Open Access

Stationary distribution and extinction of a stochastic SIRS epidemic model with information intervention

Advances in Difference Equations20172017:352

https://doi.org/10.1186/s13662-017-1406-9

Received: 28 July 2017

Accepted: 18 October 2017

Published: 2 November 2017

Abstract

In this paper, a new SIRS epidemic model which considers the influence of information intervention and environmental noise is studied. The study shows that information intervention and white noise have great effects on the disease. First, we show that there is global existence and positivity of the solution. Then, we prove that the stochastic basic production \(\mathscr{R}_{s}\) is a threshold which determines the extinction or persistence of the disease. When the intensity of noise is large, we obtain \(\mathscr{R}_{s}<1\) and the disease will die out. When the intensity of noise is small, then \(\mathscr{R}_{s}>1\) and a sufficient condition for the existence of stationary distribution is obtained, which means the disease is prevalent. Finally, the main results are illustrated by numerical simulations.

Keywords

SIRS epidemic modelinformation interventionenvironmental noisestationary distributionextinction

1 Introduction

It is well known that diseases have a great effect on people’s health. For example, according to the report, the H7N9 bird flu, which is the world’s first new subtype of influenza virus with symptoms of fever and fever in the early days of the virus, was discovered in Shanghai and Anhui in 2013. Until now, H7N9 bird flu has caused many infections or deaths. When the infectious disease appears, how to prevent and treat disease is one of the hot issues that people care about. Awareness is raised by increasing media coverage and health education, which can prevent or delay disease occurrence to a certain extent. For instance, since the outbreak of the H7N9 flu, the public media has been reporting the daily number of infected people, the number of deaths, the symptoms and prevention measures of the disease. Media coverage has greatly reduced the rate of infection and has had a significant impact on disease control. For treatment of the disease, in addition to the drug treatment, combining with non-drug therapy can be effective in half the effort, and psychotherapy is a very effective method. A large number of studies have shown that many diseases are related to our thoughts in varying degrees [1]. Positive psychological suggestion will make positive emotions generate, so as to avoid a negative effect of negative feelings caused by negative emotion suggestion on the disease, which is of a great help in controlling disease and restoring healthy life. Therefore, information intervention (media coverage, health education, psychological suggestion) as a kind of non-drug treatment is a very important method for the prevention and treatment of diseases.

Currently, there have been many studies that considered taking information intervention into series infectious diseases models including SIR, SEIR, SIRS, etc. For example, Joshi et al. [2] considered SIR infectious disease model which was based on information intervention and found that information could reduce the level of infection. In addition, Buonomo et al. [3] studied the influence of information on vaccination at the time of new birth in an SEIR model, and he found that information induced vaccination may trigger oscillation which is different from others. Xiao et al. [4] found that the impact of the media could not only delay the peak of disease, but also could reduce the severity of disease outbreak. In further work, Joshi et al. [5] studied the effects of information or education on the dynamic control of disease. Recently, Kumar et al. [6] combined treatment and information influence as control intervention, as well as put forward the following SIRS epidemic model:
$$ \textstyle\begin{cases} \frac{dS}{dt}=\Lambda -\beta SI-\mu S-\mu_{1}mZS+\delta_{0}R, \\ \frac{dI}{dt}=\beta SI-(\mu +\delta +\gamma )I, \\ \frac{dR}{dt}=\gamma I+\mu_{1}mZS-(\mu +\delta_{0})R, \\ \frac{dZ}{dt}=\frac{aI}{1+bI}-a_{0}Z, \end{cases} $$
(1)
here S denotes the susceptible population, I denotes the infective population, R denotes the removed population and Z denotes the density of information in the population. The parameters in the model are summarized in the following list:
Λ:: 

the birth or inflow rate of the susceptible population;

γ:: 

the recovery rate of the infected population;

μ:: 

the natural death rate;

δ:: 

the disease caused mortality rate;

β:: 

the contact transmission coefficient;

\(\delta_{0}\) (\(=\delta_{1}+\delta_{2}\)):: 

the rate of losing their total immunity including both the loss of natural immunity \({(\delta_{1})}\) and the loss of immunity of safeguard measure (\({\delta_{2}}\));

m:: 

the information interaction rate by which individuals change their behavior;

\(\mu_{1}\) (\({0\leq \mu_{1}\leq 1}\)):: 

response intensity;

a:: 

the growth rate of information;

b:: 

the saturation constant;

\({a_{0}}\):: 

the natural degradation rate of information.

All parameters in model (1) are assumed to be non-negative.

We know the basic reproduction number \(\mathscr{R}_{0}\) is a threshold which represents how many secondary infections result from the introduction of one infected individual into a population of susceptible [7]. In model (1), the basic reproduction number \(\mathscr{R}_{0}=\frac{\Lambda \beta }{\mu ( \mu +\delta +\gamma )}\) is a threshold of extinction and persistence of disease. If \(\mathscr{R}_{0}<1\), model (1) has a disease-free equilibrium \(E_{0}=(\frac{\Lambda }{\mu },0,0,0)\) and it is globally stable in \(\mathbb{\bar{R}}_{+}^{4}= \{ (S, I, R, Z): S\geq 0, I \geq 0, R\geq 0, Z\geq 0, S+I+R\leq \frac{\Lambda }{\mu } \} \), it means that the disease dies out. If \(\mathscr{R}_{0}>1\), then \(E_{0}\) is unstable and there exists a globally asymptotically stable endemic equilibrium \(E^{*}=(S^{*}, I^{*}, R^{*},Z^{*})\), where \(S^{*}=\frac{\mu +\delta +\gamma }{\beta }\), \(R^{*}=\frac{1}{\mu + \delta_{0}} ( \gamma I^{*}+\frac{\mu_{1}ma(\mu +\delta +\gamma )I ^{*}}{a_{0}\beta (1+bI^{*})} ) \), \(Z^{*}=\frac{aI^{*}}{a_{0}(1+bI ^{*})}\) and \(I^{*}\) is the unique positive solution of equation \(AI^{2}+BI+C=0\), here \(A=b(\mu +\delta )+\frac{b\mu \gamma }{\mu + \delta }\), \(B=b\Lambda ( \frac{1}{\mathscr{R}_{0}}-1 ) +\frac{ \mu (\mu +\delta +\gamma )+\delta_{0}(\mu +\delta )}{\mu +\delta_{0}} + \frac{\mu \mu_{1}ma(\mu +\delta +\gamma )}{\beta a_{0}(\mu +\delta _{0})}\), \(C=\Lambda ( \frac{1}{\mathscr{R}_{0}}-1 ) \), it means that the disease is prevalent.

However, model (1) is just a deterministic model. In fact, ambient white noise has a big impact on the infectious disease [812]. Many scholars have studied random infectious disease models [1320]. May [21] pointed out that due to environmental fluctuation, the birth rates, death rates, transmission coefficient and other parameters of a deterministic system give a greater or lesser extent of random fluctuations. Dalal, Greenhalgh and Mao [22] found that the introduction of stochastic noise changes the basic reproduction number of the disease.

In this paper, taking into account the effect of randomly fluctuating environment, we assume that the rates m and γ are subject to random fluctuations, \(m\rightarrow m+\sigma_{1}\,dB_{1}(t)\), \(\gamma \rightarrow \gamma +\sigma_{2}\,dB_{2}(t)\). Thus, deterministic model (1) is given by the following new stochastic SIRS epidemic model with information intervention:
$$ \textstyle\begin{cases} dS=[\Lambda -\beta SI-\mu S-\mu_{1}mZS+\delta_{0}R]\,dt-\sigma_{1}\mu _{1}ZS\,dB_{1}(t), \\ dI=[\beta SI-(\mu +\delta +\gamma )I]\,dt-\sigma_{2}I\,dB_{2}(t), \\ dR=[\gamma I+\mu_{1}mZS-(\mu +\delta_{0})R]\,dt+\sigma_{1}\mu_{1}ZS\,dB _{1}(t)+\sigma_{2}I\,dB_{2}(t), \\ dZ= [ \frac{aI}{1+bI}-a_{0}Z ]\,dt, \end{cases} $$
(2)
where \(B_{1}(t)\), \(B_{2}(t)\) are mutually independent Brownian motions (white noise) and \(\sigma^{2}_{1}\), \(\sigma^{2}_{2}\) are their intensities.
Now, on the basis of deterministic model (1), we add the effect of white noise. As a result, we get a new comprehensive model (2) which considers the influence of information intervention and white noise. For model (2), the following questions may be proposed:
  1. (Q1)

    What is the impact of environmental noise on the transmission of disease?

     
  2. (Q2)

    What role does information intervention play in the transmission of disease?

     
  3. (Q3)

    What conditions are required in the existence of stationary distribution?

     

This paper mainly solves the above mentioned problems, and the paper is organized as follows. In Section 2, we show that model (2) has an existence and unique positive solution. In Section 3, we prove the extinction, and that the disease is prevalent in Section 4. In Section 5, we conclude there is a sufficient condition of the existence of stationary distribution for model (2). In Section 6, we make simulations to confirm our results. Finally, we finish the paper with conclusions and future directions.

