Extinction and stationary distribution of a novel SIRS epidemic model with general incidence rate and Ornstein–Uhlenbeck process

Abstract

We propose, in this paper, a novel stochastic SIRS epidemic model to characterize the effect of uncertainty on the distribution of infectious disease, where the general incidence rate and Ornstein–Uhlenbeck process are also introduced to describe the complexity of disease transmission. First, the existence and uniqueness of the global nonnegative solution of our model is obtained, which is the basis for the discussion of the dynamical behavior of the model. And then, we derive a sufficient condition for exponential extinction of infectious diseases. Furthermore, through constructing a Lyapunov function and using Fatou’s lemma, we obtain a sufficient criterion for the existence and ergodicity of a stationary distribution, which implies the persistence of the disease. In addition, the specific form of the density function of the model near the quasiendemic equilibrium is proposed by solving the corresponding Fokker–Planck equation and using some relevant algebraic equation theory. Finally, we explain the above theoretical results through some numerical simulations.

1 Introduction

According to the World Health Organisation (WHO), infectious diseases are a major cause of global health losses and the second leading cause of human mortality [1]. The epidemics of infectious diseases pose serious threats to human life, health and safety, cause huge losses to people’s property, hinder the country’s social and economic development, and take a serious challenge to national public health security. By establishing reasonable mathematical models of these infectious diseases and analyzing the properties of these models, we can accurately reveal the epidemiological pattern, identify the main causes of the diseases, and explore the interrelationships among the factors, etc. In addition, analyzing infectious disease models can predict the development trend of the disease and provide theoretical guidance for seeking the best strategy to control and prevent the spread and outbreak of these diseases. Nowadays, mathematical models have become an important tool for both qualitative and quantitative analyses of the spread and control of infectious diseases.

Since 1927, the classical SIR epidemic model and the corresponding threshold theory were proposed by Kermack and McKendrick [2], various deterministic mathematical models (the main reference here is to the ordinary differential equation models) and theoretical approaches [3–6] have been proposed by many scholars to study the dynamical behavior of infectious diseases and the epidemiological characteristics. In the process of establishing a model of disease transmission, the incidence rate is a very important parameter, which represents the number of new cases per unit time. The bilinear incidence rate [4] is often used in describing highly infectious diseases, such as influenza and SARS. However, due to the complexity of disease transmission, the number of susceptible individuals does not always increase linearly with the number of infected individuals. Therefore, it is necessary to investigate other incidence rate models, including the saturation incidence rate, the semi-saturated incidence rate, the general incidence rate, among others [5, 6].

Recently, May [7] has pointed out that, due to random disturbances in the environment, stochastic noise can affect the growth rate, environmental capacity and other parameters of the system to varying degrees. Moreover, the transmission of infectious disease is influenced by several random factors, including weather factors, government policies, psychological factors, and population movements. Therefore, in order to investigate the dynamical properties in different environments, many researchers have introduced random perturbations into their models, creating stochastic epidemic models, and have studied the dynamic behavior of these models [8, 9]. Currently, there are two main forms of infectious disease models described by stochastic differential equations. One form is to assume that the parameter such as β (the transmission rate of pathogen from infected to susceptible individuals) is perturbed by a standard Gaussian white noise, that is, $\beta (t)\sim \mathcal{N}(\bar{\beta}, \frac{\sigma ^{2}}{t})$, which can be described by the equation $\mathrm{d}\beta (t)=\bar{\beta} \mathrm{{d}t}+\mathrm{\sigma \mathrm{d}B(t)}$, see [10, 11] for more details. Many papers and references [12–18] therein adopt this method to simulate the environment noise, so as to better approach the reality of biological systems, gain insight into the impact of environmental change on the ecosystem, and obtain some good results. Of course, this assumption has some unrealistic aspects, for example, as the time interval shortens, the variance of the average value of the corrected parameters becomes infinite, which is inconsistent with reality.

Another form is the hypothesis that the parameter like β follows a mean-reverting process [19] (Ornstein–Uhlenbeck process), that is, $\mathrm{d}\beta (t)=k(\bar{\beta}-\beta (t))\mathrm{d}t+\sigma \mathrm{d}B(t)$. It is straightforward to show that the expected value of $\beta (t)$ is $E(\beta (t))=\bar{\beta}+(\beta ^{0}-\bar{\beta}){\text{e}}^{-kt}$ and the variance value of $\beta (t)$ is $\mathrm{Var}(\beta (t))=\frac{\sigma ^{2}}{2k}(1-{\text{e}}^{-2kt})$. This means $E(\beta (t))\rightarrow \beta ^{0}$, $\mathrm{Var}(\beta (t))\rightarrow 0$ as $t \rightarrow 0$. Obviously, this method can effectively avoid the unboundedness problem as well and obtain more realistic results. Furthermore, Cai et al. [20] introduced the Ornstein–Uhlenbeck process into a stochastic infectious disease model. They found that, compared to the linear function of white noise, the Ornstein–Uhlenbeck process exhibits characteristics such as nonnegativity, practicality, continuity, and asymptotic distribution. These characteristics enable it to better characterize environmental changes in biological systems. Therefore, the mean-reverting process is considered a more reasonable perturbation method than ordinary linear white noise perturbation, from both the biological and mathematical perspectives. In the context of infectious disease modeling, it can effectively capture the temporal dynamics of the transmission rate, leading to a more detailed exploration of the disease dynamics. Very recently, some scholars have developed some stochastic epidemic models with Ornstein–Uhlenbeck process to simulate the environment noise and thus explain the complex of the transmission and control of infectious diseases [21–28].

Inspired by the work described above, in this paper, an SIRS epidemic model with general incidence rate and Ornstein–Uhlenbeck process is proposed to discuss the effects of environment noise on the distribution of a disease. The rest of the paper is arranged as follows. The epidemic model and two useful lemmas are introduced, and the existence and uniqueness of global nonnegative solution of our model is obtained in Sect. 2. The extinction and persistence of disease are discussed respectively in Sect. 3. In Sect. 4, using the four-dimensional Fokker–Planck equation and some solution theory for the relevant algebraic equations, the specific form of the probability density function around the quasiequilibrium is calculated. In Sect. 5, the theoretical results are explained through some numerical examples and simulations, and the effect of random disturbances is analyzed. Finally, the corresponding conclusion and discussion are given.

2 Model formulation and preliminaries

Based on the pattern of transmission of pathogens between humans, people in a certain area are divided into three classes: susceptible, infected, and recovered individuals, which are expressed by $S(t)$, $I(t)$, and $R(t)$, respectively. Additionally, Λ denotes the recruitment rate of individuals, β represents the transmission rate among the susceptible and infected individuals. The natural death rate is μ and α denotes the death rate due to the disease of infected, γ indicates the recovery rate of the infected individuals and δ is the rate at which the recovered population loses immunity and returns to the susceptible group. In addition, to enhance the applicability of the model, we employ the general incidence rate $\beta Sg(I)$ to describe new infected patients per unit of time. From the above assumptions, the model reads as

$$ \left \{ \begin{aligned} &\frac{\mathrm{d}S}{\mathrm{d}t}=\Lambda -\beta Sg(I)- \mu S+\delta R, \\ &\frac{\mathrm{d}I}{\mathrm{d}t}=\beta Sg(I)- (\mu +\alpha +\gamma ) I, \\ &\frac{\mathrm{d}R}{\mathrm{d}t}=\gamma I-(\mu +\delta )R, \end{aligned} \right . $$

(1)

where Λ, β, μ, α, γ, and δ are positive constants, and function $g(I)$ satisfies the following three conditions for any $I\in [0, +\infty )$: $(i)$ $g(0)=0$; $(ii)$ $g'(0)>0$; $(iii)$ $0\leq (\frac{I}{g(I)})'\leq m$, where m is a positive constant.

By using the next-generation matrix formulated in [29], the basic reproduction number of model (1) is

$$ \mathcal{R}_{0}= \frac{\beta \Lambda g'(0)}{\mu (\mu +\gamma +\alpha )}. $$

(2)

Obviously, model (1) has the disease-free equilibrium $P^{0}=(S^{0},I^{0},R^{0})=(\frac{\Lambda}{\mu},0,0)$ and the endemic equilibrium $P^{*}=(S^{*},I^{*},R^{*})$ for $\mathcal{R}_{0}>1$.

Due to various forms of random disturbance in the environment, stochastic models can more accurately describe the properties of infectious diseases. Based on the research conclusions of Cai et al. [20], we assume that the key parameter β in our model (1) is fluctuated by the Ornstein–Uhlenbeck process as follows:

$$ \mathrm{d}\beta =k(\bar{\beta}-\beta )\mathrm{d}t+\sigma \mathrm{d}B(t), $$

where k represents the speed of reversion, $\sigma ^{2}$ denotes the noise intensity, $\bar{\beta}>0$ is the long-time average level of β. Let us define $\beta ^{+}:=\max \{0,\beta \}$ [22, 23] to ensure that the parameter β is nonnegative. Then model (1) can be described as follows:

$$ \left \{ \begin{aligned} &\mathrm{d}S=(\Lambda -\beta ^{+} Sg(I)- \mu S+\delta R)\mathrm{d}t, \\ &\mathrm{d}I=(\beta ^{+} Sg(I)- (\mu +\alpha +\gamma ) I)\mathrm{d}t, \\ &\mathrm{d}R=(\gamma I-(\mu +\delta )R)\mathrm{d}t, \\ &\mathrm{d}\beta =k(\bar{\beta}-\beta )\mathrm{d}t+\sigma \mathrm{d}B(t). \end{aligned} \right . $$

(3)

Let $(\Omega ,~\mathcal{F},~\{\mathcal{F}_{t}\}_{t\geq 0},~\text{P})$ be a complete probability space with a filtration $\{\mathcal{F}_{t}\}_{t\geq 0}$ satisfying the usual conditions, while $B(t)$ is a standard Brownian motion defined on this space. Denote $\mathbb{R}^{n}=\{(x_{1},x_{2},\dots ,x_{n})\mid x_{i}\in \mathbb{R},i=1,2, \dots , n\}$ and $\mathbb{R}^{n}_{+}=\{(x_{1},x_{2},\dots ,x_{n})\mid x_{i}>0,i=1,2, \dots , n\}$. Let $\mathbf{I}_{A}$ be the indicator function of the set A. Also denote $\hat{a}\wedge \hat{b}:=\min \{\hat{a},\hat{b}\}$ and $\check{a}\vee \check{b}:=\max \{\check{a},\check{b}\}$ for $\check{a}, \check{b}\in \mathbb{R}$.

