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Dynamics analysis of stochastic epidemic models with standard incidence

Advances in Difference Equations20192019:22

https://doi.org/10.1186/s13662-019-1972-0

  • Received: 9 July 2018
  • Accepted: 15 January 2019
  • Published:

Abstract

In this paper, two stochastic SIRS epidemic models with standard incidence were proposed and investigated. For the non-autonomous periodic model, the sufficient criteria for extinction of the disease are obtained firstly. Then we show that the stochastic system has at least one nontrivial positive T-periodic solution under some conditions. For the model that are both disturbed by the white noise and telephone noise, we construct a suitable Lyapunov functions to verify the existence of a unique ergodic stationary distribution. Meanwhile, the sufficient condition for the extinction of the disease is also established. Finally, examples are introduced to illustrate the theoretical analysis.

Keywords

  • Stochastic SIRS epidemic model
  • Standard incidence
  • Extinction
  • Periodic solution
  • Stationary distribution and ergodicity

MSC

  • 60H10
  • 65C30
  • 91B70

1 Introduction

In the natural world, various systems are inevitably affected by the random factors [13]. Stochastic differential equations are important tools for studying random phenomena (see e.g. [411]). May [12] has pointed out that because of the continuous interference of the environment, the biological parameters in the ecosystem such as birth rates, intraspecific competition coefficients, death rates and other parameters may have some degree of random fluctuation. Parameter perturbation induced by white noise is an important and common form to describe the effect of stochasticity [1317]. In recent years, many famous susceptible–infected–recovered–susceptible stochastic models have been formulated (see e.g. [1824]). In [25], a stochastic SIRS model with environment noise was proposed as follows:
$$ \textstyle\begin{cases} dS(t)= [A-dS(t)- \frac{\beta S(t)I(t)}{N(t)}+\delta R(t) ]\,dt+\sigma _{1}S(t)\,dB_{1}(t), \\ dI(t)= [\frac{\beta S(t)I(t)}{N(t)}-(\gamma+d+\alpha)I(t) ]\,dt+\sigma_{2}I(t)\,dB_{2}(t), \\ dR(t)=[\gamma I(t)-(\delta+d)R(t)]\,dt+\sigma_{3}R(t)\,dB_{3}(t), \end{cases} $$
(1)
where \(B_{i}(t)\), denoting the white noise, are independent standard Brownian motions, \(\sigma_{i}^{2}\) are the intensities of the white noise, \(i=1,2,3\). For the meaning of detailed parameters, please see [25]. By using stochastic Lyapunov functions, the authors proved that the system (1) has a unique global positive solution for any initial value \((S(0),I(0),R(0))\in\mathbb{R}_{+}^{3}\) and also has an ergodic stationary distribution under some conditions.
In fact, owing to the season alternation, the life cycle of the individual, the mating habits, the food supply and so on, the birth rate, incidence rate of disease and other parameters in system (1) will exhibit more or less periodicity rather than being constant [2630]. So it is more realistic to discuss the model (1) with periodic coefficients. In 2003, Greenhalgh and Moneim [31] studied a SIRS epidemic model with general seasonal variation in the contact rate. In 2009, Martcheva [32] studied a non-autonomous multi-strain SIS epidemic model with periodic coefficients. In 2015, Lin et al. [33] proposed a stochastic SIR epidemic model with seasonal variation and analyzed the existence of a nontrivial positive periodic solution. Motivated by the above work, we shall investigate the stochastic non-autonomous SIRS epidemic model which takes the form as follows:
$$ \textstyle\begin{cases} dS(t)= [A(t)-d(t)S(t)- \frac{\beta(t) S(t)I(t)}{N(t)}+\delta(t) R(t) ]\,dt+\sigma _{1}(t)S(t)\,dB_{1}(t), \\ dI(t)= [\frac{\beta(t) S(t)I(t)}{N(t)}-(\gamma(t)+d(t)+\alpha (t))I(t) ]\,dt+\sigma_{2}(t)I(t)\,dB_{2}(t), \\ dR(t)=[\gamma(t) I(t)-(\delta(t)+d(t))R(t)]\,dt+\sigma_{3}(t)R(t)\,dB_{3}(t), \end{cases} $$
(2)
where \(A(t)\), \(d(t)\), \(\beta(t)\), \(\delta(t)\), \(\gamma(t)\), \(\alpha(t)\) and \(\sigma_{i}(t)\) stand for the continuous positive periodic functions of period T, \(i=1,2,3\). In this paper, we intend to prove the existence of nontrivial positive T-periodic solutions under sufficient conditions of system (2).
In the real world, besides white noise, the system may be disturbed by many other noises. For example, telephone noise often causes the system to switch from one state to another [34, 35]. Recently, a large number of researchers have widely concerned the stochastic models with regime switching (see e.g. [3640]). In 1978, Slatkin [41] developed and analyzed a population model in a markovian environment. In 1999, Mao [42] investigated the stability of stochastic differential equations with markovian switching. In 2016, Zhao et al. [43] have studied a stochastic phytoplankton allelopathy model under regime switching. The telephone noise is usually described by Markov chains. Let \((r(t))_{t\geq0}\) be a continuous-time Markov chain taking values in a finite state space \(\mathbb{S}=\{1,2,\ldots,m\}\). Coupling the Markov chain \(r(t)\) into model (1), we get
$$ \textstyle\begin{cases} dS(t)= [A(r(t))-d(r(t))S(t)- \frac{\beta(r(t)) S(t)I(t)}{N(t)}+\delta(r(t)) R(t) ]\,dt \\ \hphantom{dS(t)={}}{} +\sigma_{1}(r(t))S(t)\,dB_{1}(t), \\ dI(t)= [\frac{\beta(r(t)) S(t)I(t)}{N(t)}-(\gamma (r(t))+d(r(t))+\alpha(r(t)))I(t) ]\,dt \\ \hphantom{dI(t)={}}{}+\sigma_{2}(r(t))I(t)\,dB_{2}(t), \\ dR(t)=[\gamma(r(t)) I(t)-(\delta(r(t))+d(r(t)))R(t)]\,dt+\sigma _{3}(r(t))R(t)\,dB_{3}(t). \end{cases} $$
(3)
When the model is affected by severe stochastic interference such as rainfall or nutrition, etc., the parameter switch one state \(r(t)=i\) into another state \(r(t)=j\) and it will switch into the next regime until the next major environmental change. For any \(k\in\mathbb{S}\), \(A(k)\), \(d(k)\), \(\beta(k)\), \(\delta(k)\), \(\gamma(k)\), \(\alpha(k)\) and \(\sigma_{i}(k)\) (\(i=1,2,3\)) are positive constants. Another goal of this paper is to prove the existence of a unique ergodic stationary distribution of the positive solution to the system (3).

This paper is arranged as follows. In Sect. 2, we give some basic knowledge which are used in this paper. In Sect. 3, for the system (2), the criteria for extinction of the disease are obtained and then we show that the system has at least one nontrivial positive T-periodic solution under some conditions. In Sect. 4, for the system (3), we proposed a sufficient condition of the disease extinction. Meanwhile, the existence of a unique ergodic stationary distribution is proved. Finally, we conclude the main result briefly and make some numerical simulations in Sect. 5.