2 Existence and uniqueness of positive solution

First, consider the d-dimensional stochastic differential equation
$$ dX(t)=f\bigl(X(t),t\bigr)\,dt+g\bigl(X(t),t\bigr)\,dB(t), \quad \mbox{for } t\geq t_{0}, $$
with the initial value \(X(0)=X_{0}\in \mathbb{R}^{d}\). Define the differential operator L associated with the above equation by
$$ L=\frac{\partial }{\partial t} + \sum^{d}_{i=1}f_{i}(X,t) \frac{\partial }{\partial X_{i}} + \frac{1}{2}\sum^{d}_{i,j=1} \bigl[g^{T}(X,t)g(X,t)\bigr]_{ij}\frac{ \partial^{2}}{\partial X_{i}\partial X_{j}}. $$
If L acts on a function \(V\in C^{2,1}(\mathbb{R}^{d}\times [t_{0}, \infty ];\mathbb{R}_{+})\), then
$$ LV(X,t)=V_{t}(X,t)+V_{X}(X,t)f(X,t)+\frac{1}{2}\operatorname{trace} \bigl[g^{T}(X,t)V_{XX}(X,t)g(X,t)\bigr], $$
where \(V_{t}=\frac{\partial V}{\partial t}\), \(V_{X}=(\frac{\partial V}{ \partial X_{1}},\ldots,\frac{\partial V}{\partial X_{d}})\), \(V_{XX}=(\frac{ \partial^{2} V}{\partial X_{i}\partial X_{j}})_{d\times d}\). Using Itô’s formula [23], if \(X(t)\in \mathbb{R}^{d}\), then
$$ dV\bigl(X(t),t\bigr)=LV\bigl(X(t),t\bigr)\,dt + V_{X}\bigl(X(t),t \bigr)g\bigl(X(t),t\bigr)\,dB(t). $$
Next, we show that the solution of model (2) is global and positive by using the Lyapunov analysis method [10, 22]. To get this result, define a bounded set Γ as follows:
$$ \Gamma = \biggl\{ (S, I, R, Z)\in \mathbb{R}_{+}^{4}: \frac{\Lambda }{ \mu +\delta }\leq S+I+R\leq \frac{\Lambda }{\mu }, 0\leq Z\leq \frac{a \Lambda }{a_{0}(\mu +b\Lambda )} \biggr\} \subset \mathbb{R}_{+}^{4}. $$

The main results of this section are given by the following two lemmas.

Lemma 2.1

For any initial value \((S_{0}, I_{0}, R_{0}, Z_{0})\in \mathbb{R}_{+} ^{4}\), there exists a unique positive solution to system (2) on \(t\geq 0\), and the solution will remain in \(\mathbb{R}_{+}^{4}\) with probability one.

Proof

Since the coefficients of system (2) satisfy the local Lipschitz condition [23], then for any initial value \((S_{0}, I_{0}, R_{0}, Z_{0})\in \mathbb{R}_{+}^{4}\), there is a unique local solution \((S(t), I(t), R(t), Z(t))\) on \([0, \tau_{e})\), where \(\tau_{e}\) is the explosion time [23]. If we can prove \(\tau_{e}=\infty \) a.s., which means the solution is global. Let \(k_{0}\geq 0\) be sufficiently large so that \(S_{0}\), \(I_{0}\), \(R_{0}\), and \(Z_{0}\) all lie within the interval \([\frac{1}{k_{0}},k_{0}]\). For each integer \(k\geq k_{0}\), define the following stopping time:
$$ \tau_{k}=\inf \biggl\{ t\in [0, \tau_{e}): \min \bigl\{ S(t), I(t), R(t),Z(t) \bigr\} \leq \frac{1}{k} \mbox{ or } \max \bigl\{ S(t), I(t), R(t), Z(t)\bigr\} \geq k \biggr\} . $$
In this paper, we set \(\inf \varnothing =\infty \) (as usual is the empty set). By the definition, \(\tau_{k}\) is increasing as \(k\rightarrow \infty \). Let \(\tau_{\infty }= \lim_{k\rightarrow \infty }\tau_{k}\), then \(\tau_{\infty }\leq \tau _{e}\) a.s. In the following, we need to verify \(\tau_{\infty }=\infty \) a.s. If this assertion is false, then there exist a constant \(T>0\) and \(\epsilon \in (0, 1)\) such that \(\mathbb{P}\{\tau_{\infty }\leq T\}> \epsilon \). As a consequence, there exists an integer \(k_{1}\geq k _{0}\) such that
$$ \mathbb{P}\{\tau_{k}\leq T\}\geq \epsilon ,\quad \forall k\geq k_{1}. $$
For \(t\leq \tau_{k}\) and each k,
$$ d(S+I+R)=\bigl[\Lambda -\mu (S+I+R)-\delta I\bigr]\,dt\leq \bigl[\Lambda -\mu (S+I+R)\bigr]\,dt. $$
Then
$$ S(t)+I(t)+R(t)\leq \textstyle\begin{cases} \frac{\Lambda }{\mu } &\mbox{ if } S_{0}+I_{0}+R_{0}\leq \frac{ \Lambda }{\mu } \\ S_{0}+I_{0}+R_{0} &\mbox{ if } S_{0}+I_{0}+R_{0}>\frac{\Lambda }{ \mu }, \end{cases}\displaystyle \quad :=M $$
where \(M=\max [ \frac{\Lambda }{\mu }, S_{0}+I_{0}+R_{0} ] \).
Define a \(C^{2}\)-function V: \(\mathbb{R}_{+}^{4}\rightarrow \mathbb{\bar{R}}_{+}\) by
$$ V(S, I, R, Z)=(S-1-\ln S)+(I-1-\ln I)+\biggl(\frac{1}{2}R^{2}+R \biggr)+(Z-1- \ln Z). $$
By Itô’s formula, we have
$$ dV(S, I, R, Z)=LV(S, I, R, Z)\,dt+\sigma_{1}(1+RS)\mu_{1}Z\,dB_{1}(t)+ \sigma_{2}(1+RI)\,dB_{2}(t), $$
where
$$\begin{aligned} LV&= \biggl( 1-\frac{1}{S} \biggr) ( \Lambda - \beta SI-\mu S-\mu_{1} mZS+ \delta_{0} R ) + \biggl( 1- \frac{1}{I} \biggr) \bigl( \beta SI-( \mu +\delta +\gamma )I \bigr) \\ &\quad {}+ ( 1+R ) \bigl( \gamma I+ \mu_{1}mZS-(\mu +\delta_{0})R \bigr) + \biggl( 1-\frac{1}{Z} \biggr) \biggl( \frac{aI}{1+bI} -a_{0}Z \biggr) \\ &\quad {}+\frac{1}{2}\sigma_{1}^{2}\mu_{1}^{2}Z^{2} +\frac{1}{2}\sigma_{1} ^{2}\mu_{1}^{2}Z^{2}S^{2} + \frac{1}{2}\sigma_{2}^{2}\bigl(1+I^{2} \bigr) \\ &= \Lambda +\beta I+\mu_{1}mZ +(\mu +\delta +\gamma ) +\mu +\gamma RI+ \mu_{1}mZSR + \frac{aI}{1+bI}+a_{0} \\ &\quad {}+\frac{1}{2}\sigma_{1}^{2}\mu_{1}^{2}Z^{2} +\frac{1}{2}\sigma_{1} ^{2}\mu_{1}^{2}Z^{2}S^{2} +\frac{1}{2}\sigma_{2}^{2}\bigl(1+I^{2} \bigr) -\mu S-\frac{ \Lambda }{S} -\frac{\delta_{0}R}{S} \\ & \quad {}- \mu I - \delta I-\beta S -\mu R-(\mu +\delta_{0})R^{2}-a_{0}Z- \frac{aI}{Z(1+bI)} \\ &\leq \Lambda +2\mu +\gamma +\delta +a_{0}+\frac{a}{b} + \frac{\mu _{1}ma}{a_{0}b} +\beta M + \biggl( \gamma + \frac{\mu_{1}ma}{a_{0}b} \biggr) M ^{2} \\ &\quad {}+\frac{\sigma_{1}^{2}\mu_{1}^{2}a^{2}}{2a_{0}^{2}b^{2}}\bigl(1+M^{2}\bigr)+ \frac{1}{2} \sigma_{2}^{2}\bigl(1+M^{2}\bigr) \\ &:=K, \end{aligned}$$
where K is a positive constant which is independent of S, I, R, Z and t. This completes the proof. □

Lemma 2.2

For any initial value \((S_{0}, I_{0}, R_{0}, Z_{0})\in \mathbb{R}_{+} ^{4}\), the unique solution of epidemic model (2) on \(t\geq 0\) will enter Γ and will remain in Γ with probability one.