Now, we will prove the globality and positivity of the solution of model (3).

Theorem 1

For any initial value $X(0)=(S(0), I(0), R(0),\beta (0))\in \mathbb{R}^{3}_{+}\times \mathbb{R}$, there is a unique solution $X(t)=(S(t), I(t), R(t),\beta (t))$ of model (3) for $t\geq 0$, and the solution will remain in $\mathbb{R}^{3}_{+}\times \mathbb{R}$ with probability one.

Proof

For any initial value $X(0)=(S(0), I(0), R(0),\beta (0))\in \mathbb{R}^{3}_{+} \times \mathbb{R}$, all the coefficients of model (3) are locally Lipschitz continuous, so there is a unique local solution $X(t)=(S(t), I(t), R(t),\beta (t))$ on $t\in [0, \tau _{e})$, where $\tau _{e}$ is the explosion time [10]. In order to establish that this solution is global, one only need to show that $\tau _{e}=+\infty$ almost surely (a.s.). Set $l_{0}\geq 0$ be large enough to ensure that $S(0)$, $I(0)$, $R(0)$, and ${\mathrm{e}}^{\beta (0)}$ all lie within the interval $[\frac{1}{ l_{0}}, l_{0}]$. Let us define the stopping time $\tau _{l}$ for each integer $l>l_{0}$ as

$$ \tau _{l}=\inf \left \{t\in [0,\tau _{e}): \max \{S(t),I(t), R(t),{ \mathrm{e}}^{\beta (t)}\} \geq l ~\text{or}~\min \{S(t), I(t), R(t),{\mathrm{e}}^{ \beta (t)}\}\leq \frac{1}{l} \right \}, $$

and set $\inf \emptyset =+\infty$. Apparently, $\tau _{l}$ is increasing as $l\rightarrow +\infty$. Let $\tau _{\infty }= \lim _{l\rightarrow + \infty }\tau _{l}$, then $\tau _{\infty}\leq \tau _{e}$ a.s. If $\tau _{\infty }=+\infty$ a.s., then one can get that $\tau _{e}=+\infty$ a.s. and $(S(t), I(t), R(t),\mathrm{e}^{\beta (t)})\in \mathbb{R}^{3}_{+} \times \mathbb{R}$ a.s. for all $t>0$. If $\tau _{e}<+\infty$, then there exists a pair of constants $T_{0}>0$ and $\varepsilon _{0} \in (0, 1)$ such that $\mathrm{P}\{\tau _{\infty}\leq T_{0}\}>\varepsilon _{0}$. Hence there is an integer $l_{1} > l_{0}$ such that

$$\begin{aligned} \mathrm{P}\{\tau _{l}\leq T_{0}\}\geq \varepsilon _{0} ,~~~ \text{for all}~ l>l_{1}. \end{aligned}$$

(4)

For $t\leq \tau _{l}$, one can see for each l that

$$ {\mathrm{d}}(S + I + R) = [\Lambda - \mu (S + I + R)-\alpha I] {\mathrm{d}}t \leq [\Lambda -\mu (S + I +R)]{\mathrm{d}}t, $$

and

$$ S + I + R\leq \left \{ \textstyle\begin{array}{ll} \frac{\Lambda}{\mu},&if ~S(0)+I(0)+R(0)\leq \frac{\Lambda}{\mu}, \\ S(0)+I(0)+R(0),& if ~S(0)+I(0)+R(0)>\frac{\Lambda}{\mu}. \end{array}\displaystyle \right . $$

(5)

Define a $C^{2}$-function $U : \mathbb{R}^{4}_{+} \rightarrow \mathbb{R}_{+}$ by

$$ U=\Psi (S)+\Psi (I)+\Psi (R)+\frac{1}{2}\beta ^{2}, $$

where $\Psi (u)=u-1-\ln u>0$ for all $u>0$. Using Itô’s formula [10], one can obtain $\mathrm{d}U=\mathcal{L}U\mathrm{d}t+\sigma \beta \mathrm{d}B(t)$, where

$$\begin{aligned} \mathcal{L}U &=\Lambda -\beta ^{+} S g(I)-\mu S+\delta R- \frac{\Lambda}{S}+\beta ^{+}g(I)+\mu -\delta \frac{R}{S}+\beta ^{+}Sg(I)-( \mu +\gamma +\alpha )I \\ &\quad -\beta ^{+}S\frac{g(I)}{I}+(\mu +\gamma +\alpha )+\gamma I-( \mu +\delta )R-\gamma \frac{I}{R}+ \mu +\delta +k (\bar{\beta} - \beta )\beta +\frac{1}{2}\sigma ^{2} \\ &\leq \Lambda +\beta ^{+}g(I)+3\mu +\alpha +\gamma +\delta + \frac{1}{2} \sigma ^{2}+k(\bar{\beta}-\beta )\beta \\ &\leq \Lambda +|\beta |g(I)+3\mu +\alpha +\gamma +\delta +\frac{1}{2} \sigma ^{2}+k\beta \bar{\beta} -k\beta ^{2}. \end{aligned}$$

(6)

Note that for any $I>0$, $\big(\frac{I}{g(I)}\big)'\geq 0$, which means $\frac{I}{g(I)}$ is an increasing function. From this condition, we conclude that $g(I)\leq g'(0)I$. Combining with (5), inequality (6) can be written as follows:

$$ \mathcal{L}U \leq \Lambda +|\beta |g'(0)K+3\mu +\alpha +\gamma + \delta +\frac{1}{2}\sigma ^{2}+k\beta \bar{\beta} -k\beta ^{2}. $$

Letting $A=\sup _{\beta \in \mathbb{R}}\{k\bar{\beta} \beta -k\beta ^{2}+| \beta |g'(0)K\}$, where $K:=\max \big\{\frac{\Lambda}{\mu}, S(0)+I(0)+R(0)\big\}$, we obtain

$$\begin{aligned} \mathcal{L}U\leq \Lambda +3\mu +\alpha +\gamma +\delta +\frac{1}{2} \sigma ^{2}+A:=M, \end{aligned}$$

where M is a positive constant, independent of S, I, and R. Then

$$ \mathrm{d}U\leq M\mathrm{d}t+\sigma \beta \mathrm{d}B(t). $$

(7)

Integrating both sides of the above equation (7) from 0 to $\tau _{l}\wedge T_{0}$ and taking expectation, one gets

$$ \mathbb{E}U(S(\tau _{l}\wedge T_{0}),I(\tau _{l}\wedge T_{0}),R(\tau _{l} \wedge T_{0}),\beta (\tau _{l}\wedge T_{0}))\leq U(S(0),I(0),R(0), \beta (0))+MT_{0}. $$

For $l\geq l_{1}$, let $\Omega _{l}=\{\tau _{l}\leq T_{0}\}$, and then from (4) one obtains $\mathrm{P}(\Omega _{l})\geq \varepsilon _{0}$ with $\varepsilon _{0}\in (0,1)$. According to the definition of the stopping time, we can get that at least one of $S(\tau _{l}, \omega )$, $I(\tau _{l}, \omega )$, $R(\tau _{l}, \omega )$, $\beta (\tau _{l}, \omega )$ is equal to either to l or $\frac{1}{l}$ for any $\omega \in \Omega _{l}$, thus

$$\begin{aligned} &\mathbb{E}[U(S(\tau _{l}\wedge T_{0}),I(\tau _{l}\wedge T_{0}),R( \tau _{l}\wedge T_{0})),\beta (\tau _{l} \wedge T_{0})] \\ &=\mathbb{E}[\mathbf{I}_{\Omega _{l}}U(S(\tau _{l}, \omega ),I(\tau _{l}, \omega ),R(\tau _{l}, \omega ),\beta (\tau _{l}, \omega ))] \\ &\geq \varepsilon _{0}\left ((l-1-{\mathrm{\ln}} l )\wedge \left ( \frac{1}{l}-1+{\mathrm{\ln}} l\right )\wedge \frac{({\mathrm{\ln }} l)^{2}}{2} \right ), \end{aligned}$$

Then taking $l\rightarrow +\infty$, one obtains

$$ +\infty > U(S(0),I(0),R(0),\beta (0))+M T_{0}=+\infty , $$

which is a contradiction. Hence, we conclude that $\tau _{\infty }=+\infty$ a.s. □

Remark 1

From Theorem 1, for any initial value $X(0)=(S(0),I(0),R(0),\beta (0))\in \Gamma ^{*}$, there exists a unique global solution $X(t)= (S(t),I(t),R(t),\beta (t))$ that remains in $\Gamma ^{*}$ almost surely, where $\Gamma ^{*}: =\big\{(S,I,R,\beta )\in \mathbb{R}^{3}_{+}\times \mathbb{R}: \ S+I+R\leq \frac{\Lambda}{\mu}\big\}$.

Consider the stochastic differential equation

$$ \mathrm{d}Z(t)=f(Z(t))\mathrm{d}t+g(Z(t))\mathrm{d}\mathbf{B}(t), \quad Z(0)\in \mathbb{R}^{n}, $$

(8)

where $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ and $g:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n\times d}$ are Borel measurable, $\mathbf{B}(t)$ is a standard n-dimensional Brownian motion. Lemmas 2.1 and 2.2 provide relative theory for the existence of a stationary distribution and probability density function of the model.

Lemma 2.1

([30, 31])

Assume that there exists a bounded closed domain $\mathbb{A}\subset \mathbb{R}^{n}$ with a regular boundary Γ such that, for any initial value $Z(0)\in \mathbb{R}^{n}$,

$$ \liminf _{t\rightarrow +\infty}\frac{1}{t}\int ^{t}_{0}\mathbb{P}(s,Z(s), \mathbb{A})\mathrm{d}s>0 ~\textit{a.s.}, $$

where $\mathbb{P}(s,Z(s),\cdot )$ represents the transition probability of $Z(t)$. Then the solution of system (8) admits the Feller property, and there is at least one stationary distribution $\eta (\cdot )$ on $\mathbb{R}^{n}$.

