array}$ Obviously, Equation (6(6) $\begin{array}{rl}0& =\underset{t\to \mathrm{\infty }}{lim}\frac{v\left(t\right)\left(S\right(t)+I(t\left)\right)+u\left(t\right)R\left(t\right)}{t}\\ & =\underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^t\mathrm{\Lambda }\left(s\right)v\left(s\right)\mathrm{d}s-\underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^t\beta \left(s\right)S\left(s\right)\mathrm{d}s-\underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^{t}h\left(s\right)I\left(s\right)\mathrm{d}s\\ & =⟨\mathrm{\Lambda }v{⟩}_{\omega }-\underset{t\to \mathrm{\infty }}{lim}⟨\beta S{⟩}_t-\underset{t\to \mathrm{\infty }}{lim}⟨hI{⟩}_t.\end{array}$(6) ) implies that $\underset{t\to \mathrm{\infty }}{lim}\varphi \left(t\right)/t=0,a.s.$ Let ${\varphi }_1\left(t\right)=\hat{\alpha }{\int }_0^t\left(\mu \right(s)+\epsilon (s)+\gamma (s\left)\right)I\left(s\right)\mathrm{d}s-\hat{\alpha }{\int }_0^t\frac{\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}\mathrm{d}s+\varphi \left(t\right)$ and ${\varphi }_2\left(t\right)=\check{\alpha }{\int }_0^t\left(\mu \right(s)+\epsilon (s)+\gamma (s\left)\right)I\left(s\right)\mathrm{d}s-\check{\alpha }{\int }_0^t\frac{\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}\mathrm{d}s+\varphi \left(t\right).$ In view of Equation (7(7) $\begin{array}{rl}\log I\left(t\right)& =\log I\left(0\right)+{\int }_0^t[\frac{\beta \left(s\right)S\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}-\mu (s)-\epsilon (s)-\gamma (s)-\frac{{\sigma }_2^2\left(s\right)}{2}]\mathrm{d}s+{\int }_0^t{\sigma }_2\left(s\right)\mathrm{d}{B}_2\\ & =\log I\left(0\right)+{\int }_0^t\left[\beta \right(s\left)S\right(s)-\mu (s)-\epsilon (s)-\gamma (s)-\frac{{\sigma }_2^2\left(s\right)}{2}]\mathrm{d}s\\ & -{\int }_0^t\frac{\alpha \left(s\right)\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}\mathrm{d}s+{\int }_0^t{\sigma }_2\left(s\right)\mathrm{d}{B}_2\\ & =⟨R_0{⟩}_{\omega }t-{\int }_0^{t}h\left(s\right)I\left(s\right)\mathrm{d}s-{\int }_0^t\frac{\alpha \left(s\right)\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}\mathrm{d}s+\varphi \left(t\right),\end{array}$(7) ), we have (8) $\begin{array}{rl}\log I\left(t\right)& \le ⟨R_0{⟩}_{\omega }t-\hat{h}{\int }_0^{t}I\left(s\right)\mathrm{d}s-\hat{\alpha }{\int }_0^t\frac{\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}\mathrm{d}s+\varphi \left(t\right)\\ & =⟨R_0{⟩}_{\omega }t-\hat{h}{\int }_0^{t}I\left(s\right)\mathrm{d}s-\hat{\alpha }{\int }_0^t\left(\mu \right(s)+\epsilon (s)+\gamma (s\left)\right)I\left(s\right)\mathrm{d}s+{\varphi }_1\left(t\right)\\ & \le ⟨R_0{⟩}_{\omega }t-\hat{h}{\int }_0^{t}I\left(s\right)\mathrm{d}s-\hat{\alpha }\hat{k}{\int }_0^{t}I\left(s\right)\mathrm{d}s+{\varphi }_1\left(t\right)\end{array}$(8) and (9) $\begin{array}{rl}\log I\left(t\right)& \ge ⟨R_0{⟩}_{\omega }t-\check{h}{\int }_0^{t}I\left(s\right)\mathrm{d}s-\check{\alpha }{\int }_0^t\frac{\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}\mathrm{d}s+\varphi \left(t\right)\\ & =⟨R_0{⟩}_{\omega }t-\check{h}{\int }_0^{t}I\left(s\right)\mathrm{d}s-\check{\alpha }{\int }_0^t\left(\mu \right(s)+\epsilon (s)+\gamma (s\left)\right)I\left(s\right)\mathrm{d}s+{\varphi }_2\left(t\right)\\ & \ge ⟨R_0{⟩}_{\omega }t-\check{h}{\int }_0^{t}I\left(s\right)\mathrm{d}s-\check{\alpha }\check{k}{\int }_0^{t}I\left(s\right)\mathrm{d}s+{\varphi }_2\left(t\right),\end{array}$(9) where $k\left(t\right)=\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)$.