Proof

From (2), we obtain that the total size of population \(N=S+I+R\), then
$$ \frac{dN}{dt}=\Lambda -\mu N-\delta I. $$
This implies that \(\Lambda -(\mu +\delta )N\leq \frac{dN}{dt}\leq \Lambda -\mu N\), then we have
$$ \frac{\Lambda }{\mu +\delta }\leq \liminf_{t\rightarrow \infty } N \leq \limsup _{t\rightarrow \infty } N\leq \frac{\Lambda }{\mu }. $$
As we can see, all solutions S, I and R of epidemic model (2) are bounded by \(\frac{\Lambda }{\mu }\). From the last equation of model (2) and \(I\leq \frac{\Lambda }{\mu }\), we get \(\limsup_{t\rightarrow \infty }Z\leq \frac{a\Lambda }{a_{0}(\mu +b \Lambda )}\). Therefore, we obtain that Γ is the positively invariant bounded set. Then the trajectories of all solutions initiating anywhere of \(\mathbb{R}_{+}^{4}\) will enter Γ and then remain in Γ with probability one. □

3 Extinction of disease

In model (1), the value of the basic reproductive number \(\mathscr{R}_{0}\) determines extinction and persistence of disease. But for epidemic model (2), we will show that the stochastic basic reproduction number \(\mathscr{R}_{s}\) is a condition for the extinction and persistence of disease, where \(\mathscr{R}_{s}\) is denoted
$$ \mathscr{R}_{s} = \mathscr{R}_{0} - \frac{\sigma_{2}^{2}}{2(\mu + \delta +\gamma )}. $$

Theorem 3.1

Let \((S_{t}, I_{t}, R_{t}, Z_{t})\) be the solution of epidemic model (2) with the initial value \((S_{0}, I_{0}, R_{0}, Z_{0})\in \mathbb{R}_{+}^{4}\). If
$$ \mathscr{R}_{s}< 1, $$
(3)
then the solution of epidemic model (2) obeys
$$\begin{aligned}& \limsup_{t\rightarrow \infty }\frac{\log I_{t}}{t}\leq -c< 0, \quad \textit{a.s.}, \\& \limsup_{t\rightarrow \infty }Z(t)=0, \quad \textit{a.s.}, \\& \limsup_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}R_{s}\,ds=0, \quad \textit{a.s.}, \\& \liminf_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}S_{s}\,ds=\frac{ \Lambda }{\mu },\quad \textit{a.s.}, \end{aligned}$$
namely, the disease in epidemic model (2) will go to extinction with probability one, where \(c=(\mu +\delta +\gamma )(1-\mathscr{R}_{s})\) corresponding to condition (3).

Proof

Itô’s formula yields that
$$ d\log I= \biggl( \beta S-(\mu +\delta +\gamma )-\frac{1}{2} \sigma_{2} ^{2} \biggr)\,dt - \sigma_{2}\,dB_{2}(t). $$
(4)
Integrating both sides from 0 to t and dividing by t, we have
$$ \frac{\log I_{t}-\log I_{0}}{t}=\frac{1}{t} \int_{0}^{t} \biggl( \beta S _{s}-(\mu + \delta +\gamma )-\frac{1}{2}\sigma_{2}^{2} \biggr)\,ds - \frac{1}{t} \int_{0}^{t}\sigma_{2}\,dB_{2}(s). $$
(5)
According to the large number theorem for local martingales [23], we can get
$$ \limsup_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}\sigma_{2}\,dB_{2}(s)=0, \quad \mbox{a.s.}$$
Since \(S(t)\leq \frac{\Lambda }{\mu }\) for all \(t\geq 0\), we have
$$\begin{aligned} \beta S-(\mu +\delta +\gamma )-\frac{1}{2}\sigma_{2}^{2}& \leq \frac{\beta \Lambda }{\mu }-(\mu +\delta +\gamma )-\frac{1}{2} \sigma_{2}^{2} \\ & = (\mu +\delta +\gamma ) (\mathscr{R}_{s}-1). \end{aligned}$$
Therefore, we obtain
$$ \limsup_{t\rightarrow \infty }\frac{\log I_{t}}{t}\leq (\mu +\delta + \gamma ) ( \mathscr{R}_{s}-1)< 0, \quad \mbox{a.s.} $$
(6)
Denote \(\Omega_{1}= \{ \omega \in \Omega: \limsup_{t\rightarrow \infty }I(\omega , t)=0 \} \). In view of (6), we have
$$ \mathbb{P}(\Omega_{1})=1. $$
(7)
It means, for any given \(\varepsilon_{1}>0\), there exists a constant \(T_{1}=T_{1}(\omega , \varepsilon_{1})\) such that \(I(t)<\varepsilon _{1}\), a.s. for \(t>T_{1}\).
From the last equation of epidemic model (2), we obtain
$$\begin{aligned} dZ(\omega , t)&= \biggl[ \frac{aI(\omega , t)}{1+bI(\omega , t)}-a_{0}Z( \omega , t) \biggr]\,dt \\ &\leq \bigl[a\varepsilon_{1}-a_{0}Z(\omega , t)\bigr]\,dt \quad \mbox{for } \omega \in \Omega_{1}, t\geq T_{1}. \end{aligned}$$
(8)
According to the comparison theorem [24], one can get that
$$\begin{aligned} Z(\omega , t)&\leq e^{-a_{0}t}\cdot \biggl( Z(T_{1})+ \int_{T_{1}}^{t}a \varepsilon_{1}\cdot e^{a_{0}s}\,ds \biggr) \\ &\leq Z(T_{1})e^{-a_{0}t}+\frac{a\varepsilon_{1}}{a_{0}} \quad \mbox{for } \omega \in \Omega_{1}, t\geq T_{1}. \end{aligned}$$
Then
$$ \limsup_{t\rightarrow \infty }Z(\omega , t)\leq \frac{a\varepsilon _{1}}{a_{0}} \quad \mbox{for } \omega \in \Omega_{1}, t\geq T_{1}. $$
(9)
By the arbitrariness of \(\varepsilon_{1}\), we have \(\limsup_{t\rightarrow \infty }Z(\omega , t)\leq 0\). On the other hand, \(\limsup_{t\rightarrow \infty }Z(\omega , t)\geq 0\). Therefore, \(\limsup_{t\rightarrow \infty }Z(t)=0\), a.s. Let \(\Omega_{2}= \{ \omega \in \Omega: \limsup_{t\rightarrow \infty }Z(\omega , t)=0 \} \subset \Omega_{1}\), then for any given \(\varepsilon_{2}>0\), there exists a constant \(T_{2}=T_{2}(\omega , t)\geq T_{1}\) such that \(Z(t)<\varepsilon_{2}\), a.s. for \(t>T_{2}\).
For the third equation of (2), integrating both sides from 0 to t and dividing by t, we have
$$\begin{aligned} \frac{(\mu +\delta_{0})}{t} \int_{0}^{t}R_{s}\,ds &\leq \frac{\gamma }{t} \int_{0}^{t}I_{s}\,ds + \frac{\mu_{1}m\Lambda }{\mu t} \int_{0}^{t}Z_{s}\,ds - \frac{R_{t}-R_{0}}{t}+ \frac{1}{t} \int_{0}^{t}\sigma_{1} \mu_{1}Z_{s}S _{s}\,dB_{1}(s) \\ &\quad {}+\frac{1}{t} \int_{0}^{t}\sigma_{2}I_{s}\,dB_{2}(s) \\ &= \frac{\gamma }{t} \int_{0}^{T_{1}}I_{s}\,ds + \frac{\mu_{1}m\Lambda }{ \mu t} \int_{0}^{T_{2}}Z_{s}\,ds + \frac{\gamma }{t} \int_{T_{1}}^{t}I _{s}\,ds + \frac{\mu_{1}m\Lambda }{\mu t} \int_{T_{2}}^{t}Z_{s}\,ds \\ &\quad {}- \frac{R_{t}-R_{0}}{t}+\frac{1}{t} \int_{0}^{t}\sigma_{1}\mu_{1}Z _{s}S_{s}\,dB_{1}(s)+\frac{1}{t} \int_{0}^{t}\sigma_{2}I_{s}\,dB_{2}(s) \\ &\leq \frac{\gamma \Lambda T_{1}}{\mu t} + \frac{a\mu_{1}m\Lambda ^{2}T_{2}}{a_{0}\mu (\mu +b\Lambda )t} + \gamma \varepsilon_{1} + \frac{ \mu_{1}m\Lambda }{\mu }\varepsilon_{2}-\frac{\gamma \varepsilon_{1}T _{1}}{t}- \frac{\mu_{1}m\Lambda \varepsilon_{2}T_{2}}{\mu t} \\ &\quad {}- \frac{R_{t}-R_{0}}{t}+\frac{1}{t} \int_{0}^{t}\sigma_{1}\mu_{1}Z _{s}S_{s}\,dB_{1}(s)+\frac{1}{t} \int_{0}^{t}\sigma_{2}I_{s}\,dB_{2}(s). \end{aligned}$$
Since \(S(t)\leq \frac{\Lambda }{\mu }\) and \(Z(t)\leq \frac{a\Lambda }{a _{0}(\mu +b\Lambda )}\) for all \(t\geq 0\), according to the large number theorem for local martingales, we get
$$ \limsup_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}\sigma_{1}\mu _{1}Z_{s}S_{s}\,dB_{1}(s)=\limsup _{t\rightarrow \infty }\frac{1}{t} \int _{0}^{t}\sigma_{2}I_{s}\,dB_{2}(s)=0, \quad \mbox{a.s.} $$
Then, according to the arbitrariness of \(\varepsilon_{1}\) and \(\varepsilon_{2}\), we have
$$ \limsup_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}R_{s}\,ds\leq 0, \quad \mbox{a.s.} $$
Thus, from the fact that \(R(t)\geq 0\), we can get that \(\limsup_{t\rightarrow \infty }\frac{1}{t}\int_{0}^{t}R_{s}\,ds = 0\), a.s. Let \(\Omega_{3} = \{ \omega \in \Omega:\lim\sup_{t\rightarrow \infty }R(\omega , t) = 0 \} \subset \Omega _{2}\), then for any given \(\varepsilon_{3}\geq 0\), there exists a constant \(T_{3}=T_{3}(\omega , t)\geq T_{2}\) such that \(R(t)< \varepsilon_{3}\), a.s. for \(t>T_{3}\).
From the first three equations of epidemic model (2), for any \(\omega \in \Omega_{3}\), we have
$$ d(S_{t}+I_{t}+R_{t})=\bigl[\Lambda -\mu (S_{t}+I_{t}+R_{t})-\delta I_{t}\bigr]\,dt. $$
Integrating this from 0 to t and dividing by t yield
$$\begin{aligned} \frac{1}{t} \int_{0}^{t}S_{s}\,ds&= \frac{\Lambda }{\mu } - \frac{1}{t} \int_{0}^{t}I_{s}\,ds-\frac{\delta }{\mu t} \int_{0}^{t}I_{s}\,ds - \frac{1}{t} \int_{0}^{t}R_{s}\,ds-\phi (t) \\ &= \frac{\Lambda }{\mu } - \frac{\mu +\delta }{\mu t} \int_{T_{3}}^{t}I _{s}\,ds - \frac{1}{t} \int_{T_{3}}^{t}R_{s}\,ds-\frac{1}{t} \int_{0}^{T _{3}} \biggl( \frac{\mu +\delta }{\mu }I_{s} +R_{s} \biggr)\,ds-\phi (t) \\ &\geq \frac{\Lambda }{\mu } - \frac{\mu +\delta }{\mu }\varepsilon _{1}- \varepsilon_{3} + \biggl( \frac{\mu +\delta }{\mu }\varepsilon_{1} +\varepsilon_{3} \biggr) \frac{T_{3}}{t}- \biggl( \frac{\Lambda (\mu + \delta )}{\mu^{2}}+\frac{\Lambda }{\mu } \biggr) \frac{T_{3}}{t}-\phi (t), \end{aligned}$$
where \(\phi (t)=\frac{1}{\mu } ( \frac{S_{t}+I_{t}+R_{t}}{t}-\frac{S _{0}+I_{0}+R_{0}}{t} ) \), and \(\lim_{t\rightarrow \infty }\phi (t)=0\) a.s. By the arbitrariness of \(\varepsilon_{1}\) and \(\varepsilon_{3}\), we obtain
$$ \liminf_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}S_{s}\,ds\geq \frac{ \Lambda }{\mu }, \quad \mbox{a.s.} $$
Note that \(S\leq \frac{\Lambda }{\mu }\). Thereby,
$$ \liminf_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}S_{s}\,ds=\frac{ \Lambda }{\mu }, \quad \mbox{a.s.} $$
This completes the proof. □