Lemma 2.2

([32])

Consider the four-dimensional real algebraic equation $\Theta ^{2}+B\Sigma +\Sigma B^{T}=0$, where $\Theta =\mathrm{diag}(1,0,0,0)$, and B is a standard matrix given by

$$ {B=} \begin{pmatrix} -\varrho _{1}&-\varrho _{2}&-\varrho _{3}&-\varrho _{4} \\ 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \end{pmatrix} . $$

If $\varrho _{1}>0$, $\varrho _{3}>0$, $\varrho _{4}>0$, and $\varrho _{1}\varrho _{2}\varrho _{3}-\varrho ^{2}_{3}-\varrho ^{2}_{1} \varrho _{4}>0$, then Σ is a positive definite matrix having the form

$$ {\Sigma =} \begin{pmatrix} \frac{\varrho _{2}\varrho _{3}-\varrho _{1}\varrho _{4}}{2(\varrho _{1}\varrho _{2}\varrho _{3}-\varrho ^{2}_{3}-\varrho ^{2}_{1}\varrho _{4})}&0&- \frac{\varrho _{3}}{2(\varrho _{1}\varrho _{2}\varrho _{3}-\varrho ^{2}_{3}-\varrho ^{2}_{1}\varrho _{4})}&0 \\ 0& \frac{\varrho _{3}}{2(\varrho _{1}\varrho _{2}\varrho _{3}-\varrho ^{2}_{3}-\varrho ^{2}_{1}\varrho _{4})}&0&- \frac{\varrho _{1}}{2(\varrho _{1}\varrho _{2}\varrho _{3}-\varrho ^{2}_{3}-\varrho ^{2}_{1}\varrho _{4})} \\ - \frac{\varrho _{3}}{2(\varrho _{1}\varrho _{2}\varrho _{3}-\varrho ^{2}_{3}-\varrho ^{2}_{1}\varrho _{4})}&0& \frac{\varrho _{1}}{2(\varrho _{1}\varrho _{2}\varrho _{3}-\varrho ^{2}_{3}-\varrho ^{2}_{1}\varrho _{4})}&0 \\ 0&- \frac{\varrho _{1}}{2(\varrho _{1}\varrho _{2}\varrho _{3}-\varrho ^{2}_{3}-\varrho ^{2}_{1}\varrho _{4})}&0& \frac{\varrho _{1}\varrho _{2}-\varrho _{3}}{2\varrho _{4}(\varrho _{1}\varrho _{2}\varrho _{3}-\varrho ^{2}_{3}-\varrho ^{2}_{1}\varrho _{4})} \end{pmatrix} . $$

3 Extinction and stationary distribution

First, we give the threshold for the extinction of the disease in this section, and then prove the existence of the stationary distribution of model (3). Define

$$ \mathcal{R}_{0}^{e}= \frac{\bar{\beta}\Lambda g'(0)}{\mu (\mu +\gamma +\alpha )}+ \frac{ g'(0)\Lambda \sigma}{\mu (\mu +\gamma +\alpha )\sqrt{\pi k}}. $$

Theorem 2

Let $X(t)=(S(t),I(t),R(t),\beta (t))$ be a solution of the model (3) with initial value $X(0)=(S(0),I(0),R(0),\beta (0))\in \Gamma ^{*}$. If $\mathcal{R}_{0}^{e}<1$, then

$$ \limsup _{t\rightarrow +\infty}\frac{{\ln I(t)}}{t} \leq (\mathcal{R}^{e}_{0}-1)( \mu +\gamma +\alpha )< 0 ~\textit{a.s.}, $$

that is, the disease will go extinct with probability one.

Proof

Define a $C^{2}$-function V as $V(I)=\ln I$. Then, applying the Itô’s formula, one obtains

$$ \begin{aligned} \mathrm{d}V&=\frac{1}{I}[\beta ^{+}Sg(I)-(\mu +\gamma + \alpha )I]\mathrm{d}t \\ &\leq \left (\beta ^{+}\frac{\Lambda}{\mu}g'(0)-(\mu +\gamma +\alpha ) \right )\mathrm{d}t \\ &\leq \left (\bar{\beta}\frac{\Lambda}{\mu}g'(0)-(\mu +\gamma + \alpha )+\frac{\Lambda}{\mu}g'(0)(\beta ^{+}-\bar{\beta})\right ) \mathrm{d}t. \end{aligned} $$

(9)

Note that $\beta ^{+}=\frac{|\beta |+\beta}{2}$, which yields

$$ \beta ^{+}-\bar{\beta}= \frac{|\beta |-\bar{\beta}+\beta -\bar{\beta}}{2}\leq |\beta - \bar{\beta}|. $$

(10)

Combining with (10), (9) can be written as follows:

$$\begin{aligned} \mathrm{d}V\leq \left (\bar{\beta}\frac{\Lambda}{\mu}g'(0)-(\mu + \gamma +\alpha )+\frac{\Lambda}{\mu}g'(0)|\beta -\bar{\beta}|\right ) \mathrm{d}t. \end{aligned}$$

(11)

Integrating (11) from 0 to t, then dividing by t on both sides, the above yields that

$$\begin{aligned} \frac{{\ln I(t)-\ln I(0)}}{t}\leq \bar{\beta}\frac{\Lambda}{\mu}g'(0)-( \mu +\gamma +\alpha )+\frac{\Lambda}{\mu}g'(0)\frac{1}{t}\int _{0}^{t}| \beta (\theta )-\bar{\beta}|{\mathrm{d}}\theta . \end{aligned}$$

(12)

Because of

$$ \begin{aligned} &\lim _{t\rightarrow +\infty}\mathbb{E}\frac{1}{t}\int ^{t}_{0}| \beta (\theta )-\bar{\beta} |\mathrm{d}\theta \\ &=\int _{-\infty}^{+\infty}|x-\bar{\beta} |\pi (x)\mathrm{d}x \\ &=\int _{-\infty}^{+\infty}|x-\bar{\beta} | \frac{\sqrt{k}}{\sqrt{\pi}\sigma}\exp \left \{- \frac{k(x-\bar{\beta} )^{2}}{\sigma ^{2}}\right \}\mathrm{d}x \\ &=\int _{-\infty}^{\bar{\beta} }(\bar{\beta} -x) \frac{\sqrt{k}}{\sqrt{\pi}\sigma}\exp \left \{- \frac{k(x-\bar{\beta} )^{2}}{\sigma ^{2}}\right \}\mathrm{d}x \\ &\quad +\int _{\bar{\beta} }^{+\infty}(x-\bar{\beta} ) \frac{\sqrt{k}}{\sqrt{\pi}\sigma}\exp \left \{- \frac{k(x-\bar{\beta} )^{2}}{\sigma ^{2}}\right \}\mathrm{d}x \\ &=-\int ^{0}_{-\infty} y\frac{\sqrt{k}}{\sqrt{\pi}\sigma}\exp \left \{-\frac{ky^{2}}{\sigma ^{2}}\right \}\mathrm{d}y+\int ^{+\infty}_{0}y \frac{\sqrt{k}}{\sqrt{\pi}\sigma}\exp \left \{- \frac{ky^{2}}{\sigma ^{2}}\right \}\mathrm{d}y \\ &=\frac{\sigma}{\sqrt{\pi k}}\quad \text{a.s.}, \end{aligned} $$

(13)

where $\pi (x)=\frac{\sqrt{k}}{\sqrt{\pi}\sigma}\exp \left \{- \frac{k(x-\bar{\beta})^{2}}{\sigma ^{2}}\right \}$ is the invariant density of $\beta (t)$. Taking the superior limit on each side of the inequality (12), and combining with (13), we have

$$\begin{aligned} \limsup _{t\rightarrow +\infty}\frac{{\ln I(t)}}{t} &\leq \bar{\beta} \frac{\Lambda}{\mu}g'(0)-(\mu +\gamma +\alpha )+\frac{\Lambda}{\mu}g'(0) \frac{\sigma}{\sqrt{\pi k}} \\ &=(\mathcal{R}^{e}_{0}-1)(\mu +\gamma +\alpha )\quad \text{a.s.} \end{aligned}$$

Therefore, if $\mathcal{R}_{0}^{e}<1$, then $\limsup _{t\rightarrow +\infty}\frac{{\ln I(t)}}{t}\leq 0$ a.s. It is easy to see that $\lim _{t\rightarrow +\infty}I(t)=0$ a.s. This indicates that the disease will get extinct exponentially. □

Remark 2

From the expression of $\mathcal{R}_{0}^{e}$, it follows that $\mathcal{R}_{0}\leq \mathcal{R}_{0}^{e}$, with equality if and only $\sigma =0$. This implies that if the disease goes extinct in the stochastic model (3), it will also go extinct in the corresponding deterministic model (1).

In order to discuss the existence of stationary distribution of model (3), define the threshold value as

$$ \mathcal{R}^{s}_{0}= \frac{\tilde{\beta}\Lambda g'(0)}{\mu \left (\mu +\gamma +\alpha +a_{1}g'(0)\frac{\Lambda}{\mu}\frac{\sigma}{\sqrt{\pi k}}\right )}, $$

where $\tilde{\beta}=(\int _{0}^{+\infty} x^{\frac{1}{3}}\pi (x)dx)^{3}$, $\pi (x)=\frac{\sqrt{k}}{\sqrt{\pi}\sigma}{\mathrm{e}}^{- \frac{k(x-\bar{\beta})}{\sigma ^{2}}}$, and $\pi (x)$ is the invariant density of $\beta (t)$.

Theorem 3

If $\mathcal{R}_{0} ^{s}> 1$, then model (3) at least has one ergodic stationary distribution $\eta (\cdot )$.

Proof

We will divide into three steps to complete the proof of Theorem 3.