2 Preliminaries

In this paper, let \((\varOmega,\mathcal{F}, \{\mathcal{F}\}_{t\geq0}, \mathbb{P})\) be a complete probability space and let \(r(t)\), \(t\geq0\) be a right-continuous Markov chain on Ω taking values in the finite state space \(\mathbb{S}=\{1,2,\ldots,m\}\). For each vector \(g=(g(1),\ldots, g(m))\), set \(\hat{g}=\min_{k\in\mathbb{S}}\{g(k)\}\) and \(\check{g}=\max_{k\in\mathbb{S}}\{g(k)\}\). Supposed that the generator \(\varGamma=(q_{ij})_{m\times m}\) of the Markov chain is given by
$$ \mathbb{P}\bigl(r(t+\Delta t)=j|r(t)=i\bigr)= \textstyle\begin{cases} {q_{ij}\Delta t+o(\Delta t)}, &\mbox{if }i\neq j, \\ {1+q_{ij}\Delta t+o(\Delta t)}, &\mbox{if }i=j, \end{cases} $$
where \(\Delta t>0\), \(q_{ij}>0\), \(i\neq j\) is the transition rate from state i to j while \(\sum_{j=1}^{m}q_{ij}=0\). Suppose further that the Markov chain \(r(t)\) is irreducible and has a unique stationary distribution \(\pi=(\pi_{1},\pi_{2},\ldots,\pi_{m})\), which is the solution of the system of linear equations \(\pi\varGamma=0\) subject to \(\sum_{h=1}^{m}\pi_{h}=1\) and \(\pi_{h}>0\) for all \(h\in\mathbb{S}\). For any vector \(\varpi=(\varpi(1),\varpi(2),\ldots,\varpi(m))^{T}\), we have
$$\lim_{t\rightarrow\infty}\frac{1}{t} \int_{0}^{t}\varpi\bigl(r(s)\bigr)\,ds=\sum _{k\in\mathbb{S}}\pi_{k}\varpi(k). $$
Consider the following equation:
$$ dx(t)=f\bigl(t,x(t)\bigr)\, dt+g\bigl(t,x(t)\bigr)\, dB(t),\quad x \in \mathbb{R}^{n}, $$
(4)
where functions f and g are T-periodic in t.

Lemma 2.1

([37])

Assume that system (4) has a unique global solution. If there has a function \(V(t,x)\in C^{2}\) which is T-periodic in t such that
$$ (\mathrm{i})\quad \inf_{|x|>M}V(t,x)\rightarrow \infty\quad \textit{as }M\rightarrow\infty $$
(5)
and
$$ (\mathrm{ii})\quad LV(t,x)\leq-1 \quad \textit{outside some compact set}, $$
(6)
where we define the operator L by
$$LV(t,x)=V_{t}(t,x)+V_{x}(t,x)f(t,x)+\frac{1}{2} \operatorname{trace}\bigl(g^{T}(t,x)V_{xx}(t,x)g(t,x)\bigr). $$
Then for system (4) there exists a T-periodic solution.

Lemma 2.2

The following differential equations:
$$ \textstyle\begin{cases} {m_{1}'(t)=d(t)m_{1}(t)}, \\ {m_{2}'(t)=(\delta(t)+d(t))m_{2}(t)-\delta(t)m_{1}(t)}, \end{cases} $$
(7)
have a unique positive T-periodic solution \((m_{1}(t),m_{2}(t))^{T}\), where \(d(t)\), \(\delta(t)\) are continuous, positive and non-constant functions of period T.

The proof is similar to Lemma 3.1 in Liu et al. [28], here we omit it.

Now we are in the position to give some results of the stationary distribution for stochastic system under regime switching. Let \((X(t),r(t))\) be the diffusion process defined by the equation as follows:
$$ \textstyle\begin{cases} d X(t)=b(X(t),r(t))\,dt+\tau(X(t),r(t))\,dB(t), \\ X(0)=x_{0},\qquad r(0)=r_{0}, \end{cases} $$
(8)
where \(B(\cdot)\) denotes the p-dimensional Brownian motion and \(r(\cdot)\) is the right-continuous Markov chain in the above discussion, and \(b(\cdot, \cdot): \mathbb{R}^{n}\times\mathbb {S}\rightarrow\mathbb{R}^{n}\), \(\tau(\cdot,\cdot):\mathbb{R}^{n}\times \mathbb{S}\rightarrow\mathbb{R}^{n\times p}\), satisfying \(\tau(x,k)\tau ^{T}(x,k)=(d_{ij}(x,k))_{n\times n}\triangleq D(x,k)\). For any \(k\in \mathbb{S}\), let \(V(\cdot,k)\) be any twice continuously differentiable function, the operator \(\mathcal{L}\) is defined
$$\mathcal{L}V(x,k)=\sum_{i=1}^{n}b_{i}(x,k) \frac{\partial V(x,k)}{\partial x_{i}}+\frac{1}{2}\sum_{i,j=1}^{n}d_{ij}(x,k) \frac{\partial ^{2}V(x,k)}{\partial x_{i}\, \partial x_{j}}+\sum_{l=1}^{m}q_{kl}V(x,l). $$

Lemma 2.3

([37])

Assume that system (8) satisfies the conditions as follows:
  1. (i)

    \(q_{ij}>0\) for each \(i\neq j\);

     
  2. (ii)
    for any \(k\in\mathbb{S}\), \(D(x,k)=(d_{ij}(x,k))_{n\times n}\) is symmetric and obey
    $$ \kappa_{0}|\xi|^{2}\leq\bigl\langle D(x,k)\xi,\xi\bigr\rangle \leq\kappa_{0}^{-1}|\xi |^{2} \quad \textit{for all }\xi\in\mathbb{R}^{n}, $$
    with some constant \(\kappa_{0}\in(0,1]\) for any \(x\in\mathbb{R}^{n}\);
     
  3. (iii)
    there exists a nonempty open set \(\mathcal{D}\) with compact closure, and for any \(k\in\mathbb{S}\), there is a nonnegative function \(V(\cdot,k):\mathcal{D}^{C}\rightarrow\mathbb{R}\) such that
    $$\mathcal{L}V(x,k)\leq-1\quad \textit{for any }(x,k)\in\mathcal{D}^{C} \times\mathbb{S}. $$
     
Then \((X(t),r(t))\) of system (8) is positive recurrent and ergodic. Moreover, the system has a unique stationary distribution \(\mu (\cdot,\cdot)\) such that, for any Borel measurable function \(f(\cdot ,\cdot):\mathbb{R}^{n}\times\mathbb{S}\rightarrow\mathbb{R}\) satisfying
$$\sum_{k=1}^{m} \int_{\mathbb{R}^{n}} \bigl\vert f(x,k) \bigr\vert \mu(dx,k)< +\infty, $$
we have
$$\mathbb{P}\Biggl(\lim_{t\rightarrow+\infty}\frac{1}{t} \int _{0}^{t}f\bigl(X(s),r(s)\bigr)\,ds=\sum _{k=1}^{m} \int_{\mathbb{R}^{n}}f(x,k)\mu(dx,k)\Biggr)=1. $$

3 Extinction of the disease and the periodic solution for system (2)

For the non-autonomous stochastic system (2), we investigate the extinction criteria of the disease, firstly. Define \(R_{1}(t)=\beta(t)-(\gamma(t)+d(t)+\alpha(t)+\frac{\sigma_{2}^{2}(t)}{2})\) and \(\langle f\rangle_{T}=\frac{1}{T}\int_{0}^{T}f(s)\,ds\), where f is an integral function on \([0,+\infty)\). Then we have the following conclusion.

Theorem 3.1

The disease \(I(t)\) will go to extinction exponentially almost surely when \(\langle R_{1}(t)\rangle_{T}<0\).