Remark 3.2

It is worthy to note that if \(\mathscr{R}_{0}>1\), the disease will be persistent, while the disease dies out whenever \(\mathscr{R}_{0}<1\) in deterministic model (1). However, we can easily find an example that \(\mathscr{R}_{0}>1\) but \(\mathscr{R}_{s}<1\), which implies that the disease in stochastic model (2) dies out since \(\mathscr{R}_{s} = \mathscr{R}_{0} - \frac{\sigma_{2}^{2}}{2(\mu +\delta +\gamma )}\). Then we obtain that environmental noises can suppress the outbreak of disease.

4 Persistence of disease

In this section, we will give a condition for the persistence of disease in epidemic model (2), and our main result is presented by the following theorem.

Theorem 4.1

Let \((S_{t}, I_{t}, R_{t}, Z_{t})\) be the solution of epidemic model (2) with the initial value \((S_{0}, I_{0}, R_{0}, Z_{0})\in \mathbb{R}_{+}^{4}\). If \(\mathscr{R}_{s}>1\), then
$$\begin{aligned}& \liminf_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}S_{s}\,ds\geq \frac{ \Lambda }{\frac{\Lambda \beta }{\mu }+\mu +\frac{\mu_{1}ma\Lambda }{a _{0}(\mu +b\Lambda )}}>0,\quad \textit{a.s.}, \\& \liminf_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}I_{s}\,ds\geq \frac{a _{0}\mu^{2}(\mu +\delta +\gamma )(\mathscr{R}_{s}-1)}{\beta \Lambda (a _{0}\beta +a\mu_{1}m)}\equiv H>0, \quad \textit{a.s.}, \\& \liminf_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}R_{s}\,ds\geq \frac{ \gamma H}{(\mu +\delta_{0})}>0, \quad \textit{a.s.}, \\& \liminf_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}Z_{s}\,ds\geq \frac{a \mu H}{a_{0}(\mu +b\Lambda )}>0, \quad \textit{a.s.} \end{aligned}$$