From Equations (1(1) $\begin{array}{rl}\mathrm{d}S\left(t\right)& =\left(\mathrm{\Lambda }\right(t)-\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-\mu (t\left)S\right(t)+\delta (t\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_1\left(t\right)S\left(t\right)\mathrm{d}{B}_1\left(t\right),\\ \mathrm{d}I\left(t\right)& =(\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\left)I\right(t\left)\right)\mathrm{d}t+{\sigma }_2\left(t\right)I\left(t\right)\mathrm{d}{B}_2\left(t\right),\\ \mathrm{d}R\left(t\right)& =\left(\gamma \right(t\left)I\right(t)-(\delta \left(t\right)+\mu \left(t\right)\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_3\left(t\right)R\left(t\right)\mathrm{d}{B}_3\left(t\right),\end{array}$(1) ) and (5(5) $\begin{array}{rl}& \underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^{t}v\left(s\right){\sigma }_1\left(s\right)S\left(s\right)\mathrm{d}{B}_1\left(s\right)=0,\mathrm{a}.\mathrm{s}.\\ & \underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^{t}v\left(s\right){\sigma }_2\left(s\right)I\left(s\right)\mathrm{d}{B}_2\left(s\right)=0,\mathrm{a}.\mathrm{s}.\\ & \underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}{\int }_0^{t}u\left(s\right){\sigma }_3\left(s\right)R\left(s\right)\mathrm{d}{B}_3\left(s\right)=0,\mathrm{a}.\mathrm{s}.\\ & \underset{t\to +\mathrm{\infty }}{lim}\frac{S\left(t\right)}{t}=0,\underset{t\to +\mathrm{\infty }}{lim}\frac{I\left(t\right)}{t}=0,\underset{t\to +\mathrm{\infty }}{lim}\frac{R\left(t\right)}{t}=0,\mathrm{a}.\mathrm{s}.\end{array}$(5) ), the following holds: $\begin{array}{rl}0& =\underset{t\to \mathrm{\infty }}{lim}\frac{I\left(t\right)-I\left(0\right)}{t}\\ & =\underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}\left[{\int }_0^t\right(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\left)I\right(s)\mathrm{d}s-{\int }_0^t\frac{\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}\mathrm{d}s],\end{array}$ which together with $\underset{t\to \mathrm{\infty }}{lim}\varphi \left(t\right)/t=0,\mathrm{a}.\mathrm{s}.$ implies that $\underset{t\to \mathrm{\infty }}{lim}\frac{{\varphi }_1\left(t\right)}{t}=0,\mathrm{a}.\mathrm{s}.\mathrm{and}\underset{t\to \mathrm{\infty }}{lim}\frac{{\varphi }_2\left(t\right)}{t}=0,\mathrm{a}.\mathrm{s}.$ According to Lemma 17 in [13] and Lemma A.2 in [15], it follows from Equations (8(8) $\begin{array}{rl}\log I\left(t\right)& \le ⟨R_0{⟩}_{\omega }t-\hat{h}{\int }_0^{t}I\left(s\right)\mathrm{d}s-\hat{\alpha }{\int }_0^t\frac{\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}\mathrm{d}s+\varphi \left(t\right)\\ & =⟨R_0{⟩}_{\omega }t-\hat{h}{\int }_0^{t}I\left(s\right)\mathrm{d}s-\hat{\alpha }{\int }_0^t\left(\mu \right(s)+\epsilon (s)+\gamma (s\left)\right)I\left(s\right)\mathrm{d}s+{\varphi }_1\left(t\right)\\ & \le ⟨R_0{⟩}_{\omega }t-\hat{h}{\int }_0^{t}I\left(s\right)\mathrm{d}s-\hat{\alpha }\hat{k}{\int }_0^{t}I\left(s\right)\mathrm{d}s+{\varphi }_1\left(t\right)\end{array}$(8) ) and (9(9) $\begin{array}{rl}\log I\left(t\right)& \ge ⟨R_0{⟩}_{\omega }t-\check{h}{\int }_0^{t}I\left(s\right)\mathrm{d}s-\check{\alpha }{\int }_0^t\frac{\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}\mathrm{d}s+\varphi \left(t\right)\\ & =⟨R_0{⟩}_{\omega }t-\check{h}{\int }_0^{t}I\left(s\right)\mathrm{d}s-\check{\alpha }{\int }_0^t\left(\mu \right(s)+\epsilon (s)+\gamma (s\left)\right)I\left(s\right)\mathrm{d}s+{\varphi }_2\left(t\right)\\ & \ge ⟨R_0{⟩}_{\omega }t-\check{h}{\int }_0^{t}I\left(s\right)\mathrm{d}s-\check{\alpha }\check{k}{\int }_0^{t}I\left(s\right)\mathrm{d}s+{\varphi }_2\left(t\right),\end{array}$(9) ) that $\frac{⟨R_0{⟩}_{\omega }}{\check{h}+\check{\alpha }\check{k}}\le \underset{t\to \mathrm{\infty }}{lim inf}\frac{1}{t}{\int }_0^{t}I\left(s\right)\mathrm{d}s\le \underset{t\to \mathrm{\infty }}{lim sup}\frac{1}{t}{\int }_0^{t}I\left(s\right)\mathrm{d}s\le \frac{⟨R_0{⟩}_{\omega }}{\hat{h}+\hat{\alpha }\hat{k}}.$