Proof

Applying the generalized Itô formula to model (2) yields
$$\begin{aligned} d \ln I =& \biggl[\frac{\beta(t)S}{S+I+R}- \biggl(\gamma(t)+d(t)+\alpha (t)+ \frac{\sigma_{2}^{2}(t)}{2} \biggr) \biggr]\,dt+\sigma_{2}(t) \,dB_{2}(t) \\ \leq& \biggl[\beta(t)- \biggl(\gamma(t)+d(t)+\alpha(t)+\frac{\sigma _{2}^{2}(t)}{2} \biggr) \biggr]\,dt+\sigma_{2}(t)\,dB_{2}(t). \end{aligned}$$
Let \(M(t):=\int_{0}^{t}\sigma_{2}(t)\,dB_{2}(t)\), based on the strong law of large numbers for martingales (see [44]), then \(\lim_{t\rightarrow\infty}\frac{M(t)}{t}=0\) a.s. Thus,
$$\begin{aligned} \limsup_{t\rightarrow\infty}\frac{\ln I}{t} \leq&\limsup _{t\rightarrow \infty}\frac{1}{t} \int_{0}^{t} \biggl[\beta(s)- \biggl(\gamma(s)+d(s)+ \alpha (s)+\frac{\sigma_{2}^{2}(s)}{2} \biggr) \biggr]\,ds \\ =&\frac{1}{T} \int_{0}^{T} \biggl[\beta(s)- \biggl(\gamma(s)+d(s)+ \alpha (s)+\frac{\sigma_{2}^{2}(s)}{2} \biggr) \biggr]\,ds \\ =&\bigl\langle R_{1}(t)\bigr\rangle _{T}< 0, \end{aligned}$$
therefore
$$\lim_{t\rightarrow\infty}I(t)=0\quad \mbox{a.s.} $$
 □

Next, we consider the existence of nontrivial positive T-periodic solution of system (2). To simplify, we denote \(g^{u}=\sup_{t\in[0,+\infty)}g(t)\), \(g^{l}=\inf_{t\in[0,+\infty)}g(t)\), where g is a bounded function on \([0,+\infty)\).

Define
$$R_{2}(t)=3\sqrt[3]{A(t)\beta(t)d(t)}+\bigl(m_{1}(t)-1 \bigr)A(t)- \biggl(\gamma (t)+2d(t)+\alpha(t)+\frac{\sigma_{1}^{2}(t)+\sigma_{2}^{2}(t)}{2} \biggr), $$
where \(m_{1}(t)\) is the solution of system (7). Then we get the following theorem.

Theorem 3.2

If \(\langle R_{2}(t)\rangle_{T}>0\), then system (2) admits at least one positive T-periodic solution.

Proof

To prove Theorem 3.2, we should construct a \(C^{2}\)-function \(V(t,x)\) which is T-periodic in t and a closed set \(U\in\mathbb {R}_{+}^{3}\) satisfy the conditions in Lemma 2.1.