Proof

For the first equation of epidemic model (2), we have
$$ dS_{t}\geq \biggl[ \Lambda - \biggl( \frac{\Lambda \beta }{\mu }+\mu + \frac{ \mu_{1}ma\Lambda }{a_{0}(\mu +b\Lambda )} \biggr) S_{t} \biggr]\,dt-\sigma _{1} \mu_{1}Z_{t}S_{t}\,dB_{1}(t). $$
Integrating this from 0 to t, we get
$$ \biggl[ \frac{\Lambda \beta }{\mu }+\mu +\frac{\mu_{1}ma\Lambda }{a_{0}( \mu +b\Lambda )} \biggr] \frac{1}{t} \int_{0}^{t}S_{s}\,ds\geq \Lambda - \frac{S _{t}-S_{0}}{t} -\frac{\mu_{1}\sigma_{1}}{t} \int_{0}^{t}Z_{s}S_{s}\,dB _{1}(s). $$
(10)
Since \(S(t)\leq \frac{\Lambda }{\mu }\) and \(I(t)\leq \frac{\Lambda }{ \mu }\), an application of the strong law of large numbers gives
$$ \limsup_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}Z_{s}S_{s}\,dB_{1}(s)=0, \quad \mbox{a.s.} $$
Therefore
$$ \liminf_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}S_{s}\,ds\geq \frac{ \Lambda }{\frac{\Lambda \beta }{\mu }+\mu +\frac{\mu_{1}ma\Lambda }{a _{0}(\mu +b\Lambda )}}>0,\quad \mbox{a.s.} $$
From the first and third equations of epidemic model (2), we can obtain
$$\begin{aligned}& d \biggl( \frac{\mu +\delta_{0}}{\mu }S_{t} + \frac{\delta_{0}}{\mu }R _{t} \biggr) \\& \quad = \biggl[\frac{\mu +\delta_{0}}{\mu }(\Lambda - \beta S_{t}I _{t}-\mu S_{t}-\mu_{1}mZ_{t}S_{t} + \delta_{0}R_{t})\,dt + \frac{\delta _{0}}{\mu }\bigl(\gamma I_{t} + \mu_{1}mZ_{t}S_{t} \\& \quad \quad {} -(\mu +\delta_{0})R_{t}\bigr) \biggr]\,dt - \sigma_{1}\mu_{1}Z_{t}S_{t}\,dB_{1}(t) +\frac{\delta_{0}}{\mu }\sigma_{2}I_{t}\,dB_{2}(t) \\& \quad = \biggl[(\mu +\delta_{0}) \biggl(\frac{\Lambda }{\mu }-S_{t} \biggr) - \frac{\mu + \delta_{0}}{\mu }\beta S_{t}I_{t} - \frac{\mu_{1}m(\mu +\delta_{0})}{ \mu }Z_{t}S_{t} \\& \quad \quad {} +\frac{\mu_{1}m\delta_{0}}{\mu }Z_{t}S_{t} + \frac{\gamma \delta _{0}}{\mu }I_{t} \biggr]\,dt - \sigma_{1}\mu_{1}Z_{t}S_{t}\,dB_{1}(t)+ \frac{ \delta_{0}}{\mu }\sigma_{2}I_{t}\,dB_{2}(t) \\& \quad \geq \biggl[ (\mu +\delta_{0}) \biggl(\frac{\Lambda }{\mu }-S_{t} \biggr) - \frac{ \Lambda \beta (\mu +\delta_{0})}{\mu^{2}}I_{t} - \frac{\Lambda \mu _{1}m(\mu +\delta_{0})}{\mu^{2}}Z_{t} \biggr]\,dt \\& \quad \quad {} -\sigma_{1}\mu_{1}Z_{t}S_{t}\,dB_{1}(t)+ \frac{\delta_{0}}{\mu }\sigma _{2}I_{t}\,dB_{2}(t). \end{aligned}$$
(11)
By the last equation of model (2), we have
$$ Z_{t}\,dt=\frac{aI_{t}\,dt}{a_{0}(1+bI_{t})}-\frac{1}{a_{0}}\,dZ_{t}\leq \frac{a}{a _{0}}I_{t}\,dt-\frac{1}{a_{0}}\,dZ_{t}. $$
Substituting the above inequality into (11), we have
$$\begin{aligned}& d \biggl( \frac{\mu +\delta_{0}}{\mu }S_{t} + \frac{\delta_{0}}{\mu }R _{t} \biggr) \\& \quad \geq \biggl[ (\mu + \delta_{0}) \biggl(\frac{\Lambda }{\mu }-S_{t}\biggr) - \frac{\Lambda (\mu +\delta_{0})(a_{0}\beta +a\mu_{1}m)}{a_{0}\mu ^{2}}I_{t} \biggr]\,dt \\& \quad \quad {} + \frac{\Lambda \mu_{1}m(\mu +\delta_{0})}{a_{0}\mu^{2}}\,dZ_{t} - \sigma_{1} \mu_{1}Z_{t}S_{t}\,dB_{1}(t)+ \frac{\delta_{0}}{\mu }\sigma _{2}I_{t}\,dB_{2}(t). \end{aligned}$$
(12)
Integrating both sides from 0 to t, we can get that
$$\begin{aligned}& \int_{0}^{t} \biggl( \frac{\Lambda }{\mu }-S_{s} \biggr)\,ds \\& \quad \leq \frac{1}{ \mu }(S_{t}-S_{0}) + \frac{\delta_{0}}{\mu (\mu +\delta_{0})}(R_{t}-R _{0}) + \frac{\Lambda (a_{0}\beta +a\mu_{1}m)}{a_{0}\mu^{2}} \int_{0} ^{t}I_{s}\,ds \\& \quad \quad {}-\frac{\Lambda \mu_{1}m}{a_{0}\mu^{2}}(Z_{t}-Z_{0}) + \frac{\sigma _{1}\mu_{1}}{\mu +\delta_{0}} \int_{0}^{t}Z_{s}S_{s}\,dB_{1}(s) - \frac{ \sigma_{2}\delta_{0}}{\mu (\mu +\delta_{0})} \int_{0}^{t}I_{s}\,dB_{2}(s). \end{aligned}$$
(13)
From Theorem 3.1, let \(h(S)=\beta S-(\mu +\delta +\gamma )- \frac{1}{2}\sigma_{2}^{2}\), we can obtain
$$\begin{aligned} h(S)-h\biggl(\frac{\Lambda }{\mu }\biggr)&= \biggl[ \beta S-(\mu +\delta +\gamma )- \frac{1}{2}\sigma_{2}^{2} \biggr] - \biggl[ \frac{\Lambda \beta }{\mu }-( \mu +\delta +\gamma ) - \frac{1}{2} \sigma_{2}^{2} \biggr] \\ &= \beta \biggl(S-\frac{\Lambda }{\mu }\biggr). \end{aligned}$$
This implies that
$$ h(S)\geq (\mu +\delta +\gamma ) (\mathscr{R}_{s}-1)-\beta \biggl( \frac{ \Lambda }{\mu }-S \biggr). $$
(14)
Substituting (14) into (5), we have
$$ \log I_{t}\geq \log I_{0}+(\mu +\delta +\gamma ) ( \mathscr{R}_{s}-1)t- \beta \int_{0}^{t} \biggl( \frac{\Lambda }{\mu }-S_{s} \biggr)\,ds - \int _{0}^{t}\sigma_{2}\,dB_{2}(s). $$
(15)
Then substituting (13) into (15), we can get that
$$\begin{aligned} \log I_{t}&\geq \log I_{0} + (\mu +\delta +\gamma ) (\mathscr{R}_{s}-1)t - \frac{\beta \Lambda (a_{0}\beta +a\mu_{1}m)}{a_{0}\mu^{2}} \int_{0} ^{t}I_{s}\,ds \\ & \quad {}- \frac{\beta }{\mu }(S_{t}-S_{0})-\frac{\beta \delta_{0}}{\mu ( \mu +\delta_{0})}(R_{t}-R_{0})+ \frac{\beta \Lambda \mu_{1}m}{a_{0}\mu ^{2}}(Z_{t}-Z_{0}) \\ &\quad {}-\frac{\beta \sigma_{1}\mu_{1}}{(\mu +\delta_{0})} \int_{0}^{t}Z_{s}S _{s}\,dB_{1}(s) + \frac{\beta \sigma_{2}\delta_{0}}{\mu (\mu +\delta_{0})} \int_{0}^{t}I _{s}\,dB_{2}(s) - \sigma_{2} \int_{0}^{t}\,dB_{2}(s). \end{aligned}$$
Let
$$\begin{aligned}& \varphi_{1}(t)=\log I_{0} - \frac{\beta }{\mu }(S_{t}-S_{0})-\frac{ \beta \delta_{0}}{\mu (\mu +\delta_{0})}(R_{t}-R_{0})+ \frac{\beta \Lambda \mu_{1}m}{a_{0}\mu^{2}}(Z_{t}-Z_{0}), \\& \varphi_{2}(t)=-\frac{\beta \sigma_{1}\mu_{1}}{(\mu +\delta_{0})} \int_{0}^{t}Z_{s}S_{s}\,dB_{1}(s) +\frac{\beta \sigma_{2}\delta_{0}}{ \mu (\mu +\delta_{0})} \int_{0}^{t}I_{s}\,dB_{2}(s) - \sigma_{2} \int_{0} ^{t}\,dB_{2}(s). \end{aligned}$$
So
$$ \log I_{t}\geq (\mu +\delta +\gamma ) (\mathscr{R}_{s}-1)t - \frac{ \beta \Lambda (a_{0}\beta +a\mu_{1}m)}{a_{0}\mu^{2}} \int_{0}^{t}I_{s}\,ds + \varphi_{1}(t)+\varphi_{2}(t). $$
Since \(S_{t}\), \(I_{t}\), \(R_{t}\) all have positive upper bound \(\frac{ \Lambda }{\mu }\) and \(Z_{t}\leq \frac{a\Lambda }{a_{0}(\mu +b\Lambda )}\), according to the large number theorem for local martingales, we can obtain that
$$ \lim_{t\rightarrow \infty }\frac{\varphi_{1}(t)}{t}= \lim_{t\rightarrow \infty } \frac{\varphi_{2}(t)}{t}=0, \quad \mbox{a.s.} $$
Therefore, dividing by t of the above inequality and letting \(t\rightarrow \infty \), we have
$$ \liminf_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}I_{s}\,ds\geq \frac{( \mu +\delta +\gamma )(\mathscr{R}_{s}-1)}{\frac{\beta \Lambda (a_{0} \beta +a\mu_{1}m)}{a_{0}\mu^{2}}}\equiv H>0, \quad \mbox{a.s.} $$
(16)
Then, for any \(\xi >0\) (\(\xi < H\)), there exists a constant \(T(\omega )>0\) such that
$$ \frac{1}{t} \int_{0}^{t}I_{s}\,ds\geq H-\xi \quad \mbox{for } t \geq T. $$
(17)
From the third equation of (2), we have
$$ dR_{t}\geq \bigl[\gamma I_{t}-(\mu +\delta_{0})R_{t} \bigr]\,dt+\sigma_{1}\mu_{1}Z _{t}S_{t}\,dB_{1}(t)+ \sigma_{2}I_{t}\,dB_{2}(t). $$
(18)
Integrating both sides from 0 to t and dividing by t, we can get that for \(t>T\),
$$\begin{aligned} \frac{1}{t} \int_{0}^{t}R_{s}\,ds&\geq \frac{\gamma }{(\mu +\delta_{0})t} \int_{0}^{t}I_{s}\,ds-\frac{R_{t}-R _{0}}{(\mu +\delta_{0})t} + \frac{1}{(\mu +\delta_{0})t} \int_{0}^{t} \sigma_{1} \mu_{1}Z_{s}S_{s}\,dB_{1}(s) \\ &\quad {}+ \frac{1}{(\mu +\delta_{0})t} \int_{0}^{t}\sigma_{2}I_{s}\,dB_{2}(s) \\ &\geq \frac{\gamma }{(\mu +\delta_{0})}(H-\xi )-\frac{R_{t}-R_{0}}{( \mu +\delta_{0})t} + \frac{1}{(\mu +\delta_{0})t} \int_{0}^{t}\sigma _{1} \mu_{1}Z_{s}S_{s}\,dB_{1}(s) \\ &\quad {}+\frac{1}{(\mu +\delta_{0})t} \int_{0}^{t}\sigma_{2}I_{s}\,dB_{2}(s). \end{aligned}$$
Since \(S\leq \frac{\Lambda }{\mu }\), \(I\leq \frac{\Lambda }{\mu }\), \(R\leq \frac{\Lambda }{\mu }\) and \(Z\leq \frac{a\Lambda }{a_{0}( \mu +b\Lambda )}\), according to the large number theorem for local martingales, we can obtain that
$$ \limsup_{t\rightarrow \infty }\frac{1}{(\mu +\delta_{0})t} \int_{0} ^{t}\sigma_{1} \mu_{1}Z_{s}S_{s}\,dB_{1}(s) = \limsup _{t\rightarrow \infty }\frac{1}{(\mu +\delta_{0})t} \int_{0}^{t}\sigma_{2}I_{s}\,dB_{2}(s)=0. $$
Then, by the arbitrariness of ξ, we have
$$ \liminf_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}R_{s}\,ds\geq \frac{ \gamma H}{(\mu +\delta_{0})}>0,\quad \mbox{a.s.} $$
Integrating both sides from 0 to t and dividing by t of the last equation of (2), we have
$$ \frac{Z_{t}-Z_{0}}{t}\geq \frac{a\mu }{(\mu +b\Lambda )t} \int_{0}^{t}I _{s}\,ds-\frac{a_{0}}{t} \int_{0}^{t}Z_{s}\,ds. $$
This implies that for \(t>T\)
$$ \frac{1}{t} \int_{0}^{t}Z_{s}\,ds\geq \frac{a\mu }{a_{0}(\mu +b\Lambda )}(H-\xi )- \frac{Z_{t}-Z_{0}}{a_{0}t}. $$
By the arbitrariness of ξ, we obtain
$$ \liminf_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t}Z_{s}\,ds\geq \frac{a \mu H}{a_{0}(\mu +b\Lambda )}>0, \quad \mbox{a.s.} $$
This completes the proof. □