Theorem 3.2

Assume $\left(S\right(0),I(0),R(0\left)\right)\in {\mathbb{R}}_{+}^3$. The disease will become extinct exponentially almost surely when $⟨R_0{⟩}_{\omega }<0$ and ${\int }_0^{\omega }(\mu \left(t\right)-({\sigma }_1^2\left(t\right)\vee {\sigma }_2^2\left(t\right)\vee {\sigma }_3^2\left(t\right)\left)/2\right)\mathrm{d}t>0$.

Proof.

It follows from Equation (7(7) $\begin{array}{rl}\log I\left(t\right)& =\log I\left(0\right)+{\int }_0^t[\frac{\beta \left(s\right)S\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}-\mu (s)-\epsilon (s)-\gamma (s)-\frac{{\sigma }_2^2\left(s\right)}{2}]\mathrm{d}s+{\int }_0^t{\sigma }_2\left(s\right)\mathrm{d}{B}_2\\ & =\log I\left(0\right)+{\int }_0^t\left[\beta \right(s\left)S\right(s)-\mu (s)-\epsilon (s)-\gamma (s)-\frac{{\sigma }_2^2\left(s\right)}{2}]\mathrm{d}s\\ & -{\int }_0^t\frac{\alpha \left(s\right)\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}\mathrm{d}s+{\int }_0^t{\sigma }_2\left(s\right)\mathrm{d}{B}_2\\ & =⟨R_0{⟩}_{\omega }t-{\int }_0^{t}h\left(s\right)I\left(s\right)\mathrm{d}s-{\int }_0^t\frac{\alpha \left(s\right)\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha \left(s\right)I\left(s\right)}\mathrm{d}s+\varphi \left(t\right),\end{array}$(7) ) that $\log I\left(t\right)\le ⟨R_0{⟩}_{\omega }t+\varphi \left(t\right),\mathrm{a}.\mathrm{s}.$ Noting $\underset{t\to \mathrm{\infty }}{lim}\varphi \left(t\right)/t=0$ a.s., we get $\underset{t\to \mathrm{\infty }}{lim sup}\frac{\log I\left(t\right)}{t}\le ⟨R_0{⟩}_{\omega },\mathrm{a}.\mathrm{s}.,$ which completes the proof.