(i):

Construct stochastic Lyapunov function

In order to construct an appropriate Lyapunov function, we first use the Itô’s formula to calculate the following:

$$ \begin{aligned} \mathcal{L}(-\ln S)&=\mu -\frac{\Lambda}{S}+\beta ^{+}g(I)-\delta \frac{R}{S}, \\ \mathcal{L}(-\ln I)&=-\frac{\beta ^{+}S g(I)}{I}+(\mu +\gamma + \delta ), \\ \mathcal{L}(-\ln R)&=\mu +\delta -\gamma \frac{I}{R}, \\ \mathcal{L}\Big(-\ln \Big(\frac{\Lambda}{\mu}-(S+I+R)\Big)\Big) &= \mu -\frac{\alpha I}{\frac{\Lambda}{\mu}-(S+I+R)}. \end{aligned} $$

(14)

Due to

$$\begin{aligned} \left (\frac{I}{g(I)}\right )' = \frac{\frac{I}{g(I)}-\lim \limits _{t\rightarrow +\infty}\frac{I}{g(I)}}{I}= \frac{\frac{I}{g(I)}-\frac{1}{g'(0)}}{I}\leq m, \end{aligned}$$

one obtains

$$ \frac{I}{g(I)}\leq mI+\frac{1}{g'(0)}. $$

(15)

First defining the function $W_{1}=-\ln I-a_{1}\ln S$, where $a_{1}$ is determined by (17), and then combining (14) and (15), one has

$$ \mathcal{L}W_{1} =-\frac{\beta ^{+}Sg(I)}{I}+(\mu +\gamma +\alpha )- \frac{a_{1}\Lambda}{S}+a_{1}\mu +a_{1}\beta ^{+}g(I)-a_{1} \frac{\delta R}{S}. $$

As $g(I)\leq g'(0)I$, it is further deduced that

$$ \begin{aligned} \mathcal{L}W_{1} &\leq -\frac{\beta ^{+}Sg(I)}{I}+(\mu + \gamma +\alpha )-\frac{a_{1}\Lambda}{S}+a_{1}\mu +a_{1}\beta ^{+}g'(0)I-a_{1} \frac{\delta R}{S} \\ &\leq -\frac{\beta ^{+}Sg(I)}{I}-\frac{a_{1}\Lambda}{S}- \frac{a_{2}I}{g(I)}+\frac{a_{2}I}{g(I)}+a_{1}\mu +(\mu +\gamma + \alpha )+a_{1}\beta ^{+}g'(0)I \\ &\leq -3\sqrt[3]{\beta ^{+}\Lambda a_{1}a_{2}}+a_{1}\mu + \frac{a_{2}}{g'(0)}+a_{2}mI+(\mu +\gamma +\alpha )+a_{1}\beta ^{+}g'(0)I \\ &\leq -3\sqrt[3]{\tilde{\beta}\Lambda a_{1}a_{2}}+a_{1}\mu + \frac{a_{2}}{g'(0)}+a_{2}mI+(\mu +\gamma +\alpha )+a_{1}\bar{\beta}g'(0)I \\ &\quad +3\left (\sqrt[3]{\tilde{\beta}\Lambda a_{1}a_{2}}- \sqrt[3]{\beta ^{+}\Lambda a_{1}a_{2}}\right )+a_{1}g'(0)I(\beta ^{+}- \bar{\beta}), \end{aligned} $$

(16)

where $\tilde{\beta}=(\int _{0}^{+\infty} x^{\frac{1}{3}} \pi (x){\mathrm{d}}x)^{3}$. Choosing

$$ a_{1}=\frac{\tilde{\beta}\Lambda g'(0)}{\mu ^{2}},\quad a_{2}= \frac{\tilde{\beta}\Lambda g^{\prime \,2}(0)}{\mu}, $$

(17)

and then substituting (10) and (17) into (16), we have

$$\begin{aligned} \mathcal{L}W_{1} \leq& - \frac{\tilde{\beta}\Lambda g'(0)}{\mu}+a_{2}mI+\mu +\gamma +\alpha +a_{1} \bar{\beta} g'(0)I+a_{1}g'(0)I(\beta ^{+}-\bar{\beta} ) \\ & +3\left (\sqrt[3]{\tilde{\beta}\Lambda a_{1}a_{2}}- \sqrt[3]{\beta ^{+}\Lambda a_{1}a_{2}}\right ) \\ \leq& -\frac{\tilde{\beta}\Lambda g'(0)}{\mu}+a_{2}mI+\mu +\gamma + \alpha +a_{1}\bar{\beta} g'(0)I+a_{1}g'(0)\frac{\Lambda}{\mu} \frac{\sigma}{\sqrt{\pi k}} \\ & +a_{1}g'(0)\frac{\Lambda}{\mu}|\beta -\bar{\beta} |-a_{1}g'(0) \frac{\Lambda}{\mu}\frac{\sigma}{\sqrt{\pi k}}+3\left ( \sqrt[3]{\tilde{\beta}\Lambda a_{1}a_{2}}- \sqrt[3]{\beta ^{+}\Lambda a_{1}a_{2}}\right ) \\ \leq& -\frac{\tilde{\beta}\Lambda g'(0)}{\mu}+\mu +\gamma +\alpha +a_{1}g'(0) \frac{\Lambda}{\mu}\frac{\sigma}{\sqrt{\pi k}}+(a_{2}m+a_{1} \bar{\beta} g'(0))I \\ & +a_{1}g'(0)\frac{\Lambda}{\mu}\left (|\beta -\bar{\beta} |- \frac{\sigma}{\sqrt{\pi k}}\right )+3\left ( \sqrt[3]{\tilde{\beta}\Lambda a_{1}a_{2}}- \sqrt[3]{\beta ^{+}\Lambda a_{1}a_{2}}\right ) \\ =&-(\mathcal{R}^{s}_{0}-1)\left (\mu +\gamma +\alpha +a_{1}g'(0) \frac{\Lambda}{\mu}\frac{\sigma}{\sqrt{\pi k}}\right )+(a_{2}m+a_{1} \bar{\beta} g'(0))I \\ &+g_{1}(\beta )+g_{2}(\beta ^{+}), \end{aligned}$$

(18)

where

$$ g_{1}(\beta )=a_{1}g'(0)\frac{\Lambda}{\mu}\left (|\beta -\bar{\beta} |- \frac{\sigma}{\sqrt{\pi k}}\right ),\qquad g_{2}(\beta ^{+})=3\left ( \sqrt[3]{\tilde{\beta}\Lambda a_{1}a_{2}}- \sqrt[3]{\beta ^{+}\Lambda a_{1}a_{2}}\right ), $$

and one also used the fact

$$ \begin{aligned} \mathcal{R}^{s}_{0}&= \frac{\tilde{\beta}\Lambda g'(0)}{\mu \left (\mu +\gamma +\alpha +a_{1}g'(0)\frac{\Lambda}{\mu}\frac{\sigma}{\sqrt{\pi k}}\right )} \\ &= \frac{\Lambda g'(0)\left (\int ^{+\infty}_{0}x^{\frac{1}{3}}\pi (x)\mathrm{d}x\right )^{3}}{\mu \left (\mu +\gamma +\alpha +g^{\prime \,2}(0)\frac{\Lambda ^{2}}{\mu ^{3}}\frac{\sigma}{\sqrt{\pi k}}\left (\int ^{+\infty}_{0}x^{\frac{1}{3}}\pi (x)\mathrm{d}x\right )^{3}\right )}. \end{aligned} $$

Next, we choose

$$ W_{2}=-\ln S-\ln R-\ln \left (\frac{\Lambda}{\mu}-(S+I+R)\right )+ \frac{\beta ^{2}}{2}, $$

and then

$$ \mathrm {d}W_{2}=\mathcal{L}W_{2}\mathrm{d}t+\sigma \beta \mathrm{d}B(t), $$

by the Itô’s formula, where $\mathcal{L}W_{2}$ satisfies

$$\begin{aligned} \mathcal{L}W_{2}&=-\frac{\Lambda}{S}+\mu +\beta ^{+}g(I)- \delta \frac{R}{S}-\gamma \frac{I}{R}+\mu +\delta +\mu - \frac{\alpha I}{\frac{\Lambda}{\mu}-(S+I+R)} \\ &\quad +\frac{\sigma ^{2}}{2}+\beta k(\bar{\beta} -\beta ) \\ &\leq -\frac{\Lambda}{S}+3\mu +\beta ^{+}g'(0)\frac{\Lambda}{\mu}- \delta \frac{R}{S}-\gamma \frac{I}{R}+\delta - \frac{\alpha I}{\frac{\Lambda}{\mu}-(S+I+R)}\\ &\quad +\frac{\sigma ^{2}}{2}+ \beta k(\bar{\beta} -\beta ) \\ &\leq -\delta \frac{R}{S}-\gamma \frac{I}{R}- \frac{\alpha I}{\frac{\Lambda}{\mu}-(S+I+R)}-\frac{k\beta ^{2}}{2}+B, \end{aligned}$$

(19)

and

$$ B=\sup \limits _{\beta \in \mathbb{R}}\left \{-\frac{k\beta ^{2}}{2}+ k \bar{\beta} \beta +|\beta |g'(0)\frac{\Lambda}{\mu}+3\mu +\delta + \frac{\sigma ^{2}}{2}\right \}. $$

Finally, let $W_{3}=M_{0}W_{1}+W_{2}$, where $M_{0}$ is large enough so that

$$ -M_{0}(\mathcal{R}^{s}_{0}-1)\left (\mu +\gamma +\alpha +a_{1}g'(0) \frac{\Lambda}{\mu}\frac{\sigma}{\sqrt{\pi k}}\right )+B\leq -2. $$

From the above discussion, we have no difficulty in obtaining

$$ \mathrm{d}W_{3}=\mathcal{L}W_{3}\mathrm{d}t+\sigma \beta \mathrm{d}B(t), $$

where $\mathcal{L}W_{3}$ satisfies, from (18) and (19),

$$ \begin{aligned} \mathcal{L}W_{3} &\leq -\delta \frac{R}{S}-\gamma \frac{I}{R}-\frac{\alpha I}{\frac{\Lambda}{\mu}-(S+I+R)}- \frac{k\beta ^{2}}{2}+B+M_{0}(a_{2}m+a_{1}\bar{\beta} g'(0))I \\ &\quad -M_{0}(\mathcal{R}^{s}_{0}-1)\left (\mu +\gamma +\alpha +a_{1}g'(0) \frac{\Lambda}{\mu}\frac{\sigma}{\sqrt{\pi k}}\right )+M_{0}g_{1}( \beta )+M_{0}g_{2}(\beta ^{+}) \\ &\leq -\delta \frac{R}{S}-\gamma \frac{I}{R}- \frac{\alpha I}{\frac{\Lambda}{\mu}-(S+I+R)}-\frac{k\beta ^{2}}{2}-2+M_{0}(a_{2}m+a_{1} \bar{\beta} g'(0))I \\ &\quad +M_{0}g_{1}(\beta )+M_{0}g_{2}(\beta ^{+}) \\ &= g_{3}(S,I,R,\beta )+M_{0}g_{1}(\beta )+M_{0}g_{2}(\beta ^{+}), \end{aligned} $$

and

$$ \begin{aligned} g_{3}(S,I,R,\beta (t))&=-\delta \frac{R}{S}-\gamma \frac{I}{R}- \frac{\alpha I}{\frac{\Lambda}{\mu}-(S+I+R)}-\frac{k\beta ^{2}}{2}\\ &\quad -2+M_{0}(a_{2}m+a_{1} \bar{\beta} g'(0))I. \end{aligned} $$

(20)

(ii):