Remark 4.2

The results of Theorem 4.1 mean that when the noise is small, then the value of \(\mathscr{R}_{s} = \mathscr{R}_{0} - \frac{\sigma_{2}^{2}}{2( \mu +\delta +\gamma )}>1\), which implies that the disease is prevalent. Therefore, from Theorem 3.1 and Theorem 4.1, we can see that \(\mathscr{R}_{s}\) is a threshold which determines the extinction or persistence of the disease.

5 Stationary distribution and ergodicity

In this section, based on the known result of Has’minskii (see [25], Theorem 4.1, p.119 and Lemma 9.4, p.138), we prove that there is an ergodic stationary distribution for the solution of epidemic model (2), which shows that the disease will prevail.

Let \(X(t)\) be a regular time-homogeneous Markov process in \(\mathbb{R}^{d}\) described by
$$ dX(t)=b(X)\,dt+\sum_{r=1}^{k} \sigma_{r}(X)\,dB_{r}(t). $$
(19)
The diffusion matrix is defined as follows:
$$ A(x)= \bigl(a_{ij}(x) \bigr),\quad a_{ij}(x)=\sum _{r=1}^{k}\sigma_{r}^{i}(x) \sigma_{r}^{j}(x). $$

Lemma 5.1

The Markov process \(X(t)\) has a unique ergodic stationary distribution \(\pi (\cdot )\) if there exists a bounded open domain \(U\subset \mathbb{R}^{d}\) with regular boundary Ū and the following hold:
  1. (i)

    There is a positive number ϖ such that \(\sum_{i,j=1} ^{d}a_{ij}(x)\xi_{i}\xi_{j}\geq \varpi \vert \xi \vert ^{2}\), \(x\in U\), \(\xi \in \mathbb{R}^{d}\) (see [26] and Rayleigh’s principle in [27]).

     
  2. (ii)
    There exists a non-negative \(C^{2}\)-function V such that LV is negative for every \(x\in \mathbb{R}^{d}\setminus U\) (see[28]). Let \(\rho (\cdot )\) be a function integrable with respect to the measure \(\pi (\cdot )\), then for all \(x\in \mathbb{R}^{d}\setminus U\)
    $$ \mathbb{P} \biggl\{ \lim_{T\rightarrow \infty }\frac{1}{T} \int_{0}^{T} \rho \bigl(X(t)\bigr)\,dt= \int_{\mathbb{R}^{d}}\rho (x)\pi (dx) \biggr\} =1. $$
     

The following result is concerned with the stationary distribution and ergodicity.

Theorem 5.2

Consider epidemic model (2) with the initial value \((S_{0}, I_{0}, R _{0}, Z_{0})\in \mathbb{R}_{+}^{4}\). If \(\mathscr{R}_{s}>1\), then there exists a stationary distribution \(\pi (\cdot )\), and it has the ergodic property.