4. Existence of ω-periodic solution

Theorem 4.1

Assume $\left(S\right(0),I(0),R(0\left)\right)\in {\mathbb{R}}_{+}^3$. If $⟨R_0{⟩}_{\omega }>0,$ then there exists a ω-periodic solution for system (1(1) $\begin{array}{rl}\mathrm{d}S\left(t\right)& =\left(\mathrm{\Lambda }\right(t)-\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-\mu (t\left)S\right(t)+\delta (t\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_1\left(t\right)S\left(t\right)\mathrm{d}{B}_1\left(t\right),\\ \mathrm{d}I\left(t\right)& =(\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\left)I\right(t\left)\right)\mathrm{d}t+{\sigma }_2\left(t\right)I\left(t\right)\mathrm{d}{B}_2\left(t\right),\\ \mathrm{d}R\left(t\right)& =\left(\gamma \right(t\left)I\right(t)-(\delta \left(t\right)+\mu \left(t\right)\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_3\left(t\right)R\left(t\right)\mathrm{d}{B}_3\left(t\right),\end{array}$(1) ).

Proof.

Since for any $\left(S\right(0),I(0),R(0\left)\right)\in {\mathbb{R}}_{+}^3$ system (1(1) $\begin{array}{rl}\mathrm{d}S\left(t\right)& =\left(\mathrm{\Lambda }\right(t)-\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-\mu (t\left)S\right(t)+\delta (t\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_1\left(t\right)S\left(t\right)\mathrm{d}{B}_1\left(t\right),\\ \mathrm{d}I\left(t\right)& =(\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\left)I\right(t\left)\right)\mathrm{d}t+{\sigma }_2\left(t\right)I\left(t\right)\mathrm{d}{B}_2\left(t\right),\\ \mathrm{d}R\left(t\right)& =\left(\gamma \right(t\left)I\right(t)-(\delta \left(t\right)+\mu \left(t\right)\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_3\left(t\right)R\left(t\right)\mathrm{d}{B}_3\left(t\right),\end{array}$(1) ) has a unique global positive solution, we take ${\mathbb{R}}_{+}^3$ as the whole space. It is obvious that the coefficients of system (1(1) $\begin{array}{rl}\mathrm{d}S\left(t\right)& =\left(\mathrm{\Lambda }\right(t)-\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-\mu (t\left)S\right(t)+\delta (t\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_1\left(t\right)S\left(t\right)\mathrm{d}{B}_1\left(t\right),\\ \mathrm{d}I\left(t\right)& =(\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\left)I\right(t\left)\right)\mathrm{d}t+{\sigma }_2\left(t\right)I\left(t\right)\mathrm{d}{B}_2\left(t\right),\\ \mathrm{d}R\left(t\right)& =\left(\gamma \right(t\left)I\right(t)-(\delta \left(t\right)+\mu \left(t\right)\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_3\left(t\right)R\left(t\right)\mathrm{d}{B}_3\left(t\right),\end{array}$(1) ) satisfy the local Lipschitz condition. According to Lemma 1 and Remark 2.2, in order to prove Theorem 4.1 it suffices to find a $C^2$-function $V(t,x)$ and a closed set $U\subset {\mathbb{R}}_{+}^3$ such that Equations (3(3) $\underset{|x|>R}{inf}V(t,x)\to \mathrm{\infty }asR\to \mathrm{\infty }$(3) ) and (4(4) $\mathcal{L}V(t,x)\le -1\mathrm{outside}\mathrm{some}\mathrm{compact}\mathrm{set},$(4) ) hold.