Construction of a compact set

Define a compact set $\mathbb{D}$ as follows:

$$ \mathbb{D}=\left \{(S,I,R,\beta )\in \Gamma ^{*}|\varepsilon \leq I,~ \varepsilon ^{2}\leq R,~\varepsilon ^{3}\leq S,~\varepsilon ^{2}\leq \frac{\Lambda}{\mu}-(S+I+R),~|\beta |\leq \frac{1}{\varepsilon} \right \}, $$

then $\mathbb{D}^{c}=\bigcup ^{5}_{i=1} \mathbb{D}^{c}_{i}$, where,

$$\begin{aligned} &\mathbb{D}^{c}_{1}=\{(S,I,R,\beta )\in \Gamma ^{*}|I< \varepsilon \},~~~~~~~~~~~~~~~ \mathbb{D}^{c}_{2}=\{(S,I,R,\beta )\in \Gamma ^{*}|R< \varepsilon ^{2},I \geq \varepsilon \}, \\ &\mathbb{D}^{c}_{3}=\{(S,I,R,\beta )\in \Gamma ^{*}|S< \varepsilon ^{3},R \geq \varepsilon ^{2}\},~~~\mathbb{D}^{c}_{4}=\left \{(S,I,R,\beta ) \in \Gamma ^{*}||\beta |>\frac{1}{\varepsilon}\right \}, \\ &\mathbb{D}^{c}_{5}=\left \{(S,I,R,\beta )\in \Gamma ^{*}\Big| \frac{\Lambda}{\mu}-(S+I+R)< \varepsilon ^{2},I\geq \varepsilon \right \}, \end{aligned}$$

and ε is a small enough positive constant satisfying the following inequalities:

$$\begin{aligned} M_{0}(a_{2}m+a_{1}\bar{\beta} g'(0))\varepsilon &\leq 1, \end{aligned}$$

(21)

$$\begin{aligned} -\min \left \{\frac{\delta}{\varepsilon},\frac{\gamma}{\varepsilon}, \frac{\alpha}{\varepsilon},\frac{k}{2\varepsilon ^{2}}\right \}+M_{0}(a_{2}m+a_{1} \bar{\beta} g'(0))\frac{\Lambda}{\mu}&\leq 1. \end{aligned}$$

(22)

Now, we need to prove that $g_{3}(S,I,R,\beta )\leq -1$ for $(S,I,R,\beta )\in \mathbb{D}^{c}$. That is, one needs to verify that the latter inequality it valid on all five domains.

• Case 1: $(S,I,R,\beta )\in \mathbb{D}_{1}^{c}$

  If $(S,I,R,\beta )\in \mathbb{D}_{1}^{c}$, from inequality (21), for $(S,I,R,\beta )\in \mathbb{D}_{1}^{c}$ and ε small enough, we have

  $$\begin{aligned} g_{3}(S,I,R,\beta )&\leq -2+M_{0}(a_{2}m+a_{1}\bar{\beta} g'(0))I \\ &\leq -2+M_{0}(a_{2}m+a_{1}\bar{\beta} g'(0))\varepsilon \\ &\leq -1. \end{aligned}$$

• Case 2: $(S,I,R,\beta )\in \mathbb{D}_{2}^{c}$

  If $(S,I,R,\beta )\in \mathbb{D}_{2}^{c}$, according to (22), it is clear that

  $$\begin{aligned} g_{3}(S,I,R,\beta )&\leq -\gamma \frac{I}{R}+M_{0}(a_{2}m+a_{1} \bar{\beta} g'(0))\frac{\Lambda}{\mu}-2 \\ &\leq -\frac{\gamma}{\varepsilon}-2+M_{0}(a_{2}m+a_{1}\bar{\beta} g'(0)) \frac{\Lambda}{\mu} \\ &\leq -1. \end{aligned}$$

• Case 3: $(S,I,R,\beta )\in \mathbb{D}_{3}^{c}$

  If $(S,I,R,\beta )\in \mathbb{D}_{3}^{c}$, by condition (22), it is easy to get that

  $$\begin{aligned} g_{3}(S,I,R,\beta )&\leq -\delta \frac{R}{S}+M_{0}(a_{2}m+a_{1} \bar{\beta} g'(0))\frac{\Lambda}{\mu}-2 \\ &\leq -\frac{\delta }{\varepsilon}-2+M_{0}(a_{2}m+a_{1}\bar{\beta} g'(0)) \frac{\Lambda}{\mu} \\ &\leq -1. \end{aligned}$$

• Case 4: $(S,I,R,\beta )\in \mathbb{D}_{4}^{c}$

  If $(S,I,R,\beta )\in \mathbb{D}_{4}^{c}$, the inequality (22) implies that

  $$\begin{aligned} g_{3}(S,I,R,\beta )&\leq -\frac{k}{2}\beta ^{2}+M_{0}(a_{2}m+a_{1} \bar{\beta} g'(0))\frac{\Lambda}{\mu}-2 \\ &\leq -\frac{k}{2\varepsilon ^{2}}-2+M_{0}(a_{2}m+a_{1}\bar{\beta} g'(0)) \frac{\Lambda}{\mu} \\ &\leq -1. \end{aligned}$$

• Case 5: $(S,I,R,\beta )\in \mathbb{D}_{5}^{c}$

  If $(S,I,R,\beta )\in \mathbb{D}_{5}^{c}$, using condition (22), we deduce that

  $$ \begin{aligned} g_{3}(S,I,R,\beta )&\leq - \frac{\alpha I}{\frac{\Lambda}{\mu}-(S+I+R)}+M_{0}(a_{2}m+a_{1} \bar{\beta} g'(0))\frac{\Lambda}{\mu}-2 \\ &\leq -\frac{\alpha}{\varepsilon}-2+M_{0}(a_{2}m+a_{1}\bar{\beta} g'(0)) \frac{\Lambda}{\mu} \\ &\leq -1. \end{aligned} $$

In summary, for all $(S,I,R,\beta )\in \mathbb{D}^{c}$, we have $g_{3}(S,I,R,\beta )\leq -1$.

Since $W_{3}$ is a continuous function, there exists a minimum point $(\underline{S},\underline{I},\underline{R},\underline{\beta})$. Therefore, we can establish a nonnegative Lyapunov function as follows:

$$ W=W_{3}(S,I,R,\beta )-W_{3}(\underline{S},\underline{I},\underline{R}, \underline{\beta}). $$

(23)

By using the Itô’s formula for (23), we then get

$$ \mathrm{d}W=\mathcal{L}W\mathrm{d}t+\sigma \beta \mathrm{d}B(t), $$

(24)

where

$$\begin{aligned} \mathcal{L}W\leq g_{3}(S,I,R,\beta )+M_{0}g_{1}(\beta )+M_{0}g_{2}( \beta ^{+}). \end{aligned}$$

(iii):

Existence and ergodicity of a stationary distribution

For any initial value $X(0)=(S(0), I(0), R(0), \beta (0))\in \Gamma ^{*}$ and an interval $[0, t]$, taking the expectation of (24), one gets

$$ \begin{aligned} 0&\leq \frac{\mathbb{E}W(S(t),I(t),R(t),\beta (t))}{t} \\ &=\frac{\mathbb{E}W(S(0),I(0),R(0),\beta (0))}{t}+\frac{1}{t}\int ^{t}_{0} \mathbb{E}(\mathcal{L}W(S(\theta ),I(\theta ),R(\theta ),\beta ( \theta )))\mathrm{d}\theta \\ &\leq \frac{\mathbb{E}W(S(0),I(0),R(0),\beta (0))}{t}+\frac{1}{t} \int ^{t}_{0}\mathbb{E}(g_{3}(S(\theta ),I(\theta ),R(\theta ),\beta ( \theta )))\mathrm{d}\theta \\ &\quad +M_{0}a_{1}g'(0)\frac{\Lambda}{\mu}\mathbb{E}\left ( \frac{1}{t}\int ^{t}_{0}|\beta (\theta )-\bar{\beta} |\mathrm{d} \theta -\frac{\sigma}{\sqrt{\pi k}}\right ) \\ &\quad +3M_{0}\sqrt[3]{\Lambda a_{1}a_{2}}\mathbb{E}\frac{1}{t}\int ^{t}_{0} \left (\sqrt[3]{\tilde{\beta}}-\sqrt[3]{\beta ^{+}(\theta )}\right ) \mathrm{d}\theta . \end{aligned} $$

(25)

Noting that

$$\begin{aligned} \begin{aligned} &\lim _{t\rightarrow +\infty}\mathbb{E}\frac{1}{t}\int ^{t}_{0} \sqrt[3]{{\beta ^{+}}(\theta )}\mathrm{d}\theta =\int ^{+\infty}_{- \infty}\sqrt[3]{\max \{0,x\}}\pi (x)\mathrm{d}x =\int ^{+\infty}_{0}x^{ \frac{1}{3}}\pi (x)\mathrm{d}x =\sqrt[3]{\tilde{\beta}}, \\ &\lim _{t\rightarrow +\infty} \mathbb{E}\frac{1}{t}\int ^{t}_{0}| \beta (\theta )-\bar{\beta}|\mathrm{d}\theta = \frac{\sigma}{\sqrt{\pi k}}, \end{aligned} \end{aligned}$$

(26)

and then taking the inferior limit on both sides of (25) and using (26), one obtains

$$\begin{aligned} 0&\leq \liminf _{t\rightarrow +\infty} \frac{\mathbb{E}W(S(0),I(0),R(0),\beta (0))}{t}\\ &\quad +\liminf _{t \rightarrow +\infty}\frac{1}{t}\int ^{t}_{0}\mathbb{E}(g_{3}(S( \theta ),I(\theta ),R(\theta ),\beta (\theta )))\mathrm{d}\theta \\ &=\liminf _{t\rightarrow +\infty}\frac{1}{t}\int ^{t}_{0}\mathbb{E}(g_{3}(S( \theta ),I(\theta ),R(\theta ),\beta (\theta )))\mathrm{d}\theta . \end{aligned}$$