Proof

Let \(\alpha_{1}\) and \(\alpha_{2}\) be sufficiently large numbers. Let
$$ U= \biggl\{ (x_{1}, x_{2}, x_{3}, x_{4})\in \Gamma:\frac{1}{\alpha_{1}}< x _{1}, x_{2}, x_{3}< \frac{\Lambda }{\mu }-\frac{1}{\alpha_{1}},\frac{1}{ \alpha_{2}}< x_{4}< \frac{a\Lambda }{a_{0}(\mu +b\Lambda )}-\frac{1}{ \alpha_{2}} \biggr\} . $$
We can write system (2) as the form of system (19) [25]:
$$ d \begin{pmatrix} S \\ I \\ R \\ Z \end{pmatrix} = \begin{pmatrix} \Lambda -\beta SI-\mu S-\mu_{1}mZS+\delta_{0}R \\ \beta SI-(\mu +\delta +\gamma )I \\ \gamma I+\mu_{1}mZS-(\mu +\delta_{0})R \\ \frac{aI}{1+bI}-a_{0}Z \end{pmatrix}\,dt + \begin{pmatrix} -\sigma_{1}\mu_{1}ZS \\ 0 \\ \sigma_{1}\mu_{1}ZS \\ 0 \end{pmatrix} dB_{1}(t) + \begin{pmatrix} 0 \\ -\sigma_{2}I \\ \sigma_{2}I \\ 0 \end{pmatrix}\,dB_{2}(t). $$
The diffusion matrix associated to epidemic model (2) is given by
$$ A(S, I, R, Z)= \begin{pmatrix} \sigma_{1}^{2}\mu_{1}^{2}Z^{2}S^{2} & 0 & -\sigma_{1}^{2}\mu_{1}^{2}Z ^{2}S^{2} & 0 \\ 0 & \sigma_{2}^{2}I^{2} & -\sigma_{2}^{2}I^{2} & 0 \\ -\sigma_{1}^{2}\mu_{1}^{2}Z^{2}S^{2} & -\sigma_{2}^{2}I^{2} & \sigma _{1}^{2}\mu_{1}^{2}Z^{2}S^{2}+\sigma_{2}^{2}I^{2} & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}. $$
Since \(\bar{U}\subset \mathbb{R}_{+}^{4}\) and \(\xi \in \mathbb{R}^{4} _{+}\setminus \{(\xi_{1},\xi_{2},\xi_{3})\in \mathbb{R}^{4}_{+}:\xi _{1}=\xi_{2}=\xi_{3}\}\), then there is a positive number C such that
$$\begin{aligned} \sum_{i,j=1}^{4}a_{ij} \xi_{i}\xi_{j} &=\sigma_{1}^{2}\mu_{1}^{2}Z^{2}S ^{2}\xi_{1}^{2} + \sigma_{2}^{2}I^{2}\xi_{2}^{2}+ \bigl(\sigma_{1}^{2}\mu _{1}^{2}Z^{2}S^{2}+ \sigma_{2}^{2}I^{2}\bigr)\xi_{3}^{2} \\ &\quad {}- 2\sigma_{1}^{2} \mu_{1}^{2}Z^{2}S^{2} \xi_{1}\xi_{3} - 2\sigma_{2}^{2}I^{2} \xi_{2}\xi _{3} \\ &=\sigma_{1}^{2}\mu_{1}^{2}Z^{2}S^{2}( \xi_{1}-\xi_{3})^{2} + \sigma _{2}^{2}I^{2}( \xi_{2}-\xi_{3})^{2} \\ &\geq C. \end{aligned}$$
Then condition (i) of Lemma 5.1 holds.
Consider the positive function \(V_{1}(I)\), \(V_{2}(S,I)\) and \(V_{3}(S,R,Z)\) defined for \((S, I, R, Z)\in \mathbb{R}_{+}^{4}\) by
$$\begin{aligned} &V_{1}(I)=\frac{1}{q}I^{-q}, \\ &V_{2}(S, I)=\frac{1}{q}I^{-q} \biggl( \frac{\Lambda }{\mu }-S \biggr) , \\ &V_{3}(S, R, Z)=\frac{1}{S}+R+\frac{1}{Z}, \end{aligned}$$
where q is a positive number to be determined later. Hence we define the Lyapunov function
$$\begin{aligned} V(S, I, R, Z)&= V_{1}(I)+V_{2}(S, I)+V_{3}(S, R, Z) \\ &= \frac{1}{q}I^{-q}+\frac{1}{q}I^{-q} \biggl( \frac{\Lambda }{\mu }-S \biggr) +\biggl( \frac{1}{S}+R+ \frac{1}{Z}\biggr). \end{aligned}$$
By Itô’s formula and system (2), we have
$$\begin{aligned} LV_{1}(I)&= -I^{-q}\bigl(\beta S-( \mu +\delta +\gamma )\bigr)+\frac{1}{2}(q+1) \sigma_{2}^{2}I^{-q} \\ &= I^{-q} \biggl( -(\mu +\delta +\gamma ) (\mathscr{R}_{s}-1)+ \frac{q}{2} \sigma_{2}^{2} \biggr) +I^{-q} \beta \biggl( \frac{\Lambda }{\mu }-S \biggr). \end{aligned}$$
(20)
Then we compute \(LV_{2}(S, I)\)
$$\begin{aligned} LV_{2}(S, I)&= I^{-q} \biggl( \frac{\Lambda }{\mu }-S \biggr) \biggl( \mu + \delta +\gamma -\beta S+ \frac{1}{2}(q+1)\sigma_{2}^{2} \biggr) \\ &\quad {}- \frac{1}{q}I^{-q}(\Lambda -\beta SI-\mu S-\mu_{1}mZS+ \delta_{0}R) \\ &\leq I^{-q} \biggl( \frac{\Lambda }{\mu }-S \biggr) \biggl( \mu +\delta + \gamma -\frac{\mu }{q}+\frac{1}{2}(q+1)\sigma_{2}^{2} \biggr) + \frac{1}{q}\mu_{1}mZSI^{-q} + \frac{\beta \Lambda }{q\mu }I^{-q}. \end{aligned}$$
(21)
For the function \(V_{3}(S, R, Z)\), we obtain that
$$\begin{aligned} LV_{3}(S, R, Z)&= -\frac{\Lambda }{S^{2}} + \frac{\beta I}{S} + \frac{ \mu }{S}+ \frac{\mu_{1}mZ}{S} - \frac{\delta_{0}R}{S^{2}} + \frac{ \sigma_{1}^{2}\mu_{1}^{2}Z^{2}}{S} + \gamma I + \mu_{1}mZS \\ &\quad {}-(\mu +\delta_{0})R-\frac{aI}{(1+bI)Z^{2}}+\frac{a_{0}}{Z} \\ &\leq -\frac{\Lambda }{S^{2}} + \biggl( \frac{\beta \Lambda }{\mu }+ \mu + \frac{\mu_{1}ma\Lambda }{a_{0}(\mu +b\Lambda )} + \frac{\sigma _{1}^{2}\mu_{1}^{2}a^{2}\Lambda^{2}}{a_{0}^{2}(\mu +b\Lambda )^{2}} \biggr) \frac{1}{S} + \frac{\gamma \Lambda }{\mu } \\ &\quad {}+ \frac{\mu_{1}ma\Lambda^{2}}{a_{0}\mu (\mu +b\Lambda )} - \frac{a \mu I}{(\mu +b\Lambda )Z^{2}}+\frac{a_{0}}{Z}. \end{aligned}$$
(22)
Combining (21)-(23), we have
$$\begin{aligned} LV&\leq I^{-q} \biggl( -(\mu +\delta +\gamma ) (\mathscr{R}_{s}-1)+ \frac{q}{2}\sigma_{2}^{2} \biggr) \\ &\quad {}+ I^{-q} \biggl( \frac{\Lambda }{\mu }-S \biggr) \biggl( \mu + \delta +\gamma +\beta -\frac{\mu }{q}+\frac{1}{2}(q+1) \sigma_{2}^{2} \biggr) \\ &\quad {}-\frac{\Lambda }{S^{2}}+ \biggl( \frac{\beta \Lambda }{\mu }+\mu +\frac{ \mu_{1}ma\Lambda }{a_{0}(\mu +b\Lambda )} + \frac{\sigma_{1}^{2}\mu _{1}^{2}a^{2}\Lambda^{2}}{a_{0}^{2}(\mu +b\Lambda )^{2}} \biggr) \frac{1}{S} + \frac{1}{q} \mu_{1}mZSI^{-q} + \frac{\beta \Lambda }{q \mu }I^{-q} \\ &\quad {}+ \frac{\gamma \Lambda }{\mu }+ \frac{\mu_{1}ma\Lambda^{2}}{a_{0} \mu (\mu +b\Lambda )} - \frac{a\mu I}{(\mu +b\Lambda )Z^{2}}+ \frac{a _{0}}{Z}. \end{aligned}$$
(23)
According to Theorem 4.1, for all \(t\geq T\), there exists a constant \(T(\omega )>0\) such that \(I(t)>\frac{H}{2}\) a.s. Therefore, rearranging the terms of (24), we get
$$\begin{aligned} LV&\leq I^{-q} \biggl( -(\mu +\delta +\gamma ) (\mathscr{R}_{s}-1)+ \frac{q}{2}\sigma_{2}^{2} \biggr) \\ &\quad {}+ I^{-q} \biggl( \frac{\Lambda }{\mu }-S \biggr) \biggl( \mu + \delta +\gamma +\beta -\frac{\mu }{q}+\frac{1}{2}(q+1) \sigma_{2}^{2} \biggr) \\ &\quad {} -\frac{\Lambda }{2S^{2}}-\frac{a\mu H}{4(\mu +b\Lambda )Z^{2}}+ \lambda , \end{aligned}$$
(24)
where
$$\begin{aligned} \lambda& = \sup_{(S,I,R,Z)\in \Gamma } \biggl\{ - \frac{\Lambda }{2S^{2}} + \biggl( \frac{\beta \Lambda }{\mu }+\mu +\frac{\mu_{1}ma\Lambda }{a _{0}(\mu +b\Lambda )}+ \frac{\sigma_{1}^{2}\mu_{1}^{2}a^{2}\Lambda^{2}}{a _{0}^{2}(\mu +b\Lambda )^{2}} \biggr) \frac{1}{S} +\frac{\gamma \Lambda }{\mu } \\ &\quad {}+ \frac{\beta }{q} \biggl( \frac{\Lambda }{\mu } \biggr) ^{2-q} + \frac{ \mu_{1}ma\Lambda^{2}}{a_{0}\mu (\mu +b\Lambda )} \biggl( 1+\frac{2^{q}}{qH ^{q}} \biggr) - \frac{a\mu H}{4(\mu +b\Lambda )Z^{2}} + \frac{a_{0}}{Z} \biggr\} . \end{aligned}$$
Since \(\mathscr{R}_{s}>1\), and choose q sufficiently small such that
$$\begin{aligned}& -(\mu +\delta +\gamma ) (\mathscr{R}_{s}-1)+ \frac{q}{2}\sigma_{2}^{2}< 0, \\& \mu +\delta +\gamma +\beta -\frac{\mu }{q}+\frac{1}{2}(q+1)\sigma _{2}^{2}< 0. \end{aligned}$$
On the other hand, \(\frac{\Lambda }{\mu +\delta }\leq S+I+R\leq \frac{ \Lambda }{\mu }\), \(0\leq Z\leq \frac{a\Lambda }{a_{0}(\mu +b\Lambda )}\), then for \((S, I, R, Z) \in \Gamma \setminus U\), either \(S<\frac{1}{\alpha_{1}}\), \(I<\frac{1}{ \alpha_{1}}\), \(R<\frac{1}{\alpha_{1}}\) or \(Z<\frac{1}{\alpha_{2}}\). It is easy to see from (24) that for sufficiently large \(\alpha_{1}\) or \(\alpha_{2}\),
$$ LV\leq -1 \quad \mbox{ for } (S, I, R, Z)\in \Gamma \setminus U. $$
Thus, the conditions of Lemma 5.1 are met. As a consequence, epidemic model (2) admits a unique ergodic invariant distribution \(\pi ( \cdot )\). By the ergodicity of \((S_{t}, I_{t}, R_{t}, Z_{t})\), we have
$$ \mathbb{P} \biggl\{ \lim_{t\rightarrow \infty }\frac{1}{t} \int_{0}^{t} \chi_{{(S_{s},I_{s},R_{s},Z_{s})\in \Gamma }}\,ds= \int_{\mathbb{R}^{4}} \chi_{\Gamma }\pi (dx) \biggr\} =1, $$
(25)
where \(\chi_{\Gamma }\) is the characteristic function of Γ. This completes the proof of Theorem 5.2. □