Take \(0<\theta<\min \{\frac{2d^{l}}{(\sigma_{1}^{2})^{u}\vee(\sigma _{2}^{2})^{u}\vee(\sigma_{3}^{2})^{u}},1 \}\) and \(K>0\) such that
$$\begin{aligned}& \varrho=:d^{l}-\frac{\theta}{2}\bigl(\bigl( \sigma_{1}^{2}\bigr)^{u}\vee\bigl(\sigma _{2}^{2}\bigr)^{u}\vee\bigl(\sigma_{3}^{2} \bigr)^{u}\bigr)>0, \end{aligned}$$
(9)
$$\begin{aligned}& \tau=:-K\bigl\langle R_{2}(t)\bigr\rangle _{T}+\delta^{u}+2d^{u}+ \beta^{u}+H+\frac {(\sigma_{1}^{2})^{u}+(\sigma_{3}^{2})^{u}}{2}\leq-2, \end{aligned}$$
(10)
where
$$H=\sup_{(S,I,R)\in\mathbb{R}_{+}^{3}} \biggl\{ A^{u}(S+I+R)^{\theta}- \frac {\varrho}{2}(S+I+R)^{\theta+1} \biggr\} . $$
Define
$$\begin{aligned} V(S,I,R,t) =&\frac{1}{\theta+1}(S+I+R)^{\theta+1}+K\bigl(-\ln S-\ln I-m_{1}(t) (S+I) \\ &{}-m_{2}(t)R+S+I+R-\omega(t)\bigr)-\ln S-\ln R \\ =:&V_{1}+KV_{2}+V_{3}+V_{4}, \end{aligned}$$
where \(V_{1}=\frac{1}{\theta+1}(S+I+R)^{\theta+1}\), \(V_{2}=-\ln S-\ln I-m_{1}(t)(S+I)-m_{2}(t)R+S+I+R-\omega(t)\), \(V_{3}=-\ln S\), \(V_{4}=-\ln R\), \(m_{1}(t)\), \(m_{2}(t)\) are given in Lemma 2.2, \(\omega(t)\) is a T-periodic function defined on \([0,+\infty)\) satisfying \(\omega'(t)=\langle R_{2}(t)\rangle_{T}-R_{2}(t)\) and \(\omega (0)=0\). Obviously, \(V(S,I,R,t)\) is T-periodic in t and
$$\liminf_{k\rightarrow+\infty,(S,I,R)\in\mathbb{R}_{+}^{3}\setminus U_{k}}V(S,I,R,t)=+\infty, $$
where \(U_{k}=(\frac{1}{k}, k)\times(\frac{1}{k}, k)\times(\frac{1}{k}, k)\). Therefore, condition (i) of Lemma 2.1 is satisfied. Next, we prove that condition (ii) of Lemma 2.1 is true.
Using Itô’s formula, we get
$$\begin{aligned}& LV_{1} =(S+I+R)^{\theta}\bigl(A(t)-d(t)S-\bigl(d(t)+\alpha(t) \bigr)I-d(t)R\bigr) \\& \hphantom{LV_{1} ={}}{}+\frac{\theta}{2}(S+I+R)^{\theta-1}\bigl( \sigma_{1}^{2}(t)S^{2}+\sigma _{2}^{2}(t)I^{2}+ \sigma_{3}^{2}(t)R^{2}\bigr) \\& \hphantom{LV_{1}} \leq A(t) (S+I+R)^{\theta}-d(t) (S+I+R)^{\theta+1} \\& \hphantom{LV_{1} ={}}{} {}+\frac{\theta}{2}\bigl(\sigma_{1}^{2}(t) \vee\sigma_{2}^{2}(t)\vee\sigma _{3}^{2}(t) \bigr) (S+I+R)^{\theta+1} \\& \hphantom{LV_{1}} = A(t) (S+I+R)^{\theta}- \biggl(d(t)-\frac{\theta}{2} \bigl(\sigma_{1}^{2}(t)\vee \sigma_{2}^{2}(t) \vee\sigma_{3}^{2}(t)\bigr) \biggr) (S+I+R)^{\theta+1} \\& \hphantom{LV_{1}} \leq H-\frac{1}{2}\varrho(S+I+R)^{\theta+1}, \\& LV_{2} = - \biggl(\frac{A(t)}{S}-d(t)-\frac{\beta(t)I}{N}+ \frac{\delta (t)R}{S} \biggr)- \biggl(\frac{\beta(t)S}{N}-\bigl(\gamma(t)+d(t)+\alpha (t)\bigr) \biggr) \\& \hphantom{LV_{2} ={}}{}-m_{1}(t) \bigl(A(t)-d(t)S+\delta(t)R-\bigl( \gamma(t)+d(t)+\alpha (t)\bigr)I\bigr)-m_{1}'(t) (S+I) \\& \hphantom{LV_{2} ={}}{} -m_{2}(t) \bigl(\gamma(t)I-\bigl(\delta(t)+d(t) \bigr)R\bigr)-m_{2}'(t)R-\omega'(t)+ \frac {\sigma_{1}^{2}(t)+\sigma_{2}^{2}(t)}{2} \\& \hphantom{LV_{2} ={}}{} +A(t)-d(t) (S+I+R)-\alpha(t)I \\& \hphantom{LV_{2} } \leq -\frac{A(t)}{S}-\frac{\beta (t)S}{S+I+R}-d(t) (S+I+R)-m_{1}(t)A(t)+\gamma(t)+d(t)+\alpha(t) \\& \hphantom{LV_{2} ={}}{} +\frac{\sigma_{1}^{2}(t)+\sigma _{2}^{2}(t)}{2}-\bigl(m_{1}'(t)-m_{1}(t)d(t) \bigr)S-\bigl[m_{2}'(t)+m_{1}(t)\delta (t)- \bigl(\delta(t) \\& \hphantom{LV_{2} ={}}{} +d(t)\bigr)m_{2}(t)\bigr]R-\bigl[m_{1}'(t)-m_{1}(t) \bigl(\gamma(t)+d(t)+\alpha (t)\bigr)+m_{2}(t)\gamma(t)+\alpha(t) \bigr]I \\& \hphantom{LV_{2} ={}}{} -\omega'(t)+A(t)+d(t)+\frac{\beta(t)I}{N} \\& \hphantom{LV_{2} } \leq -3\sqrt[3]{A(t)\beta(t)d(t)}-m_{1}(t)A(t)+A(t)+2d(t)+ \gamma (t)+\alpha(t) \\& \hphantom{LV_{2} ={}}{}+\frac{\sigma_{1}^{2}(t)+\sigma_{2}^{2}(t)}{2}+ \biggl(m_{1}(t) \gamma(t)+m_{1}(t)\alpha(t)-m_{2}(t)\gamma(t)-\alpha(t)+ \frac{\beta (t)}{N} \biggr)I \\& \hphantom{LV_{2} ={}}{}-\omega'(t) \\& \hphantom{LV_{2}}\leq -\bigl\langle R_{2}(t)\bigr\rangle _{T}+ \biggl(m_{1}^{u}\gamma ^{u}+m_{1}^{u} \alpha^{u}-m_{2}^{l}\gamma^{l}- \alpha^{l}+\frac{\beta ^{u}}{N} \biggr)I, \\& LV_{3}=-\frac{A(t)}{S}+d(t)+\frac{\beta(t)I}{N}- \frac{\delta(t)R}{S}+\frac {\sigma_{1}^{2}(t)}{2}\leq-\frac{A^{l}}{S}+d^{u}+ \beta^{u}+\frac{(\sigma _{1}^{2})^{u}}{2}, \end{aligned}$$
and
$$ LV_{4}=-\frac{\gamma(t)I}{R}+\delta(t)+d(t)+\frac{\sigma _{3}^{2}(t)}{2}\leq- \frac{\gamma^{l}I}{R}+\delta^{u}+d^{u}+\frac{(\sigma _{3}^{2})^{u}}{2}. $$
Hence
$$\begin{aligned} LV \leq&H-\frac{1}{2}\varrho(S+I+R)^{\theta+1}+K \biggl( \frac{\beta ^{u}}{N}+m_{1}^{u}\gamma^{u}+m_{1}^{u} \alpha^{u}-m_{2}^{l}\gamma^{l}-\alpha ^{l} \biggr)I \\ &{}-K\bigl\langle R_{2}(t)\bigr\rangle _{T}- \frac{A^{l}}{S}+d^{u}+\beta^{u}+\frac {(\sigma_{1}^{2})^{u}}{2}- \frac{\gamma^{l}I}{R}+\delta^{u}+d^{u}+\frac {(\sigma_{3}^{2})^{u}}{2} \\ \leq&-\frac{\varrho}{2}S^{\theta+1}-\frac{A^{l}}{S}- \frac{\varrho }{2}I^{\theta+1}+K \biggl(\frac{\beta^{u}}{N}+m_{1}^{u} \gamma ^{u}+m_{1}^{u}\alpha^{u}-m_{2}^{l} \gamma^{l}-\alpha^{l} \biggr)I \\ &{}-\frac{\varrho}{2}R^{\theta+1}-\frac{\gamma^{l}I}{R}+\tau. \end{aligned}$$
Define a bounded closed set
$$U_{\varepsilon}= \biggl\{ (S,I,R)\in\mathbb{R}_{+}^{3}: \varepsilon\leq S\leq\frac{1}{\varepsilon} ,\varepsilon^{2}\leq I\leq \frac {1}{\varepsilon^{2}}, \varepsilon^{3}\leq R\leq\frac{1}{\varepsilon ^{3}} \biggr\} , $$
where \(\varepsilon>0\) is small enough. In the set \(\mathbb {R}_{+}^{3}\backslash U_{\varepsilon}\), one can choose ε sufficiently small and satisfying
$$\begin{aligned}& -\frac{A^{l}}{\varepsilon}+\tilde{K}+\tau\leq-1, \end{aligned}$$
(11)
$$\begin{aligned}& \tau+K\bigl(m_{1}^{u}\gamma^{u}+m_{1}^{u} \alpha^{u}-m_{2}^{l}\gamma^{l}-\alpha ^{l}\bigr)\varepsilon^{2}+K\beta^{u}\varepsilon \leq-1, \end{aligned}$$
(12)
$$\begin{aligned}& -\frac{\gamma^{l}}{\varepsilon}+\tilde{K}+\tau\leq-1, \end{aligned}$$
(13)
$$\begin{aligned}& -\frac{\varrho}{2\varepsilon^{\theta+1}}+\tilde{K}+\tau\leq-1, \end{aligned}$$
(14)
$$\begin{aligned}& -\frac{\varrho}{4\varepsilon^{2(\theta+1)}}+\tilde{K}+\tau\leq-1, \end{aligned}$$
(15)
$$\begin{aligned}& -\frac{\varrho}{2\varepsilon^{3(\theta+1)}}+\tilde{K}+\tau\leq-1, \end{aligned}$$
(16)
where is a positive constant which of the following can be found in Eq. (18). For convenience, one can divide \(U_{\varepsilon}^{C}\) into the following six domains:
$$\begin{aligned}& U_{1}=\bigl\{ (S,I,R)\in\mathbb{R}_{+}^{3}, 0< S< \varepsilon\bigr\} ,\qquad U_{2}=\bigl\{ (S,I,R)\in\mathbb{R}_{+}^{3}, 0< I< \varepsilon^{2},S\geq\varepsilon\bigr\} , \\& U_{3}=\bigl\{ (S,I,R)\in\mathbb{R}_{+}^{3}, 0< R< \varepsilon^{3}, I\geq \varepsilon^{2}\bigr\} ,\qquad U_{4}=\biggl\{ (S,I,R)\in\mathbb{R}_{+}^{3}, S> \frac {1}{\varepsilon}\biggr\} , \\& U_{5}=\biggl\{ (S,I,R)\in\mathbb{R}_{+}^{3}, I> \frac{1}{\varepsilon^{2}}\biggr\} ,\qquad U_{6}=\biggl\{ (S,I,R)\in \mathbb{R}_{+}^{3}, R>\frac{1}{\varepsilon^{3}}\biggr\} . \end{aligned}$$
Clearly, \(U_{\varepsilon}^{C}=U_{1}\cup\cdots\cup U_{6}\). Now we show that \(LV(S,I,R,t)\leq-1\) on \(U_{\varepsilon}^{C}\times\mathbb{R}\), which is equivalent to prove it on these six domains.
Case 1. If \((S,I,R,t)\in U_{1}\times\mathbb{R}\), from (11), we get
$$\begin{aligned} LV \leq&-\frac{A^{l}}{S}+K \biggl(\frac{\beta^{u}}{S+I+R}+m_{1}^{u} \gamma ^{u}+m_{1}^{u}\alpha^{u}-m_{2}^{l} \gamma^{l}-\alpha^{l} \biggr)I-\frac {\varrho}{4}I^{\theta+1}+ \tau \\ \leq&-\frac{A^{l}}{\varepsilon}+\tilde{K}+\tau\leq-1, \end{aligned}$$
(17)
where
$$ \tilde{K}=\sup_{(S,I,R)\in\mathbb{R}^{3}_{+}} \biggl\{ K\biggl( \frac{\beta ^{u}}{S+I+R}+m_{1}^{u}\gamma^{u}+m_{1}^{u} \alpha^{u}-m_{2}^{l}\gamma ^{l}- \alpha^{l}\biggr)I-\frac{\varrho}{4}I^{\theta+1} \biggr\} . $$
(18)
Case 2. If \((S,I,R,t)\in U_{2}\times\mathbb{R}\), from (12), one can see that
$$\begin{aligned} LV \leq&\frac{K\beta^{u}I}{S+I+R}+K\bigl(m_{1}^{u} \gamma^{u}+m_{1}^{u}\alpha ^{u}-m_{2}^{l} \gamma^{l}-\alpha^{l}\bigr)I+\tau \\ \leq&K\beta^{u}\varepsilon+K\bigl(m_{1}^{u} \gamma^{u}+m_{1}^{u}\alpha ^{u}-m_{2}^{l} \gamma^{l}-\alpha^{l}\bigr)\varepsilon^{2}+\tau \leq-1. \end{aligned}$$
(19)
Case 3. If \((S,I,R,t)\in U_{3}\times\mathbb{R}\), from (13), one can derive that
$$\begin{aligned} LV \leq&-\frac{\gamma^{l}I}{R}+K \biggl(\frac{\beta ^{u}}{S+I+R}+m_{1}^{u} \gamma^{u}+m_{1}^{u}\alpha^{u}-m_{2}^{l} \gamma ^{l}-\alpha^{l} \biggr)I-\frac{\varrho}{4}I^{\theta+1}+ \tau \\ \leq&-\frac{\gamma^{l}}{\varepsilon}+\tilde{K}+\tau\leq-1. \end{aligned}$$
(20)
Case 4. If \((S,I,R,t)\in U_{4}\times\mathbb{R}\), from (14), we get
$$\begin{aligned} LV \leq&-\frac{\varrho}{2}S^{\theta+1}+K \biggl(\frac{\beta ^{u}}{S+I+R}+m_{1}^{u} \gamma^{u}+m_{1}^{u}\alpha^{u}-m_{2}^{l} \gamma ^{l}-\alpha^{l} \biggr)I-\frac{\varrho}{4}I^{\theta+1}+ \tau \\ \leq&-\frac{\varrho}{2\varepsilon^{\theta+1}}+\tilde {K}+\tau\leq-1. \end{aligned}$$
(21)
Case 5. If \((S,I,R,t)\in U_{5}\times\mathbb{R}\), (15) implies that
$$\begin{aligned} LV \leq&-\frac{\varrho}{4}I^{\theta+1}-\frac{\varrho}{4}I^{\theta +1}+K \biggl(\frac{\beta^{u}}{S+I+R}+m_{1}^{u}\gamma^{u}+m_{1}^{u} \alpha ^{u}-m_{2}^{l}\gamma^{l}- \alpha^{l} \biggr)I+\tau \\ \leq&-\frac{\varrho}{4\varepsilon^{2(\theta+1)}}+\tilde{K}+\tau\leq-1. \end{aligned}$$
(22)
Case 6. If \((S,I,R,t)\in U_{6}\times\mathbb{R}\), from (16), one obtains
$$\begin{aligned} LV \leq&-\frac{\varrho}{2}R^{\theta+1}-\frac{\varrho}{4}I^{\theta +1}+K \biggl(\frac{\beta^{u}}{S+I+R}+m_{1}^{u}\gamma^{u}+m_{1}^{u} \alpha ^{u}-m_{2}^{l}\gamma^{l}- \alpha^{l} \biggr)I+\tau \\ \leq&-\frac{\varrho}{2\varepsilon^{3(\theta+1)}}+\tilde{K}+\tau\leq-1. \end{aligned}$$
(23)
By (17), (19), (20), (21), (22) and (23), one can get
$$LV(S,I,R,t)\leq-1, \quad (S,I,R,t)\in U_{\varepsilon}^{C}\times \mathbb{R}. $$
So, condition (ii) for Lemma 2.1 is true. By Lemma 2.1, Theorem 3.2 is proved. □