Since

$$ g_{3}(S,I,R,\beta )\leq -1,\quad \forall (S,I,R,\beta )\in \mathbb{D}^{c}, $$

and

$$ g_{3}(S,I,R,\beta )\leq F,\quad \forall (S,I,R,\beta )\in \Gamma ^{*}, $$

where

$$ F=\sup \limits _{(S,I,R,\beta )\in \Gamma ^{*}}\left \{-\delta \frac{R}{S}-\gamma \frac{I}{R}- \frac{\alpha I}{\frac{\Lambda}{\mu}-(S+I+R)}-\frac{k\beta ^{2}}{2}-2+M_{0}(a_{2}m+a_{1} \bar{\beta} g'(0))I\right \}, $$

we get

$$ \begin{aligned} &\liminf _{t\rightarrow +\infty}\frac{1}{t}\int ^{t}_{0} \mathbb{E}(g_{3}(S(\theta ),I(\theta ),R(\theta ),\beta (\theta )) \mathrm{d}\theta \\ &=\liminf _{t\rightarrow +\infty}\frac{1}{t}\int ^{t}_{0}\mathbb{E}(g_{3}(S( \theta ),I(\theta ),R(\theta ),\beta (\theta ))\mathbf{I}_{\{(S( \theta ),I(\theta ),R(\theta ),\beta (\theta ))\in \mathbb{D}\}} \mathrm{d}\theta \\ &\quad +\liminf _{t\rightarrow +\infty}\frac{1}{t}\int ^{t}_{0} \mathbb{E}(g_{3}(S(\theta ),I(\theta ),R(\theta ),\beta (\theta )) \mathbf{I}_{\{(S(\theta ),I(\theta ),R(\theta ),\beta (\theta ))\in \mathbb{D}^{c}\}}\mathrm{d}\theta \\ &\leq F\liminf _{t\rightarrow +\infty}\frac{1}{t}\int ^{t}_{0} \mathbf{I}_{\{(S(\theta ),I(\theta ),R(\theta ),\beta (\theta ))\in \mathbb{D}\}}\mathrm{d}\theta -\liminf _{t\rightarrow +\infty} \frac{1}{t}\int ^{t}_{0}\mathbf{I}_{\{(S(\theta ),I(\theta ),R( \theta ),\beta (\theta ))\in \mathbb{D}^{c}\}}\mathrm{d}\theta \\ &\leq (F+1)\liminf _{t\rightarrow +\infty}\frac{1}{t}\int ^{t}_{0} \mathbf{I}_{\{(S(\theta ),I(\theta ),R(\theta ),\beta (\theta ))\in \mathbb{D}\}}\mathrm{d}\theta -1. \end{aligned} $$

By calculating and simplifying, one thus obtains

$$ \liminf _{t\rightarrow +\infty}\frac{1}{t}\int ^{t}_{0}\mathbf{I}_{\{(S( \theta ),I(\theta ),R(\theta ),\beta (\theta ))\in \mathbb{D}\}} \mathrm{d}\theta \geq \frac{1}{F+1}>0\quad ~\text{a.s.} $$

According to the definition of event probability and Fatou’s lemma [30], we can deduce the following equivalent form:

$$ \liminf _{t\rightarrow +\infty}\frac{1}{t}\int ^{t}_{0}P(\theta ,(S( \theta ),I(\theta ),R(\theta ),\beta (\theta )),\mathbb{D})\mathrm{d} \theta \geq \frac{1}{F+1}>0 \quad ~\text{a.s.}, $$

where $P(t,(S(t),I(t),R(t),\beta (t)),\mathbb{D})$ represents the transition probability of $(S(t), I(t), R(t),\beta (t))$ belonging to the set $\mathbb{D}$. Therefore, when $\mathcal{R}^{s}_{0}>1$, based on Lemma 2.1, we prove that model (3) has a stationary distribution $\eta (\cdot )$ on $\mathbb{R}^{4}_{+}\times \mathbb{R}$.

 □

4 Probability density function

In this part, we study the exact expression of probability density function near the quasiendemic equilibrium. Obviously, the quasiendemic equilibrium of model (3), namely $\breve{P^{*}}=(\breve{S}^{*},\breve{I}^{*},\breve{R}^{*}, \breve{\beta}^{*})$, satisfies

$$ \left \{ \begin{aligned} &\Lambda -\mu \breve{S}^{*}-\beta \breve{S}^{*}g(\breve{I}^{*})+ \delta \breve{R}^{*}=0, \\ &\beta \breve{S}^{*}g(\breve{I}^{*})-(\mu +\alpha +\gamma )\breve{I}^{*}=0, \\ &\gamma \breve{I}^{*}-(\mu +\delta )\breve{R}^{*}=0, \\ &k(\bar{\beta} -\breve{\beta}^{*})=0. \end{aligned} \right . $$

(27)

Note that $\breve{S}^{*}=S^{*}$, $\breve{I}^{*}=I^{*}$, $\breve{R}^{*}=R^{*}$, $\breve{\beta}^{*}=\bar{\beta}$ and let $(y_{1}, y_{2}, y_{3}, y_{4})=(S-S^{*},I-I^{*},R-R^{*},\beta -\beta ^{*})$. Using Itô’s formula and by calculation, the corresponding linearized model of (3) near $\breve{P}^{*}$ takes the following form:

$$ \left \{ \begin{aligned} &\mathrm{d}y_{1}=(-a_{11}y_{1}-a_{12}y_{2}+a_{13}y_{3}-a_{14}y_{4}) \mathrm{d}t, \\ &\mathrm{d}y_{2}=(a_{21}y_{1}-a_{22}y_{2}+a_{24}y_{4})\mathrm{d}t, \\ &\mathrm{d}y_{3}=(a_{32}y_{2}-a_{33}y_{3})\mathrm{d}t, \\ &\mathrm{d}y_{4}=-a_{44}y_{4}+\sigma \mathrm{d}B(t), \end{aligned} \right . $$

(28)

where

$$\begin{array}{rlrlrl}& a_{11}=\mu +\overline{\beta }g(I^{\ast }),& a_{12}& =\overline{\beta }{S}^{\ast }{g}^{\prime }(I^{\ast }),& a_{13}& =\delta ,a_{14}=S^{\ast }g(I^{\ast }),\\ & a_{21}=\overline{\beta }g(I^{\ast }),& a_{22}& =\mu +\gamma +\alpha -\overline{\beta }{S}^{\ast }{g}^{\prime }(I^{\ast }),& a_{24}& =a_{14}=S^{\ast }g(I^{\ast }),\\ & a_{32}=\gamma ,& a_{33}& =\mu +\delta ,& a_{44}& =k.\end{array}$$

From the condition $(\frac{I}{g(I)})'>0$, combined with the second equality of equation (27), it yields that $a_{22}=\mu +\gamma +\alpha -\bar{\beta}S^{*}g'(I^{*}) = \frac{\bar{\beta}S^{*}g(I^{*})}{I^{*}}-\bar{\beta}S^{*}g'(I^{*}) = \bar{\beta}S^{*}(\frac{g(I^{*})}{I^{*}}-g'(I^{*})) >0$, so all the coefficients of the model (28) satisfy $a_{ij}>0~(i, j=1,2,3,4)$.

Denote $Y=(y_{1},y_{2},y_{3},y_{4})^{T}$, then model (28) can be equivalently transformed into the form of a matrix equation

$$ \mathrm{d}Y(t)=AY(t)\mathrm{d}t+G\mathrm{d}\mathbf{B(t)},\mathbf{ }$$

(29)

where

$$ {A =} \begin{pmatrix} -a_{11}&-a_{12}&a_{13}&-a_{14} \\ a_{21}&-a_{22}&0&a_{24} \\ 0&a_{32}&-a_{33}&0 \\ 0&0&0&-a_{44} \end{pmatrix} , $$

$G=\text{diag}(0,0,0,\sigma )$, and ${\mathbf{B(t)}}=(0,0,0,B(t))^{T}$.

Based on the theory in [33], near the quasiendemic equilibrium $\breve{P}^{*}$, model (29) has a unique probability density function and its form is determined by the Fokker–Plank equation as follows:

$$ \begin{aligned} &\frac{\partial}{\partial y_{1}}[(-a_{11}y_{1}-a_{12}y_{2}+a_{13}y_{3}-a_{14}y_{4}) \Phi ]+\frac{\partial}{\partial y_{2}}[(a_{21}y_{1}-a_{22}y_{2}+a_{24}y_{4}) \Phi ] \\ &+\frac{\partial}{\partial y_{3}}[(a_{32}y_{2}-a{33}y_{3})\Phi ]+ \frac{\partial}{\partial y_{4}}[-a_{44}y_{4}\Phi ]- \frac{\sigma ^{2}}{2}\frac{\partial ^{2}}{\partial y_{4}^{2}}\Phi =0. \end{aligned} $$

Because the matrix G of model (29) is constant, by a quasi-Gaussian distribution theory [34], $\Phi (Y)$ can be expressed as follows:

$$\begin{aligned} \Phi (Y)=C\text{e}^{-\frac{1}{2}YQY^{T}}, \end{aligned}$$

where C is a constant determined by $\int _{\mathbb{R}^{4}}\Phi (Y)\mathrm{d}Y=1$, Q is a real symmetric matrix which satisfies

$$ QG^{2}Q+A^{T}Q+QA=0. $$

(30)

If Q is a positive definite matrix, let $\Sigma =Q^{-1}$, then (30) can be written as follows:

$$ G^{2}+A\Sigma +\Sigma A^{T}=0. $$

(31)

The following Theorem 4 provides the expression of $\Phi (Y)$.

Theorem 4

If $\mathcal{R}_{0}^{s}>1$, then model (29) has a probability density function $\Phi (S,I,R,\beta )$ near $\breve{P}^{*}$, and the form is as follows:

$$ \begin{aligned} \Phi (S,I,R,\beta )&=(2\pi )^{-2}|\Sigma |^{-\frac{1}{2}}\exp \Big\{ - \frac{1}{2}(S-S^{*},I-I^{*},R-R^{*},\beta -\beta ^{*})\Sigma ^{-1} \\ &\qquad \qquad \qquad \qquad \quad \times (S-S^{*},I-I^{*},R-R^{*}, \beta -\beta ^{*})^{T}\Big\} , \end{aligned} $$

(32)

where

$$\begin{aligned} \Sigma =\rho ^{2}(J_{4}J_{3}J_{2}J_{1})^{-1}\Sigma _{2}((J_{4}J_{3}J_{2}J_{1})^{-1})^{T}, \end{aligned}$$

with $J_{1}$, $J_{2}$, $J_{3}$, $J_{4}$, and $\Sigma _{2}$ being matrices, which are found in (33), (34), (35), (37), and (40).