6 Numerical simulation

We have finished investigating the extinction and persistence of a disease. In order to illustrate the effectiveness of our results, now we will perform some numerical simulations. The numerical simulations are given by the Milstein scheme [29]. Consider the discretization equation of model (2):
$$ \textstyle\begin{cases} S_{k+1}=S_{k}+[\Lambda -\beta S_{k}I_{k}-\mu S_{k}-\mu_{1}mZ_{k}S_{k}+\delta_{0}R_{k}]\Delta t \\ \hphantom{S_{k+1}=}{}-\sigma_{1}\mu_{1}Z_{k}S_{k}\sqrt{\Delta t}\tau_{k} - \frac{\sigma_{1}^{2}}{2}\mu_{1}Z_{k}S_{k}(\tau _{k}^{2}-1)\Delta t, \\ I_{k+1}=I_{k}+[\beta S_{k}I_{k}-(\mu +\delta +\gamma )I_{k}]\Delta t-\sigma_{2}I_{k}\sqrt{\Delta t}\tau_{k} - \frac{\sigma_{2}^{2}}{2}I _{k}(\tau_{k}^{2}-1)\Delta t, \\ R_{k+1}=R_{k}+[\gamma I_{k}+\mu_{1}mZ_{k}S_{k}-(\mu +\delta_{0})R_{k}]\Delta t \\ \hphantom{R_{k+1}=}{}+\sigma_{1}\mu_{1}Z_{k}S_{k}\sqrt{\Delta t}\tau_{k} + \frac{ \sigma_{1}^{2}}{2}\mu_{1}Z_{k}S_{k}(\tau_{k}^{2}-1)\Delta t+\sigma_{2}I_{k}\sqrt{\Delta t}\tau_{k} + \frac{\sigma_{2}^{2}}{2}I_{k}(\tau_{k}^{2}-1)\Delta t, \\ Z_{k+1}=Z_{k}+ [ \frac{aI_{k}}{1+bI_{k}}-a_{0}Z_{k} ] \Delta t, \end{cases} $$
where \(\tau_{k}\) \((k=1,2,\ldots)\) are \(N(0,1)\)-distributed independent random variables. In Figures 1-4, we choose the parameter values in model (2) as follows:
$$\begin{aligned}& \Lambda =2.8, \qquad \beta =0.002, \qquad \mu =0.02, \qquad \delta_{0}=0.01, \qquad \delta = 0.1, \qquad \gamma =0.1, \\& \mu_{1}=0.0 \mbox{ to } 1.0,\qquad m=0.017, \qquad a=0.01, \qquad a_{0}=0.045,\qquad b=1.0. \end{aligned}$$
Figure 1

The path of \(\pmb{S(t)}\) , \(\pmb{I(t)}\) , \(\pmb{R(t)}\) for deterministic model ( 1 ) with initial \(\pmb{(S_{0},I_{0},R_{0},Z_{0})=(479.0,20.0,1.0,10.0)}\) .

The initial population size as: \(S(0)=479\), \(I(0)=20\), \(R(0)=1\), \(Z(0)=10\). We calculate the basic reproduction number \(\mathscr{R}_{0}=\frac{\Lambda \beta }{\mu (\mu +\gamma +\delta )}=1.2727>1\), the disease-free equilibrium \(E_{0}=(140.0000,0.0000,0.0000,0.0000)\) and the infected equilibrium \(E^{*}=(110.0000,3.1021,11.3877,0.1680)\). For a clear comparison with the path of epidemic model (2), we show the path of \(S(t)\), \(I(t)\), \(R(t)\) for deterministic model (1) in Figure 1.

Example 6.1

(Stochastic endemic dynamics)

In Figure 2, we choose \(\sigma_{1}=0.1\), \(\sigma_{2}=0.1\), note that \(\mathscr{R}_{s}=1.2500>1\). By Theorem 4.1, the disease will prevail, and we give the simulations to support our results in Figure 2. Comparing the first picture, with the noise getting smaller, the fluctuation of the solution of model (2) is getting weaker. If we increase \(\sigma_{1}\) and \(\sigma_{2}\) to 0.2 (\(\mathscr{R}_{s}=1.1818>1\)) and 0.3 (\(\mathscr{R}_{s}=1.0682>1\)), the amplitude of fluctuation becomes stronger. Running 10,000 numerical simulations, we get the histogram of probability density function for \(I(150)\). As we can see, with \(\sigma_{1}\) and \(\sigma_{2}\) increasing, the distribution of \(I(t)\) becomes skew. That is to say, noise intensities have great effect on the solution of \(I(t)\).
Figure 2

The path of \(\pmb{S(t)}\) , \(\pmb{I(t)}\) , \(\pmb{R(t)}\) for model ( 2 ) and the histogram of the probability density function of \(\pmb{I(150)}\) with initial \(\pmb{(S_{0},I_{0},R_{0},Z_{0})=(479.0,20.0,1.0,10.0)}\) under different noise intensities.

Example 6.2

(Stochastic disease-free dynamics)

Throughout the paper we shall assume that the unit of time is one day and the population sizes are measured in units of 1 million. We choose \(\sigma_{1}=0.1\), \(\sigma_{2}=0.35\), then by Theorem 3.1, \(\mathscr{R} _{s}=0.9943<1\). Then \(I(t)\) will tend to zero exponentially with probability one, we give the simulations shown in Figure 3 to support our results. We increase \(\sigma_{1}\) to 0.2 and 0.3, \(\sigma_{2}\) to 0.38 (\(\mathscr{R}_{s}=0.9445<1\)) and 0.4 (\(\mathscr{R}_{s}=0.9091<1\)), respectively. As a result, \(I(t)\) will tend to zero exponentially with probability one. That is to say, large noises can lead the disease to extinction, which is a phenomenon different from its corresponding deterministic model (1).
Figure 3

The path of \(\pmb{S(t)}\) , \(\pmb{I(t)}\) , \(\pmb{R(t)}\) for model ( 2 ) with initial \(\pmb{(S_{0},I_{0},R_{0},Z_{0})=(479.0,20.0,1.0,10.0)}\) under different noise intensities.

Running 10,000 numerical simulations, we obtain the average extinction time of \(I(t)\). The average extinction time for different noise intensities is 132.5930, 125.9268 and 119.3161, respectively. Therefore, we can assert that the average extinction time of disease decreases with the increase of noises.

Example 6.3

(Effect of various parameter \(\mu_{1}\))

In the following, we consider the effect of response intensity \(\mu_{1}\). Choosing \(\sigma_{1}=0\), \(\sigma_{2}=0\), from Figure 4, \(\mu_{1}\) has large influence on \(I(t)\), we can see that the number of infected individuals decreases with the increase of \(\mu_{1}\). Increase \(\sigma_{1}\) to 0.1 and 0.2, \(\sigma_{2}\) to 0.15 and 0.3, make 10,000 numerical simulation runs, then calculate mean value. Similarly, the increase \(\mu_{1}\) can reduce the peak value of \(I(t)\). Therefore, we can assert that information intervention can help in reducing the peak of infective population.
Figure 4

The path of \(\pmb{I(t)}\) for model ( 2 ) with initial \(\pmb{(S_{0},I_{0},R _{0},Z_{0})=(479.0,20.0,1.0,10.0)}\) under different noise intensities

7 Concluding remarks

In this paper, we have investigated the dynamic behavior of a new SIRS epidemic model which considers the influence of information intervention and environmental noise. It has been found that information intervention and white noise have great effects on the disease. Our main results can be stated as follows:
  1. (i)

    We have considered the effects of environmental white noise on the disease. Denote stochastic reproduction number \(\mathscr{R} _{s}=\mathscr{R}_{0}-\frac{\sigma_{2}^{2}}{2(\mu +\delta +\gamma )}\), we have proved that the \(\mathscr{R}_{s}\) is a threshold of model (2) for the disease to die out or persist, and noise intensities can change the value of the stochastic reproduction number \(\mathscr{R} _{s}\). If \(\mathscr{R}_{s}<1\), the disease will die out with probability one. On the other hand, if \(\mathscr{R}_{s}>1\), there is a stationary distribution for model (2), which means the disease will prevail.

     
  2. (ii)

    As an important non-pharmaceutical measure, the information intervention has a great impact on the spread of disease. It can be seen from the constant \(H=\frac{a_{0}\mu^{2}(\mu +\delta +\gamma )( \mathscr{R}_{s}-1)}{\beta \Lambda (a_{0}\beta +a\mu_{1}m)}\) (see Theorem 4.1) and it can help reduce the peak of infective population (see Figure 4). At the end of the paper, we have illustrated that information intervention can help reduce the peak of infective population and large noises can lead the disease to extinction by numerical simulations.

     
There are still a lot of interesting issues that we are going to deal with later. For example, the sudden climate change, weather warming or cooling, wetting or drying may affect the spread of the disease. Therefore, when the discontinuous random process is added to model (2), such as Lévy noise, how does it impact the spread of disease, we will study this issue later.

Declarations

Acknowledgements

The research was funded by the ‘Major Innovation Projects for Building First-class Universities in China’s Western Region’ (ZKZD2017009).

Authors’ contributions

All authors 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)
School of Mathematics and Statistics, Ningxia University

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