4 Extinction of the disease and the ergodic stationary distribution for system (3)

For the system with regime switching, we will explore the extinction of the disease and the existence of an ergodic stationary distribution. Let \((S(t),I(t),R(t),r(t))\) be the solution of system (3) with initial value \((S(0),I(0),R(0),r(0))\in\mathbb {R}_{+}^{3}\times\mathbb{S}\).

Define
$$ R_{1}^{*}=\frac{\sum_{k=1}^{m}\pi_{k}\beta(k)}{\sum_{k=1}^{m}\pi_{k} (\gamma(k)+d(k)+\alpha(k)+\frac{\sigma _{2}^{2}(k)}{2} )}, $$
then we have the following.

Theorem 4.1

If \(R_{1}^{*}<1\), then \(\lim_{t\rightarrow\infty}I(t)=0\) a.s.

Proof

By Itô’s formula, we get
$$\begin{aligned} d \ln I =& \biggl[\frac{\beta(r(t))S}{S+I+R}- \biggl(\gamma \bigl(r(t)\bigr)+d\bigl(r(t) \bigr)+\alpha\bigl(r(t)\bigr)+\frac{\sigma_{2}^{2}(r(t))}{2} \biggr) \biggr]\,dt \\ &{}+\sigma_{2}\bigl(r(t)\bigr)\,dB_{2}(t). \end{aligned}$$
(24)
Integrating both sides of Eq. (24) leads to
$$\begin{aligned} \frac{\ln I(t)-\ln I(0)}{t} \leq&\frac{1}{t} \int_{0}^{t} \biggl[\beta\bigl(r(s)\bigr)- \biggl( \gamma \bigl(r(s)\bigr)+d\bigl(r(s)\bigr)+\alpha\bigl(r(s)\bigr)+ \frac{\sigma_{2}^{2}(r(s))}{2} \biggr) \biggr]\,ds \\ &{}+\frac{1}{t} \int_{0}^{t}\sigma_{2}\bigl(r(s)\bigr) \,dB_{2}(s). \end{aligned}$$
(25)
From the ergodic property of \(r(t)\), one can get
$$\begin{aligned} &\lim_{t\rightarrow\infty}\frac{1}{t} \int_{0}^{t} \biggl[\beta \bigl(r(s)\bigr)- \biggl( \gamma\bigl(r(s)\bigr)+d\bigl(r(s)\bigr)+\alpha\bigl(r(s)\bigr)+\frac{\sigma _{2}^{2}(r(s))}{2} \biggr) \biggr]\,ds \\ &\quad =\sum_{k=1}^{m}\pi_{k} \beta(k)-\sum_{k=1}^{m}\pi_{k} \biggl(\gamma (k)+d(k)+\alpha(k)+\frac{\sigma_{2}^{2}(k)}{2} \biggr). \end{aligned}$$
So, (25) implies that
$$\limsup_{t\rightarrow\infty}\frac{\ln I(t)}{t}\leq\sum _{k=1}^{m}\pi _{k} \biggl(\gamma(k)+d(k)+ \alpha(k)+\frac{\sigma_{2}^{2}(k)}{2} \biggr) \bigl(R_{1}^{*}-1 \bigr)< 0\quad \mbox{a.s.} $$
Hence we have
$$\lim_{t\rightarrow\infty}I(t)=0 \quad \mbox{a.s.} $$
 □

Next, we shall establish sufficient conditions for the existence of an ergodic stationary distribution of system (3).

Let
$$R_{2}^{*}=\frac{\sum_{k=1}^{m}\pi_{k}[c_{1}(k)A(k)+\beta(k)]}{\sum_{k=1}^{m}\pi_{k} (\gamma(k)+d(k)+\alpha(k)+\frac{\sigma _{2}^{2}(k)}{2} )}, $$
where \(c_{1}(k)\) is the solution of the following linear system:
$$ \textstyle\begin{cases} c_{1}(k)d(k)-\sum_{l=1}^{m}q_{kl}c_{1}(l)-\beta(k)=0, \quad k=1,2,\ldots,m, \\ -c_{1}(k)\delta(k)+c_{2}(k)(\delta(k)+d(k))-\sum_{l=1}^{m}q_{kl}c_{2}(l)=0,\quad k=1,2,\ldots,m. \end{cases} $$
(26)
By the literature [34], system (26) has a unique solution
$$\bigl(c_{1}(1),c_{1}(2),\ldots,c_{1}(m),c_{2}(1),c_{2}(2), \ldots,c_{2}(m)\bigr)^{T}\gg0. $$
Then we have

Theorem 4.2

If \(R_{2}^{*}>1\), then system (3) has a unique ergodic stationary distribution.