Proof

First of all, one needs to verify that A is a Hurwitz matrix [3]. We consider its characteristic equation

$$\begin{aligned} \psi (\lambda )=(\lambda +a_{44})(\lambda ^{3}+\bar{a}_{1}\lambda ^{2}+ \bar{a}_{2}\lambda +\bar{a}_{3}), \end{aligned}$$

where

$$\begin{aligned} \bar{a}_{1}&=a_{11}+a_{22}+a_{33}, \\ \bar{a}_{2}&=a_{11}a_{22}+a_{11}a_{33}+a_{22}a_{33}+a_{12}a_{21}, \\ \bar{a}_{3}&=a_{11}a_{22}a_{33}-a_{13}a_{21}a_{32}+a_{12}a_{21}a_{33}. \end{aligned}$$

Due to $a_{i,j}>0\ (i, j=1,2,3,4)$, we have $\bar{a}_{1}>0$, $\bar{a}_{2}>0$, $\bar{a}_{1}\bar{a}_{2}-\bar{a}_{3}>0$. By calculating, $\bar{a}_{3}=\mu (\mu +\delta )(\mu +\gamma +\alpha -\bar{\beta} S^{*}g'(I^{*}))+[ \mu (\mu +\alpha +\gamma )+(\mu +\alpha )\delta ]\bar{\beta}g(I^{*})>0$. According to the Routh–Hurwitz criterion, it follows that A is a Hurwitz matrix.

Now, let us calculate the specific form of Σ. Set $A_{1}=J_{1}AJ_{1}^{-1}$ and $G_{1}=J_{1}GJ_{1}^{-1}$, where $J_{1}$ is an ordering matrix as follows:

$$ {J_{1}=} \begin{pmatrix} 0&0&0&1 \\ 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \end{pmatrix} . $$

(33)

Then, by direct calculation, we get

$$ {A_{1}=} \begin{pmatrix} -a_{44}&0&0&0 \\ -a_{14}&-a_{11}&-a_{12}&a_{13} \\ a_{14}&a_{21}&-a_{22}&0 \\ 0&0&a_{32}&-a_{33} \end{pmatrix} ,\qquad {G_{1}=} \begin{pmatrix} \sigma &0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \end{pmatrix} . $$

Further, choosing the elimination matrix

$$ {J_{2}=} \begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&1&1&0 \\ 0&0&0&1 \end{pmatrix} , $$

(34)

and letting $A_{2}=J_{2}A_{1}J_{2}^{-1}$, one obtains

$$ {A_{2}=} \begin{pmatrix} -a_{44}&0&0&0 \\ -a_{14}&-a_{11}+a_{12}&-a_{12}&a_{13} \\ 0&m_{1}&-a_{12}-a_{22}&a_{13} \\ 0&-a_{32}&a_{32}&-a_{33} \end{pmatrix} , $$

where $m_{1}=a_{21}-a_{11}+a_{22}+a_{12}=\gamma +\alpha >0$.

Next, we define $A_{3}=J_{3}A_{2}J_{3}^{-1}$, where $J_{3}$ is the standardized transformation matrix as follows:

$$ {J_{3}=} \begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&\frac{a_{32}}{m_{1}}&1 \end{pmatrix} . $$

(35)

By calculation, one gets

$$ {A_{3}=} \begin{pmatrix} -a_{44}&0&0&0 \\ -a_{14}&b_{1}&b_{2}&a_{13} \\ 0&m_{1}&b_{3}&a_{13} \\ 0&0&m_{2}&b_{4} \end{pmatrix} , $$

where $b_{1}=-a_{11}+a_{12}$, $b_{2}=-a_{12}-\frac{a_{13}a_{32}}{m_{1}}$, $b_{3}=-a_{22}+b_{2}$, and

$$\begin{aligned} &\quad b_{4}=\frac{a_{13}a_{32}}{m_{1}}-a_{33},\quad m_{2}=a_{32}- \frac{a_{32}(a_{12}+a_{22})}{m_{1}}-\frac {a_{32}b_{4}}{m_{1}} = \frac{\alpha \gamma \delta}{(\alpha +\gamma )^{2}}>0. \end{aligned}$$

Denote matrix J as $J=J_{3}J_{2}J_{1}$, then equation (31) can be rewritten as

$$\begin{aligned} JG^{2}J^{T}+(JAJ^{-1})(J\Sigma J^{T})+(J\Sigma J^{T})(JAJ^{-1})^{T}=0, \ JG^{2}J^{T}={\text{diag}}(\sigma ^{2},0,0,0). \end{aligned}$$

Set $JG^{2}J^{T}= G_{1}^{2}$, $JAJ^{-1}=A_{3}$ and $J\Sigma J^{T}=\Sigma _{1}$, therefore, equation (31) can be expressed as

$$ G^{2}_{1}+A_{3}\Sigma _{1}+\Sigma _{1}A^{T}_{3}=0. $$

(36)

Let $A_{4}=J_{4}A_{3}J_{4}^{-1}$. Then, according to the method in [35], the standard transformation matrix $J_{4}$ of $A_{3}$ has the following form:

$$ {J_{4}=} \begin{pmatrix} n_{1}&n_{2}&n_{3}&n_{4} \\ 0&m_{1}m_{2}&(b_{3}+b_{4})m_{2}&a_{13}m_{2}+b_{4}^{2} \\ 0&0&m_{2}&b_{4} \\ 0&0&0&1 \end{pmatrix} , $$

(37)

where $n_{1}=-a_{14}m_{1}m_{2}$, $n_{2}=(b_{1}+b_{3}+b_{4})m_{1}m_{2}$ and

$$\begin{aligned} &n_{3}=(b_{2}m_{1}+b_{3}^{2}+b_{3}b_{4}+a_{13}m_{2}+b_{4}^{2})m_{2}, \\ & n_{4}=(a_{13}m_{1}m_{2}+a_{13}b_{3}m_{2}+2a_{13}b_{4}m_{2}+b_{4}^{3}), \end{aligned}$$

so one obtains

$$ {A_{4}=} \begin{pmatrix} -\tau _{1}&-\tau _{2}&-\tau _{3}&-\tau _{4} \\ 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \end{pmatrix} , $$

where $\tau _{1}=\bar{a}_{1}+a_{44}$, $\tau _{2}=\bar{a}_{2}+\bar{a}_{1}a_{44}$, $\tau _{3}=\bar{a}_{3}+\bar{a}_{2}a_{44}$, $\tau _{4}=\bar{a}_{3}a_{44}$. More importantly, from $\bar{a}_{i}>0~(i=1,2,3,4)$ and $a_{44}>0$, we can deduce that $\tau _{i}>0~(i=1,2,3,4)$ and $\tau _{1}(\tau _{2}\tau _{3}-\tau _{1}\tau _{4})-\tau ^{2}_{3}=(a_{44}^{3}+ \bar{a}_{1}a_{44}^{2}+\bar{a}_{2}a_{44}+\bar{a}_{3})(\bar{a}_{1} \bar{a}_{2}-\bar{a}_{3})>0$. So the conditions of Lemma 2.2 are satisfied.

Based on the above discussion, equation (36) can be translated as

$$\begin{aligned} J_{4}G^{2}_{1}J^{T}_{4}+(J_{4}A_{3}J^{-1}_{4})(J_{4}\Sigma _{1}J^{T}_{4})+(J_{4} \Sigma _{1}J^{T}_{4})(J_{4}A_{3}J^{-1}_{4})^{T}=0. \end{aligned}$$

(38)

Also, it is known that $J_{4}G_{1}^{2}J_{4}^{T}={\text{diag}}(\sigma ^{2} n_{1}^{2},0,0,0)$. Let $\rho =\sigma n_{1}$, $\Sigma _{2}=\rho ^{-2}J_{4}\Sigma _{1}J^{T}_{4}$, and $G_{0}={\text{diag}}(1,0,0,0)$, then (38) is transformed into

$$ G^{2}_{0}+A_{4}\Sigma _{2}+\Sigma _{2}A_{4}^{T}=0. $$

(39)

Using Lemma 2.2, the form of $\Sigma _{2}$ can be given as

$$\begin{aligned} {\Sigma _{2}=} \begin{pmatrix} \frac{\tau _{2}\tau _{3}-\tau _{1}\tau _{4}}{2(\tau _{1}\tau _{2}\tau _{3}-\tau ^{2}_{3}-\tau ^{2}_{1}\tau _{4})}&0&- \frac{\tau _{3}}{2(\tau _{1}\tau _{2}\tau _{3}-\tau ^{2}_{3}-\tau ^{2}_{1}\tau _{4})}&0 \\ 0& \frac{\tau _{3}}{2(\tau _{1}\tau _{2}\tau _{3}-\tau ^{2}_{3}-\tau ^{2}_{1}\tau _{4})}&0&- \frac{\tau _{1}}{2(\tau _{1}\tau _{2}\tau _{3}-\tau ^{2}_{3}-\tau ^{2}_{1}\tau _{4})} \\ - \frac{\tau _{3}}{2(\tau _{1}\tau _{2}\tau _{3}-\tau ^{2}_{3}-\tau ^{2}_{1}\tau _{4})}&0& \frac{\tau _{1}}{2(\tau _{1}\tau _{2}\tau _{3}-\tau ^{2}_{3}-\tau ^{2}_{1}\tau _{4})}&0 \\ 0&- \frac{\tau _{1}}{2(\tau _{1}\tau _{2}\tau _{3}-\tau ^{2}_{3}-\tau ^{2}_{1}\tau _{4})}&0& \frac{\tau _{1}\tau _{2}-\tau _{3}}{2\tau _{4}(\tau _{1}\tau _{2}\tau _{3}-\tau ^{2}_{3}-\tau ^{2}_{1}\tau _{4})} \end{pmatrix} . \end{aligned}$$

(40)

This yields, due to

$$\begin{aligned} J\Sigma J^{T}=\Sigma _{1},\quad \Sigma _{1}=\rho ^{2}J^{-1}_{4} \Sigma _{2}(J^{-1}_{4})^{T}, \end{aligned}$$

that

$$\begin{aligned} \Sigma =J^{-1}\Sigma _{1}(J^{-1})^{T}=\rho ^{2}(J_{4}J)^{-1}\Sigma _{2}[(J_{4}J)^{-1}]^{T}. \end{aligned}$$

Hence, the probability density function near the quasiendemic equilibrium $\breve{P}^{*}$ is given by (32). □

5 Numerical simulations

As mentioned in the Introduction, incidence is a key parameter for describing the dynamics of disease transmission. Therefore, two numerical examples are employed to illustrate the main theoretical results of the stochastic model (3). These examples consider the bilinear incidence rate, where $g(I)=I$, and the saturated incidence rate, where $g(I)=\frac{aI}{b+I}$, respectively.

Example 1

The dynamical behaviors of the stochastic model (3) with the bilinear incidence rate.