Take $\theta \in (0,1)$ and r>0 such that (10) $\hat{\mu }-\frac{\theta }{2}{\check{\sigma }}_1^2>0,\hat{\mu }+\hat{\epsilon }-\frac{\theta }{2}{\check{\sigma }}_2^2>0,\hat{\mu }-\frac{\theta }{2}{\check{\sigma }}_3^2>0\mathrm{and}\check{f}+\check{h}-r⟨R_0{⟩}_{\omega }\le -2,$(10) where the function $f\left(x\right)$ and $g\left(x\right)$ are given in Equations (11(11) $\begin{array}{rl}f\left(S\right)& =3^{\theta }\check{\mathrm{\Lambda }}{S}^{\theta }-(\hat{\mu }-\frac{\theta }{2}{\check{\sigma }}_1^2)S^{1+\theta }-\frac{\hat{\mathrm{\Lambda }}}{S}+\check{\mu }+\frac{{\check{\sigma }}_1^2}{2}+(\check{\delta }+\check{\mu })+\frac{{\check{\sigma }}_3^2}{2},\end{array}$(11) ) and (12(12) $\begin{array}{rl}g\left(I\right)& =3^{\theta }\check{\mathrm{\Lambda }}{I}^{\theta }-(\hat{\mu }+\hat{\epsilon }-\frac{\theta }{2}{\check{\sigma }}_2^2)I^{1+\theta }+r(-⟨R_0{⟩}_{\omega }+[\check{\beta }+\check{v}(\check{\epsilon }+\check{\gamma })\left]I\right)+\check{\beta }I,\\ h\left(R\right)& =3^{\theta }\check{\mathrm{\Lambda }}{R}^{\theta }-(\hat{\mu }-\frac{\theta }{2}{\check{\sigma }}_3^2)R^{1+\theta }.\end{array}$(12) ), respectively. Define the auxiliary function $V(t,x)$ by $\begin{array}{rl}V(t,x)& =\frac{1}{\theta +1}(S+I+R{)}^{\theta +1}-r(\log I+v\left(t\right)(S+I)+u\left(t\right)R+w\left(t\right))\\ & -\log S-\log R,x=(S,I,R)\in {\mathbb{R}}_{+}^3,\end{array}$ where $v\left(t\right)$ and $u\left(t\right)$ are given in Theorem 3.1, $w\left(t\right)$ is a function defined on $[0,+\mathrm{\infty })$ satisfying $\dot{w}\left(t\right)=⟨R_0{⟩}_{\omega }-R_0\left(t\right)$ and $w\left(0\right)=0$. It is clear that $w\left(t\right)$ is a ω-periodic function on $[0,\mathrm{\infty })$. Hence $V(t,x)$ is ω-periodic in t and satisfies Equation (3(3) $\underset{|x|>R}{inf}V(t,x)\to \mathrm{\infty }asR\to \mathrm{\infty }$(3) ).

Next we will find a closed set $U\subset {\mathbb{R}}_{+}^3$ such that $\mathcal{L}V(t,x)\le -1,x\in {\mathbb{R}}_{+}^3-U.$

Denote $\begin{array}{rl}& V_1=\frac{1}{\theta +1}(S+I+R{)}^{\theta +1},\\ & V_2=-\log I-v\left(t\right)(S+I)-u\left(t\right)R-w\left(t\right),\\ & V_3=-\log S-\log R,\end{array}$ Then $\mathcal{L}V=\mathcal{L}{V}_1+r\mathcal{L}{V}_2+\mathcal{L}{V}_3.$