Proof

To prove Theorem 4.2, we just have to verify that conditions (i), (ii) and (iii) in Lemma 2.3 be satisfied. First, assumption \(q_{ij}>0\) for \(i\neq j\) in Sect. 2 implicates that the condition (i) holds. Second, the diffusion matrix \(D(S,I,R,k)=\operatorname{diag}\{\sigma_{1}^{2}(k)S^{2}, \sigma_{2}^{2}(k)I^{2}, \sigma _{3}^{2}(k)R^{2}\}\) of model (3) is positive definite, which shows that condition (ii) in Lemma 2.3 is satisfied. Next, we will show condition (iii) is satisfied by constructing suitable Lyapunov function. Let us define
$$\begin{aligned} V(S,I,R,k) =&\frac{1}{\xi+1}(S+I+R)^{\xi+1}+M\bigl(-c_{1}(k) (S+I)-c_{2}(k)R-\ln I-\omega(k)\bigr) \\ &{}-\ln S-\ln R, \end{aligned}$$
where \(c_{1}(k)\), \(c_{2}(k)\) are the solution of the system (26), \(\xi\in(0,1)\) and \(M>0\) satisfy \(\rho:=\hat{d}-\frac{\xi }{2}(\check{\sigma}_{1}^{2}\vee\check{\sigma}_{2}^{2}\vee\check{\sigma }_{3}^{2})>0\), and \(E+2\check{d}+\check{\beta}+\check{\delta}+\frac{\check {\sigma}_{1}^{2}+\check{\sigma}_{3}^{2}}{2}-M\varSigma_{k=1}^{m}\pi _{k}(\gamma(k)+d(k)+\alpha(k)+\frac{\sigma_{2}^{2}(k)}{2})(R_{2}^{*}-1)\leq -2\), E and \(\omega(k)\) will be defined later.
Denote
$$\begin{aligned}& V_{1} = \frac{1}{\xi+1}(S+I+R)^{\xi+1}, \\& V_{2} = -c_{1}(k) (S+I)-c_{2}(k)R-\ln I- \omega(k), \\& V_{3} = -\ln S, \\& V_{4} = -\ln R. \end{aligned}$$
Applying the generalized Itô formula, we have
$$ \mathcal{L}V_{1} \leq E-\frac{\rho}{2}(S+I+R)^{\xi+1}, $$
where \(E=\sup_{S+I+R\in\mathbb{R}_{+}}\{\check{A}(S+I+R)^{\xi}-\frac {\rho}{2}(S+I+R)^{\xi+1}\}\). Furthermore,
$$\begin{aligned} \mathcal{L}V_{2} =&-c_{1}(k) \bigl(A(k)-d(k)S+\delta(k)R- \bigl(\gamma(k)+d(k)+\alpha (k)\bigr)I\bigr) \\ &{}-(S+I)\sum_{l=1}^{m}q_{kl}c_{1}(l)-c_{2}(k) \bigl(\gamma(k)I-\bigl(\delta (k)+d(k)\bigr)R\bigr)-R\sum _{l=1}^{m}q_{kl}c_{2}(l) \\ &{}- \biggl(\frac{\beta(k)S}{N}- \biggl(\gamma(k)+d(k)+\alpha(k)+ \frac{\sigma _{2}^{2}(k)}{2} \biggr) \biggr)-\sum_{l=1}^{m}q_{kl} \omega(l) \\ \leq&-c_{1}(k)A(k)-\beta(k)+ \biggl(\gamma(k)+d(k)+\alpha(k)+ \frac{\sigma _{2}^{2}(k)}{2} \biggr)-\sum_{l=1}^{m}q_{kl} \omega(l) \\ &{}+ \Biggl(c_{1}(k)d(k)-\sum_{l=1}^{m}q_{kl}c_{1}(l)- \beta(k) \Biggr)S \\ &{}+ \Biggl(c_{1}(k) \bigl(\gamma(k)+d(k)+\alpha(k) \bigr)-c_{2}(k)\gamma(k)-\sum_{l=1}^{m}q_{kl}c_{1}(l) \Biggr)I \\ &{}+ \Biggl(-c_{1}(k)\delta(k)+c_{2}(k) \bigl( \delta(k)+d(k)\bigr)-\sum_{l=1}^{m}q_{kl}c_{2}(l) \Biggr)R \\ &{}+\beta(k)S+\frac{\beta(k)I}{N}+\frac{\beta(k)R}{N} \\ \leq&-R_{0k}-\sum_{l=1}^{m}q_{kl} \omega(l)+\bigl(c_{1}(k) \bigl(\gamma(k)+\alpha (k)\bigr)+ \beta(k)-c_{2}(k)\gamma(k)\bigr)I \\ &{}+\beta(k)S+\frac{\beta(k)I}{N}+\frac{\beta(k)R}{N}, \end{aligned}$$
(27)
where
$$R_{0k}=c_{1}(k)A(k)+\beta(k)- \biggl(\gamma(k)+d(k)+ \alpha(k)+\frac{\sigma _{2}^{2}(k)}{2} \biggr). $$
Let \(\omega=(\omega(1),\omega(2),\ldots,\omega(m))^{T}\) be the following Poisson system’s solution:
$$\varGamma\omega=\Biggl(\sum_{l=1}^{m} \pi_{k}R_{0k}\Biggr)\vec{1}-\tilde{R}_{0}, $$
where \(\tilde{R}_{0}=(R_{01},R_{02},\ldots R_{0m})^{T}\). This shows that
$$\begin{aligned} -R_{0k}-\sum_{l=1}^{m}q_{kl} \omega(l) =&-\sum_{k=1}^{m}\pi _{k}R_{0k} \\ =&-\sum_{k=1}^{m}\pi_{k} \biggl(\gamma(k)+d(k)+\alpha(k)+\frac{\sigma _{2}^{2}(k)}{2} \biggr) \bigl(R_{2}^{*}-1 \bigr). \end{aligned}$$
Substituting this equality into (27), one has
$$\begin{aligned}& \begin{aligned} \mathcal{L}V_{2}&\leq-\sum _{k=1}^{m}\pi_{k} \biggl(\gamma(k)+d(k)+ \alpha (k)+\frac{\sigma_{2}^{2}(k)}{2} \biggr) \bigl(R_{2}^{*}-1 \bigr)+\bigl(\check{c_{1}}(\check {\gamma}+\check{\alpha})+\check{ \beta} \\ &\quad {}-\hat{c_{2}}\hat{\gamma}\bigr)I+\check{\beta}S+ \frac{\check{\beta }I}{N}+\frac{\check{\beta}R}{N}, \end{aligned} \\& \mathcal{L}V_{3}=-\frac{A(k)}{S}+\frac{\beta(k)I}{N}- \frac{\delta (k)R}{S}+d(k)+\frac{\sigma_{1}^{2}(k)}{2}, \end{aligned}$$
and
$$ \mathcal{L}V_{4}=-\frac{\gamma(k)I}{R}+\delta(k)+d(k)+ \frac{\sigma_{3}^{2}(k)}{2}. $$
Consequently, one can get
$$\begin{aligned} \mathcal{L}V \leq& -\frac{\rho}{2}S^{\xi+1}+M\check{\beta}(S+1)- \frac{\hat{A}}{S}-\frac {\rho}{2}I^{\xi+1} +M\bigl[ \check{c_{1}}(\check{\gamma}+\check{\alpha})+\check{\beta}-\hat {c_{2}}\hat{\gamma}\bigr]I \\ &{}-\frac{\rho}{2}R^{\xi+1}-\frac{\hat{\gamma}I}{R}+E+2\check{d}+\check { \beta}+\check{\delta} +\frac{\check{\sigma_{1}^{2}}+\check{\sigma_{3}^{2}}}{2} \\ &{}-M\sum_{k=1}^{m}\pi_{k} \biggl(\gamma(k)+d(k)+\alpha(k)+\frac{\sigma _{2}^{2}(k)}{2} \biggr) \bigl(R_{2}^{*}-1 \bigr). \end{aligned}$$
Consider the bounded open set \(D= (\frac{1}{\eta}, \eta )\times (\frac{1}{\eta}, \eta )\times (\frac{1}{\eta}, \eta )\subset\mathbb{R}_{+}^{3}\), where η is a positive number. From the discussing above, we derive that, for a sufficiently large η,
$$ \mathcal{L}V(S,I,R,k)\leq-1,\quad \mbox{for all }(S,I,R,k)\in D^{C} \times\mathbb{S}. $$
By virtue of Lemma 2.3, one can see that system (3) has a solution which is a stationary Markov process. The proof is completed. □

5 Conclusions and numerical simulations

In this paper, we proposed a stochastic non-autonomous SIRS epidemic model with periodic coefficients (model (2)), and a stochastic epidemic model perturbed by telegraph noise (model (3)). Then the dynamic behaviors of the two models are studied.