We fix, firstly, the basic model parameters as follows: $\Lambda =6$, $\mu =0.5$, $\delta =0.01$, $\alpha =0.01$, $\gamma =0.4$, $k=2$, $\bar{\beta}=0.1$, $\sigma =0.09$, with the initial value $(S(0),I(0),R(0),\beta (0))=(30,1,1,0.1)$. The basic reproduction number of model (3) without stochastic perturbation (that is, model (1)) can be calculated directly as $\mathcal{R}_{0}\approx 1.3187>1$, and the endemic equilibrium of model (1) is $(S^{*},I^{*},R^{*})=(9.10, 1.61, 1.27)$. The blue lines in Fig. 1 show that the endemic equilibrium is asymptotical stable, which also indicates that the quantities of susceptible, infected, and recovered individuals tend to some fixed values and disease is persistent. Further, according to Theorem 3, we can get that the threshold value for model (3) is $\mathcal{R}_{0}^{s}\approx 1.1605>1$, and the model admits an ergodic stationary distribution, which implies the stochastic permanence of the disease. This is shown by the red line in Fig. 1. Indeed, uncertainty is omnipresent in the process of disease transmission. The quantity of infected individuals does not stabilize at a certain equilibrium, but fluctuates within a certain range, the magnitude of which depends on the strength of the uncertainty.

Figure 1

The time series diagrams of $S(t)$, $I(t)$, and $R(t)$ in the deterministic model (1) and stochastic model (3) with the bilinear incidence rate, where $\mathcal{R}_{0}\approx 1.3187$ and $\mathcal{R}_{0}^{s}\approx 1.1605$

Next, we compute the exact expression of the probability density function of model (3) with $\mathcal{R}_{0}^{s}>1$. According to Theorem 4, the covariance matrix Σ is obtained as

$$\begin{aligned} \begin{pmatrix} 2.034263589981693 & -1.232390790062459& -0.782046220822837& -0.01575513507558 \\ -1.232390790062459& 0.787970605764171& 0.433449445638679& 0.013542755934756 \\ -0.782046220822838& 0.433449445638679& 0.339960349520563& 0.002158208117093 \\ -0.015755135075588& 0.013542755934756& 0.002158208117093& 0.002025000000000 \end{pmatrix} . \end{aligned}$$

Since the endemic equilibrium of model (1) is $(S^{*},I^{*},R^{*})=(9.10, 1.61, 1.27)$, the solution $X(t)=(S(t),I(t),R(t),\beta (t))$ possesses the normal density function $\Phi (S, I, R, \beta )\sim \mathcal{N}((9.1, 1.61, 1.27, 0.1)^{T},\, \Sigma )$. This is illustrated in Figs. 2 and 3. The plots illustrate the marginal density function and compare it with the fitting curve of the frequency density histogram.

Figure 2

The marginal density functions and frequency histograms for $S(t)$, $I(t)$, and $R(t)$ in the stochastic model (3) with the bilinear incidence rate, where $\mathcal{R}_{0}^{s}\approx 1.1605$

Figure 3

The marginal density function and the fitting curve of frequency density histogram in the stochastic model (3) with the bilinear incidence rate, where $\mathcal{R}_{0}^{s}\approx 1.1605$

However, if we only change the recruitment rate $\Lambda =3$, with the noise intensity $\sigma =0.12$ and other parameters are remaining fixed as above, it is easy to get that $\mathcal{R}_{0}\approx 0.6593<1$ and $\mathcal{R}_{0}^{e}\approx 0.975<1$. Figure 4 reveals that the disease will go extinct in the long run. Furthermore, we perform a sensitivity analysis of the thresholds $\mathcal{R}_{0}^{e}$ in Fig. 5. It is observed that $\mathcal{R}_{0}^{e}$ increases as the speed of reversion k increases or as the noise intensity decreases. This conclusion aligns with the laws of nature and holds practical significance.

Figure 4

The trajectory of $S(t)$, $I(t)$, and $R(t)$ in the deterministic model (1) and stochastic model (3) with the bilinear incidence rate, where $\mathcal{R}_{0}\approx 0.6593$ and $\mathcal{R}_{0}^{e}\approx 0.975$

Figure 5

The sensitivity analysis of the threshold ${\mathcal{R}_{0}^{e}}$ in the stochastic model (3) with the bilinear incidence rate

Example 2

The dynamical behaviors of the stochastic model (3) with the saturation incidence rate.

We fix the basic parameters of the model as follows: $\Lambda =31$, $\mu =0.1$, $\delta =0.01$, $\alpha =0.01$, $\gamma =0.4$, $k=1.5$, $\bar{\beta}=0.2$, $\sigma =0.04$, $a=0.1$, $b=6$. The endemic equilibrium of model (1) is calculated as $(S^{*},I^{*},R^{*})=(285.4068, 5.1924 ,18.8815)$, and the basic reproduction number is $\mathcal{R}_{0}\approx 2.0261>1$. The asymptotic stability of the model (1) about endemic equilibrium is illustrated by the blue line in Fig. 6. This indicates that the numbers for susceptible, infected, and recovered individuals tend towards fixed values, implying persistence of the disease. From Theorem 3, we further obtain that the threshold value of the model (3) is $\mathcal{R}_{0}^{s}\approx 1.054>1$. This means that the model exhibits an ergodic stationary distribution and the disease is stochastically persistent, as depicted by the red lines in Fig. 6. In addition, the plots in Fig. 6 also imply that, under the interference of uncertain factors, the distribution of infected individuals shows an oscillatory phenomenon which is also consistent with the actual distribution.

Figure 6

The time series diagrams of $S(t)$, $I(t)$, and $R(t)$ in the deterministic model (1) and stochastic model (3) with the saturation incidence rate, where $\mathcal{R}_{0}\approx 2.0261$ and $\mathcal{R}_{0}^{s}\approx 1.054$

Next, we derive the exact expression for the probability density function of model (3) with $\mathcal{R}_{0}^{s}>1$. Utilizing Theorem 4, we can calculate the covariance matrix Σ of the model (3) as follows:

$$ {\Sigma =} \begin{pmatrix} 1.6612 &-0.4613 &-1.1692 &-0.0051 \\ -0.4613 &0.2080 &0.2470 &0.0040 \\ -1.1692 &0.2470 &0.8983 &0.0010 \\ -0.0051 &0.0040 &0.0010 &0.0005 \end{pmatrix} , $$

Since the endemic equilibrium of model (1) is $(S^{*},I^{*},R^{*}) = (285.4068, 5.1924 ,18.8815)$, the solution $X(t)=(S(t),I(t),R(t),\beta (t))$ of model (3) follows a normal density function $\Phi (S, I, R, \beta )$ $\sim \mathcal{N}((285.4068$, $5.1924, 18.8815, 0.2)^{T}, \Sigma )$. These characteristics are presented by Figs. 7 and 8, which show the marginal density function and compare it with the fitting curve of the frequency density histogram.

Figure 7

The marginal density functions and frequency histograms for $S(t)$, $I(t)$, and $R(t)$ in the stochastic model (3) with the saturation incidence rate, where $\mathcal{R}_{0}^{s}\approx 1.054$

Figure 8

The marginal density function and the fitting curve of frequency density histogram in stochastic model (3) with the saturation incidence rate, where $\mathcal{R}_{0}^{s}\approx 1.054$

However, if we only change the recruitment rate $\Lambda = 3$, while keeping the other parameters unchanged, it is straightforward to calculate that $\mathcal{R}_{0}^{e}\approx 0.8041<1$ and $\mathcal{R}_{0}\approx 0.6536<1$. Both in the deterministic and stochastic infectious disease models, the disease will get exponentially extinct in the long run, as illustrated in Fig. 9. Additionally, sensitivity analysis of the thresholds $\mathcal{R}_{0}^{e}$ is presented in Fig. 10. Numerical simulations reveal that the basic reproduction number $\mathcal{R}_{0}^{e}$ increases slowly as the speed of reversion k increases, while there is a significant increase in $\mathcal{R}_{0}^{e}$ as the intensity of the random disturbance increases. This indicates that stochastic disturbances increase the risk of infectious disease outbreaks, and the risk level increases with the increase of disturbance intensity. Therefore, uncertainty is not negligible in the prevention and control of infectious diseases.

Figure 9

The trajectory of $S(t)$, $I(t)$, and $R(t)$ in the deterministic model (1) and stochastic model (3) with the saturation incidence rate, where $\mathcal{R}_{0}\approx 0.6536$ and $\mathcal{R}_{0}^{e}\approx 0.8041$

Figure 10

The sensitivity analysis of the thresholds ${\mathcal{R}_{0}^{e}}$ in the stochastic model (3) with the saturation incidence rate

6 Conclusion

In this article, based on biological significance and mathematical rationality, we proposed a stochastic epidemic model with the general incidence rate, in which the parameter β, the transmission rate of the pathogen from infected to susceptible individuals, is assumed to follow the Ornstein–Uhlenbeck process. This kind of stochastic perturbation avoids the shortcomings that the variance of the parameter β becomes infinite as the time interval decreases, which appears in the white noise perturbation. This is one of the highlights of this article. First, for any given initial value, we verified that there is always a unique global positive solution to the model. Next, the threshold values for the extinction and persistence of the disease were gained separately. More specially, the disease gets extinct exponentially when $\mathcal {R}_{0}^{e}<1$, while if $\mathcal{R}_{0}^{s}>1$, the model has a stationary distribution and the disease is persistent. Note that when the random disturbance intensity tends to 0, one has $\mathcal{R}_{0}^{e}=\mathcal{R}_{0}^{s}=\mathcal{R}$, which is consistent with the properties of the deterministic model. This is another highlight of this article. In addition, based on the stationary distribution, an accurate expression of the probability density function of the model near the quasiendemic equilibrium was calculated. Finally, numerical simulations were conducted to explain our theoretical results, while a sensitivity analysis was also carried out on the threshold values of $\mathcal{R}_{0}^{e}$. In particular, in Figs. 3 and 8, we compared the frequency density histogram curves and marginal density curves of the model (3) with bilinear and saturated incidence rates, respectively, and found that the model with a saturated incidence rate fits better. Therefore, it is more practical to study the dynamic behavior of the model with a saturated incidence rate. This work provides a better theoretical basis for disease control.

It is important to note that many factors have not been considered in this paper and deserve further study. For example, other parameters in our model may also follow the Ornstein–Uhlenbeck process, higher-order perturbation, or Lévy jumps, etc. In addition, there are few literature studies of the infectious disease models with both Ornstein–Uhlenbeck perturbation and Lévy jumps, these issues are our next work and are expected to be well addressed in the future.