In the following, for simplicity we always use Λ to denote the function $\mathrm{\Lambda }\left(t\right)$, the other parameter functions are the same. Direct calculation implies that $\begin{array}{rl}\mathcal{L}{V}_1& =(S+I+R{)}^{\theta }(\mathrm{\Lambda }-\mu S-(\mu +\epsilon )I-\mu R)\\ & +\frac{\theta }{2}(S+I+R{)}^{\theta -1}({\sigma }_1^2{S}^2+{\sigma }_2^2{I}^2+{\sigma }_3^2{R}^2)\\ & \le 3^{\theta }\mathrm{\Lambda }(S^{\theta }+I^{\theta }+R^{\theta })-\mu S^{1+\theta }-(\mu +\epsilon )I^{1+\theta }-\mu R^{1+\theta }\\ & +\frac{\theta }{2}{\sigma }_1^2{S}^{1+\theta }+\frac{\theta }{2}{\sigma }_2^2{I}^{1+\theta }+\frac{\theta }{2}{\sigma }_3^2{R}^{1+\theta }\\ & =3^{\theta }\mathrm{\Lambda }(S^{\theta }+I^{\theta }+R^{\theta })-(\mu -\frac{\theta }{2}{\sigma }_1^2)S^{1+\theta }\\ & -(\mu +\epsilon -\frac{\theta }{2}{\sigma }_2^2)I^{1+\theta }-(\mu -\frac{\theta }{2}{\sigma }_3^2)R^{1+\theta },\\ \mathcal{L}{V}_2& =-\frac{\beta S}{1+\alpha I}+(\mu +\epsilon +\gamma )+\frac{{\sigma }_2^2}{2}-v(\mathrm{\Lambda }-\mu S-(\mu +\epsilon +\gamma )I+\delta R)\\ & -(\mu v-\beta )(S+I)-\left(\right(\delta +\mu )u-\delta v)R-u(\gamma I-(\delta +\mu \left)R\right)-\dot{w}\left(t\right)\\ & =-⟨R_0{⟩}_{\omega }+[\beta +v(\epsilon +\gamma )-\gamma u]I+\frac{\alpha \beta SI}{1+\alpha I},\end{array}$ and $\begin{array}{rl}\mathcal{L}{V}_3& =-\frac{\mathrm{\Lambda }}{S}+\frac{\beta I}{1+\alpha I}+\mu +\frac{{\sigma }_1^2}{2}-\frac{\delta R}{S}-\gamma \frac{I}{R}+(\delta +\mu )+\frac{{\sigma }_3^2}{2}\\ & \le -\frac{\mathrm{\Lambda }}{S}+\beta I+\mu +\frac{{\sigma }_1^2}{2}-\gamma \frac{I}{R}+(\delta +\mu )+\frac{{\sigma }_3^2}{2}.\end{array}$ Hence, $\mathcal{L}V\le f\left(S\right)+g\left(I\right)+h\left(R\right)-\gamma \frac{I}{R}+\frac{r\alpha \beta SI}{1+\alpha I},$ where (11) $\begin{array}{rl}f\left(S\right)& =3^{\theta }\check{\mathrm{\Lambda }}{S}^{\theta }-(\hat{\mu }-\frac{\theta }{2}{\check{\sigma }}_1^2)S^{1+\theta }-\frac{\hat{\mathrm{\Lambda }}}{S}+\check{\mu }+\frac{{\check{\sigma }}_1^2}{2}+(\check{\delta }+\check{\mu })+\frac{{\check{\sigma }}_3^2}{2},\end{array}$(11) (12) $\begin{array}{rl}g\left(I\right)& =3^{\theta }\check{\mathrm{\Lambda }}{I}^{\theta }-(\hat{\mu }+\hat{\epsilon }-\frac{\theta }{2}{\check{\sigma }}_2^2)I^{1+\theta }+r(-⟨R_0{⟩}_{\omega }+[\check{\beta }+\check{v}(\check{\epsilon }+\check{\gamma })\left]I\right)+\check{\beta }I,\\ h\left(R\right)& =3^{\theta }\check{\mathrm{\Lambda }}{R}^{\theta }-(\hat{\mu }-\frac{\theta }{2}{\check{\sigma }}_3^2)R^{1+\theta }.\end{array}$(12) In view of Equation (10(10) $\hat{\mu }-\frac{\theta }{2}{\check{\sigma }}_1^2>0,\hat{\mu }+\hat{\epsilon }-\frac{\theta }{2}{\check{\sigma }}_2^2>0,\hat{\mu }-\frac{\theta }{2}{\check{\sigma }}_3^2>0\mathrm{and}\check{f}+\check{h}-r⟨R_0{⟩}_{\omega }\le -2,$(10) ), we can obtain $f\left(S\right)+r\check{\beta }S+\check{g}+\check{h}\to -\mathrm{\infty },\mathrm{as}S\to 0\mathrm{or}S\to +\mathrm{\infty }.$ Take small $\kappa \in (0,1)$ such that (13) $\mathcal{L}V\le f\left(S\right)+r\check{\beta }S+\check{g}+\check{h}\le -1,\mathrm{on}{\mathbb{R}}_{+}^3-U_1,$(13) where $U_1=\left\{\right(S,I,R)\in {\mathbb{R}}_{+}^3:S\in [\kappa ,1/\kappa \left]\right\}$.