Firstly, for system (2), there are the following properties:
  1. (1)

    If \(\langle R_{1}(t)\rangle_{T}<0\), then the disease will go to extinction almost surely.

     
  2. (2)

    If \(\langle R_{2}(t)\rangle_{T}>0\), then system (2) has at least one positive T-periodic solution.

     
Secondly, system (3) possesses the following properties:
  1. (1)

    If \(R_{1}^{*}<1\), the disease \(I(t)\) will go to extinction exponentially with probability 1.

     
  2. (2)

    If \(R_{2}^{*}>1\), then the solution of system (3) has a unique ergodic stationary distribution.

     

To verify the correctness of the theoretical analysis, we will give some examples with computer simulations.

Example 1

First, we consider system (2) and let
$$\begin{aligned}& A(t) =0.1\sin t+1.1, \qquad d(t)=0.1\sin t+0.2,\qquad \beta(t)=0.1\sin t+0.9, \\& \alpha(t) =0.1\sin t+0.2,\qquad \gamma(t)=0.1\sin t+0.2,\qquad \delta(t)=0.1\sin t+0.2, \\& \sigma_{1}(t) =0.1\sin t+0.01, \qquad \sigma_{2}(t)=0.1 \sin t+1.2,\qquad \sigma _{3}(t)=0.1\sin t+0.01. \end{aligned}$$
Case (a). Simple calculation shows that
$$\bigl\langle R_{1}(t)\bigr\rangle _{T}=-0.4284< 0. $$
From Theorem 3.1, we know that the disease goes to extinction (see Fig. 1).
Figure 1
Figure 1

Sample paths of \((S(t),I(t),R(t))\) with initial conditions \((S(0),I(0),R(0))=(3,2.5,2)\)

Case (b). We only change the intensity of the noise \(\sigma _{2}(t)=0.1\sin t+0.2\). Then direct computation leads to \(\langle R_{2}(t)\rangle_{T}=0.2573>0\). From Theorem 3.2, we can see that system (2) has at least one positive T-periodic solution (see Fig. 2).
Figure 2
Figure 2

Sample paths of \((S(t),I(t),R(t))\) with initial conditions \((S(0),I(0),R(0))=(0.3,0.3,0.3)\)

Example 2

In model (3), if the Markov chain \(r(t)\) take values in \(\mathbb{S}=\{1,2\}\) with the generator
Γ = ( 1 1 1 1 ) .
Then the unique stationary distribution of \(r(t)\) is \((\pi_{1},\pi _{2})= (\frac{1}{2},\frac{1}{2} )\). Choose parameters
$$\begin{aligned}& A(1)=1.4,\qquad \beta(1)=1,\qquad d(1)=0.2,\qquad \gamma(1)=0.3,\qquad \alpha(1)=0.1, \\& \delta(1)=0.2,\qquad A(2)=2, \qquad \beta(2)=0.8,\qquad d(2)=0.15,\qquad \gamma(2)=0.25, \\& \alpha(2)=0.2,\qquad \delta(2)=0.1,\qquad \sigma_{1}(1)=0.05, \qquad \sigma_{2}(1)=1.3, \\& \sigma_{3}(1)=0.05,\qquad \sigma_{1}(2)=0.1,\qquad \sigma_{2}(2)=1.5, \qquad \sigma_{3}(2)=0.2. \end{aligned}$$
Case (a). By direct calculation, we get \(R_{1}^{*}=0.5678<1\). Then from Theorem 4.1, we know the disease \(I(t)\) finally go to extinction (see Fig. 3).
Figure 3
Figure 3

Sample paths of \((S(t),I(t),R(t))\) with initial conditions \((S(0),I(0),R(0))=(1.5,2.5,2.5)\)

Case (b). We only change the intensity of the noise \(\sigma_{2}(1)=0.15\), \(\sigma_{2}(2)=0.2\). Then we have \(R_{2}^{*}=2.0560>1\), one can derived that model (3) has a unique stationary distribution. Figure 4(a) shows the Markov chain switching process and Fig. 4(b) shows the process of changing system variables over time. From Fig. 4(c), Fig. 4(d) and Fig. 4(e), we can see system (3) has a stationary distribution.
Figure 4
Figure 4

Computer simulation of a single path of Markov chain \(r(t)\) and its corresponding solutions \((S(t),I(t),R(t))\) for the system (3). The last three pictures are the histograms of the path

Declarations

Acknowledgements

The authors thank the editors and the anonymous referees for their careful reading and valuable comments.

Availability of data and materials

Data sharing not applicable to this article as all data sets are hypothetical during the current study.

Funding

This work is supported by SDUST Research Fund (2014TDJH102).

Authors’ contributions

All authors worked together to produce the results and read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, P.R. China
(2)
State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao, P.R. China

