Since the pioneering work of Kermack and Mckendrick [1], mathematical modeling for the dynamics of epidemic transmission has a realistic significance in predicting and controlling the spread of infectious diseases in the field of epidemiological research [2,3,4,5,6,7,8]. Recently, stochastic differential equations have been widely applied to physics, engineering, chemistry, and biology [9,10,11,12,13,14,15,16,17,18,19,20,21], which have obtained some novel results.

In fact, with the development of modern medicine, vaccination has become an important strategy for disease control. Then numerous scholars have investigated the effect of vaccination on disease [22,23,24,25,26,27]. The epidemic model with a constant population size is relatively effective for diseases with a low mortality and short duration. However, it is clearly untenable for diseases with a high mortality and varying populations. Thus epidemic models with varying population size seem to be more reasonable, which have attracted much interest from the research scientists [28,29,30]. Moreover, many infectious diseases incubate inside the hosts for a period of time before becoming infectious, so it is very meaningful to consider the effect of the incubation period. Based on the above considerations, Sun et al. [28] studied an SEIR model with varying population size and vaccination. The system can be described by where$S\left(t\right)$,$E\left(t\right)$,$I\left(t\right)$ and$R\left(t\right)$, respectively, stand for the densities of the susceptible, the exposed, the infective and recovered individuals at timet, the total population size is denoted by$N\left(t\right)=S\left(t\right)+E\left(t\right)+I\left(t\right)+R\left(t\right)$.b represents the inflow rate (including birth and immigration),$\mu$ denotes the outflow rate (including natural death and emigration). The function$\frac{\beta SI}{N}$ stands for the standard incidence rate, here$\beta$ represents the transmission rate of disease.$\delta$$(0\le \delta <1)$ is the vaccine coverage rate of susceptible individuals,p$(0\le p\le 1)$ is the vaccine efficacy,$\alpha$ represents the rate at which the exposed individuals become infectious,$\epsilon$ is the rate of disease-related death and$\gamma$ stands for the recovery rate of infective individuals. The parameters$\delta$ andp are all non-negative constants andb,$\mu$,$\beta$,$\alpha$,$\epsilon$ and$\gamma$ are positive constants. Moreover, the differential equation of total population size$N\left(t\right)$ is given by$\dot{N}=(b-\mu )N-\epsilon I$. The authors [28] explored the proportions of individuals in the four epidemiological classes, namely

It is easy to get that the variables$\tilde{s}$,$\tilde{e}$,$\tilde{i}$ and$\tilde{r}$ satisfy the following system of differential equations

Since variable$\tilde{r}$ does not appear in the first, second, third equations of the above system. Then the above system becomes the following reduced system which is subject to the constraint$\tilde{r}=1-\tilde{s}-\tilde{e}-\tilde{i}$. In the region$△=\left\{\left(\tilde{s},\tilde{e},\tilde{i}\right)\in {\mathbb{R}}_{+}^3|0\le \tilde{s}+\tilde{e}+\tilde{i}\le 1\right\}$, they established the epidemiological threshold condition$R_0$, which determines disease extinction or permanence, where

Meanwhile, they analyzed the global dynamics of system (3) and derived the equilibria (including the disease-free equilibrium and the endemic equilibrium) and their global stability. In addition, the parameter restrictions for uniform permanence were obtained.

Nevertheless, the biological populations in the ecosystem are inevitably subjected to uncertain environmental perturbations. It is worth noting that this phenomenon is ubiquitous in the natural environment. So various stochastic epidemic models have been proposed and studied [31,32,33,34,35,36]. To the best of our knowledge, there are not too many researches on global dynamics of the stochastic SEIR epidemic model with varying population size and vaccination yet. In this paper, to make this epidemic model (1) more reasonable and realistic, we suppose the stochastic perturbations are directly proportional to$\tilde{s}$,$\tilde{e}$,$\tilde{i}$ and$\tilde{r}$ under the influence of white noise type, influenced on the$\dot{\tilde{s}}\left(t\right)$,$\dot{\tilde{e}}\left(t\right)$,$\dot{\tilde{i}}\left(t\right)$ and$\dot{\tilde{r}}\left(t\right)$ in system (1), respectively. This implies the stochastic effects of white noise on the birth and death rates of$S,E,I,R$. Then corresponding to system (1), a stochastic version can be reached by where$B_i\left(t\right)(i=1,2,3,4)$ is the standard Wiener processes with$B_i\left(0\right)=0$ a.s.${\sigma }_i\left(t\right)(i=1,2,3,4)$ stands for a continuous and bounded function for any$t\ge 0$ and${\sigma }_i^2\left(t\right)(i=1,2,3,4)$ represents the intensities of Wiener processes. Furthermore, the differential equation of total population size$N\left(t\right)$ is given by the following form

From (2), the system (4) becomes the following proportional system

It is worthy to note that, the variables$\tilde{s}$,$\tilde{e}$,$\tilde{i}$ and$\tilde{r}$ satisfy the relation$\tilde{r}=1-\tilde{s}-\tilde{e}-\tilde{i}$, we can omit analysis of the fourth equation of system (5) and explore the following reduced system with the initial value$\left(\tilde{s}\left(0\right),\tilde{e}\left(0\right),\tilde{i}\left(0\right)\right)\in {\mathbb{R}}_{+}^3$ and$\tilde{s}\left(0\right)+\tilde{e}\left(0\right)+\tilde{i}\left(0\right)<1$.

Since system (6) is a three-dimensional stochastic system with many high-order nonlinear terms, this makes the stochastic analysis novel and more complex than [34,36].

Throughout this article, unless otherwise specified, let$(\mathsf{\Omega },\mathcal{F},{\left\{\mathcal{F}\right\}}_{t\ge 0},\mathbb{P})$ be a complete probability space with a filtration${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ satisfying the usual conditions (i.e., it is increasing and right continuous while${\mathcal{F}}_0$ contains all$\mathbb{P}$-null sets). Further suppose$B_i\left(t\right)(i=1,2,3,4)$ stands for the mutually independent standard Wiener processes defined on the complete probability space$\mathsf{\Omega }$. For an integrable function$x\left(t\right)$ on$[0,+\infty )$, let us define$\langle x\left(t\right)\rangle =\frac{1}{t}{\int }_0^{t}x\left(r\right)\mathrm{d}r$.

The following Itô’s formula will be used frequently in the sequel.

To explore the dynamical behaviors of a population system, we first concern the global existence and positivity of the solutions of system (6).

For a population system, the parameter conditions of disease extinction and permanence have become an important issue that attracts more and more attention in real life. In this section, we mainly investigate the extinction of disease and leave the argument of permanence to the next section.