For $(S,I,R)\in U_1$, in view of Equation (10(10) $\hat{\mu }-\frac{\theta }{2}{\check{\sigma }}_1^2>0,\hat{\mu }+\hat{\epsilon }-\frac{\theta }{2}{\check{\sigma }}_2^2>0,\hat{\mu }-\frac{\theta }{2}{\check{\sigma }}_3^2>0\mathrm{and}\check{f}+\check{h}-r⟨R_0{⟩}_{\omega }\le -2,$(10) ) we have $\mathcal{L}V\le \check{f}+\check{h}+g\left(I\right)+\frac{r\check{\beta }}{\kappa }\to -\mathrm{\infty },\mathrm{as}I\to +\mathrm{\infty },$ and $\mathcal{L}V\le \check{f}+g\left(I\right)+\frac{r\check{\alpha }\check{\beta }I}{\kappa }+\check{h}\to \check{f}+\check{h}-r⟨R_0{⟩}_{\omega }\le -2,\mathrm{as}I\to 0.$ Taking $\rho \in (0,1)$ small enough, it follows that (14) $\mathcal{L}V\le -1,\mathrm{on}{U}_1-U_2,$(14) where $U_2=\left\{\right(S,I,R)\in {\mathbb{R}}_{+}^3:S\in [\kappa ,1/\kappa ],I\in [\rho ,1/\rho \left]\right\}$.

For $(S,I,R)\in U_2$, we obtain $\mathcal{L}V\le \check{f}+\check{g}+h\left(R\right)+\frac{r\check{\beta }}{\kappa }-\frac{\hat{\gamma }\rho }{R}\to -\mathrm{\infty },\mathrm{as}R\to 0\mathrm{or}R\to +\mathrm{\infty },$ Choosing $\tau \in (0,1)$ small enough, it is obvious that (15) $\mathcal{L}V<-1,(S,I,R)\in U_2-U_3$(15) where $U_3$ is defined by $U_3:=[\kappa ,1/\kappa ]\times [\rho ,1/\rho ]\times [\tau ,1/\tau ].$

Noting $U_3\subset U_2\subset U_1$, we obtain $({\mathbb{R}}_{+}^3-U_1)\cup (U_1-U_2)\cup (U_2-U_3)={\mathbb{R}}_{+}^3-U_3$. Combining Equations (13(13) $\mathcal{L}V\le f\left(S\right)+r\check{\beta }S+\check{g}+\check{h}\le -1,\mathrm{on}{\mathbb{R}}_{+}^3-U_1,$(13) ) – (15(15) $\mathcal{L}V<-1,(S,I,R)\in U_2-U_3$(15) ), it follows $\mathcal{L}V<-1,(S,I,R)\in {\mathbb{R}}_{+}^3-U_3.$ The proof is complete.

5. Conclusion

In this paper, we consider a stochastic SIRS epidemic model with seasonal variation. Denote $R_0^s:=⟨\mathrm{\Lambda }v{⟩}_{\omega }/⟨\mu +\epsilon +\gamma +{\sigma }_2^2/2{⟩}_{\omega }.$ According to Theorems 3.1 and 3.2, imposing some restrictions on the intensities of white noises, the disease dies out if $R_0^s<1$; whereas if $R_0^s>1$ the disease will persist. According to Theorem 4.1 we show that model (1(1) $\begin{array}{rl}\mathrm{d}S\left(t\right)& =\left(\mathrm{\Lambda }\right(t)-\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-\mu (t\left)S\right(t)+\delta (t\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_1\left(t\right)S\left(t\right)\mathrm{d}{B}_1\left(t\right),\\ \mathrm{d}I\left(t\right)& =(\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha \left(t\right)I\left(t\right)}-(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\left)I\right(t\left)\right)\mathrm{d}t+{\sigma }_2\left(t\right)I\left(t\right)\mathrm{d}{B}_2\left(t\right),\\ \mathrm{d}R\left(t\right)& =\left(\gamma \right(t\left)I\right(t)-(\delta \left(t\right)+\mu \left(t\right)\left)R\right(t\left)\right)\mathrm{d}t+{\sigma }_3\left(t\right)R\left(t\right)\mathrm{d}{B}_3\left(t\right),\end{array}$(1) ) has at least one ω-periodic solution when $R_0^s>1$.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Yuguo Lin http://orcid.org/0000-0003-2754-6698

Additional information

Funding

The work was supported by the Education Department of Jilin Province [Grant No: [2016]47] and Science and Technology Department of Jilin Province [Grant No: 20160101264JC].