References

  1. Miao, A., Wang, X., Zhang, T., Wang, W., Pradeep, B.G.S.A.: Dynamical analysis of a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis. Adv. Differ. Equ. 2017(1), 226 (2017) MathSciNetMATHView ArticleGoogle Scholar
  2. Li, F., Meng, X., Wang, X.: Analysis and numerical simulations of a stochastic SEIQR epidemic system with quarantine-adjusted incidence and imperfect vaccination. Comput. Math. Methods Med. 2018, Article ID 7873902 (2018) MathSciNetGoogle Scholar
  3. Zhang, T., Meng, X., Zhang, T.: Global dynamics of a virus dynamical model with cell-to-cell transmission and cure rate. Comput. Math. Methods Med. 2015, Article ID 758362 (2015) MathSciNetMATHGoogle Scholar
  4. Li, X., Lin, X., Lin, Y.: Lyapunov-type conditions and stochastic differential equations driven by G-Brownian motion. J. Math. Anal. Appl. 439(1), 235–255 (2016) MathSciNetMATHView ArticleGoogle Scholar
  5. Meng, X., Li, F., Gao, S.: Global analysis and numerical simulations of a novel stochastic eco-epidemiological model with time delay. Appl. Math. Comput. 339, 701–726 (2018) MathSciNetGoogle Scholar
  6. Zhang, T., Liu, X., Meng, X., Zhang, T.: Spatio-temporal dynamics near the steady state of a planktonic system. Comput. Math. Appl. 75(12), 4490–4504 (2018) MathSciNetView ArticleGoogle Scholar
  7. Jiang, Z., Zhang, W., Zhang, J., Zhang, T.: Dynamical analysis of a phytoplankton–zooplankton system with harvesting term and Holling III functional response. Int. J. Bifurc. Chaos 28(13), 1850162 (2018) MathSciNetMATHView ArticleGoogle Scholar
  8. Liu, L., Meng, X.: Optimal harvesting control and dynamics of two-species stochastic model with delays. Adv. Differ. Equ. 2017(1), 18 (2017) MathSciNetMATHView ArticleGoogle Scholar
  9. Miao, A., Zhang, J., Zhang, T., Pradeep, B.G.S.A.: Threshold dynamics of a stochastic SIR model with vertical transmission and vaccination. Comput. Math. Methods Med. 2017, Article ID 4820183 (2017) MathSciNetMATHView ArticleGoogle Scholar
  10. Liu, X., Li, Y., Zhang, W.: Stochastic linear quadratic optimal control with constraint for discrete-time systems. Appl. Math. Comput. 228, 264–270 (2014) MathSciNetMATHGoogle Scholar
  11. Lin, X., Zhang, R.: \(\mathrm{H}_{\infty}\) control for stochastic systems with Poisson jumps. J. Syst. Sci. Complex. 24(4), 683–700 (2011) MathSciNetMATHGoogle Scholar
  12. May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (2001) MATHGoogle Scholar
  13. Qi, H., Liu, L., Meng, X.: Dynamics of a non-autonomous stochastic SIS epidemic model with double epidemic hypothesis. Complexity 2017, Article ID 4861391 (2017) MATHView ArticleGoogle Scholar
  14. Fan, X., Song, Y., Zhao, W.: Modeling cell-to-cell spread of HIV-1 with nonlocal infections. Complexity 2018, Article ID 2139290 (2018) Google Scholar
  15. Chi, M., Zhao, W.: Dynamical analysis of multi-nutrient and single microorganism chemostat model in a polluted environment. Adv. Differ. Equ. 2018(1), 120 (2018) MathSciNetView ArticleGoogle Scholar
  16. Leng, X., Feng, T., Meng, X.: Stochastic inequalities and applications to dynamics analysis of a novel SIVS epidemic model with jumps. J. Inequal. Appl. 2017, 138 (2017) MathSciNetMATHView ArticleGoogle Scholar
  17. Liu, G., Wang, X., Meng, X.: Extinction and persistence in mean of a novel delay impulsive stochastic infected predator-prey system with jumps. Complexity 2017, Article ID 1950970 (2017) MathSciNetGoogle Scholar
  18. Zhao, W., Li, J., Meng, X.: Dynamical analysis of sir epidemic model with nonlinear pulse vaccination and lifelong immunity. Discrete Dyn. Nat. Soc. 2015, Article ID 848623 (2015) MathSciNetGoogle Scholar
  19. Miao, A., Zhang, T., Zhang, J., Wang, C.: Dynamics of a stochastic SIR model with both horizontal and vertical transmission. J. Appl. Anal. Comput. 2018(4), 1108–1121 (2018) MathSciNetGoogle Scholar
  20. Song, Y., Miao, A., Zhang, T.: Extinction and persistence of a stochastic SIRS epidemic model with saturated incidence rate and transfer from infectious to susceptible. Adv. Differ. Equ. 2018(1), 293 (2018) MathSciNetView ArticleGoogle Scholar
  21. Zhang, S., Meng, X., Wang, X.: Application of stochastic inequalities to global analysis of a nonlinear stochastic SIRS epidemic model with saturated treatment function. Adv. Differ. Equ. 2018(1), 50 (2018) MathSciNetMATHView ArticleGoogle Scholar
  22. Zhong, X., Guo, S., Peng, M.: Stability of stochastic SIRS epidemic models with saturated incidence rates and delay. Stoch. Anal. Appl. 35(1), 1–26 (2016) MathSciNetMATHView ArticleGoogle Scholar
  23. Chang, Z., Meng, X., Zhang, T.: A new way of investigating the asymptotic behaviour of a stochastic sis system with multiplicative noise. Appl. Math. Lett. 87, 80–86 (2019) MathSciNetView ArticleGoogle Scholar
  24. Qi, H., Leng, X., Meng, X., Zhang, T.: Periodic solution and ergodic stationary distribution of SEIS dynamical systems with active and latent patients. Qual. Theory Dyn. Syst. (2018). https://doi.org/10.1007/s12346-018-0289-9 View ArticleGoogle Scholar
  25. Liu, Q., Jiang, D., Shi, N., Hayat, T., Alsaedi, A.: Stationary distribution and extinction of a stochastic SIRS epidemic model with standard incidence. Phys. A, Stat. Mech. Appl. 469, 510–517 (2017) MathSciNetMATHView ArticleGoogle Scholar
  26. Zhang, S., Meng, X., Feng, T., Zhang, T.: Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects. Nonlinear Anal. Hybrid Syst. 26, 19–37 (2017) MathSciNetMATHView ArticleGoogle Scholar
  27. Yu, X., Yuan, S., Zhang, T.: The effects of toxin-producing phytoplankton and environmental fluctuations on the planktonic blooms. Nonlinear Dyn. 91(3), 1653–1668 (2018) MATHView ArticleGoogle Scholar
  28. Liu, Q., Jiang, D., Shi, N., Hayat, T., Alsaedi, A.: Nontrivial periodic solution of a stochastic non-autonomous SISV epidemic model. Phys. A, Stat. Mech. Appl. 462, 837–845 (2016) MathSciNetMATHView ArticleGoogle Scholar
  29. Zhang, T., Meng, X., Zhang, T.: Global analysis for a delayed SIV model with direct and environmental transmissions. J. Appl. Anal. Comput. 6(2), 479–491 (2016) MathSciNetGoogle Scholar
  30. Zhang, T., Meng, X., Song, Y., Zhang, T.: A stage-structured predator–prey SI model with disease in the prey and impulsive effects. Math. Model. Anal. 18(4), 505–528 (2013) MathSciNetMATHView ArticleGoogle Scholar
  31. Greenhalgh, D., Moneim, I.A.: SIRS epidemic model and simulations using different types of seasonal contact rate. Syst. Anal. Model. Simul. 43(5), 573–600 (2003) MathSciNetMATHView ArticleGoogle Scholar
  32. Martcheva, M.: A non-autonomous multi-strain SIS epidemic model. J. Biol. Dyn. 3(2–3), 235–251 (2009) MathSciNetMATHView ArticleGoogle Scholar
  33. Lin, Y., Jiang, D., Liu, T.: Nontrivial periodic solution of a stochastic epidemic model with seasonal variation. Appl. Math. Lett. 45, 103–107 (2015) MathSciNetMATHView ArticleGoogle Scholar
  34. Yu, X., Yuan, S., Zhang, T.: Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching. Commun. Nonlinear Sci. Numer. Simul. 59, 359–374 (2018) MathSciNetView ArticleGoogle Scholar
  35. Xu, C., Yuan, S., Zhang, T.: Average break-even concentration in a simple chemostat model with telegraph noise. Nonlinear Anal. Hybrid Syst. 29, 373–382 (2018) MathSciNetMATHView ArticleGoogle Scholar
  36. Liu, M., Wang, K.: Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment. J. Theor. Biol. 264(3), 934–944 (2010) MathSciNetView ArticleGoogle Scholar
  37. Bian, F., Zhao, W., Song, Y., Yue, R.: Dynamical analysis of a class of prey–predator model with Beddington–DeAngelis functional response, stochastic perturbation, and impulsive toxicant input. Complexity 2017, Article ID 3742197 (2017) MATHView ArticleGoogle Scholar
  38. Zhang, X., Jiang, D., Alsaedi, A., Hayat, T.: Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching. Appl. Math. Lett. 59, 87–93 (2016) MathSciNetMATHView ArticleGoogle Scholar
  39. Liu, Q., Jiang, D., Shi, N.: Threshold behavior in a stochastic SIQR epidemic model with standard incidence and regime switching. Appl. Math. Comput. 316, 310–325 (2018) MathSciNetGoogle Scholar
  40. Settati, A., Lahrouz, A.: Stationary distribution of stochastic population systems under regime switching. Appl. Math. Comput. 244, 235–243 (2014) MathSciNetMATHGoogle Scholar
  41. Slatkin, M.: The dynamics of a population in a Markovian environment. Ecology 59(2), 249–256 (1978) View ArticleGoogle Scholar
  42. Mao, X.: Stability of stochastic differential equations with Markovian switching. Stoch. Process. Appl. 79(1), 45–67 (1999) MathSciNetMATHView ArticleGoogle Scholar
  43. Zhao, Y., Yuan, S., Zhang, T.: The stationary distribution and ergodicity of a stochastic phytoplankton allelopathy model under regime switching. Commun. Nonlinear Sci. Numer. Simul. 37, 131–142 (2016) MathSciNetView ArticleGoogle Scholar
  44. Liptser, R.S.: A strong law of large numbers for local martingales. Stochastics 3(1–4), 217–228 (1980) MathSciNetMATHView ArticleGoogle Scholar

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