A note on the stationary distribution of stochastic SEIR epidemic model with saturated incidence rate

The stochastic SEIR infectious diseases model with saturated incidence rate is studied in this paper. By constructing appropriate Lyapunov functions, we show that there is a stationary distribution for the system and the ergodicity holds provided R0s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R}_{0}^{s}$$\end{document} > 1. In particular, we improve the results obtained by previous studies greatly, condition in our Theorem is more concise and elegant.

The study of the epidemic models have long been and will continue to be one of the dominant themes in mathematical biology due to its importance at both understanding the spread and control of infectious diseases in a community. Many researchers have made a significant progress on SIR models 1-6 , where S, I, R denote the fractions of the susceptibles, the infectives and the recovered hosts in the population respectively. SIR models assume the disease has no latent period. However, for some diseases, such as hepatitis B, AIDS, sometimes has to be passed before an infected individual becomes infectious. Therefore, an extra class, the class of exposed hosts (E), should be added to the system, where E denotes the fraction of the exposed population. The model is called SEIR (susceptible-exposed-infected-removed) model, and SEIR models were investigated by many researchers [7][8][9] .
On the other hand, the incidence rate palys an important role in the epidemics models. some authors employ the bilinear incidence rate βSI 10,11 . After studying the cholera epidemic spread in Bari in 1973, Capasso and Serio 12 introduced a saturated incidence rate g(I)S into epidemic models, where g(I) tends to a saturation level when I gets large, i.e. measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals. Then the SEIR model with a saturated incidence rate can be described as follows:

dS
The dynamical behavior of model (1.1) is as follows ref. 13: , which is a global attractor in the first octant.
If R 0 > 1, then model (1.1) has two equilibria, a disease-free equilibrium P 0 and an endemic equilibrium P* = {S*, E*, I*, R*}. P 0 is unstable and P* is a global attractor in the interior of the first octant.
Some authors take stochastic perturbation into account when they investigate the epidemics system [14][15][16][17][18][19][20][21] . In this paper, we assume that the perturbation is of white noise type, that is, 1 4 , then we get the following stochastic system are standard one-dimensional independent Wiener processes, are the intensity of the white noise. In ref. 17,Yang et al. show that there is a stationary distribution μ(·) for system (1.2) and it has ergodic property provided the following conditions hold: where (S*, E*, I*, R*) is the interior equilibrium of system of (1.1), The main aim of this paper is to deal with the existence of stationary distribution of system (1.2). By construct new Lyapunov functions and rectangular set, instead of elliptical region, our results do not depend on the equilibrium P* of the deterministic system (1.1), which will improve the above result to a great extent.

Ergodic Properties
In order to show the existence of a stationary distribution, firstly, we cite a known result from ref. 22 as a lemma.
Let X(t) be a homogeneous Markov Process in E l (E l denotes l dimensional Euclidean space), and is described by the following stochastic equation: The diffusion matrix is defined as follows:

Lemma 2.1. The Markov process X(t) has a unique ergodic stationary distribution μ(·) if there exists a bounded domain U ∈ E l with regular boundary Γ and
is a function integrable with respect to the measure μ.
then for any initial value 4 , there is a stationary distribution μ(·) for system (1.2) and the ergodicity holds.
It is easy to see that b > 0. Constructing a C 2 -function → + + Q R R : 4 in the following form p and ρ are constants satisfying the following condition respectively and A is determined in the following proof. It is easy to check that .  Consider the following bounded subset where   > , 0 1 2 are sufficiently small numbers satisfying the following conditions    , without other conditions imposed on the coefficients. That is to say, Theorem 2.2 in large improves Theorem 3.3 in ref. 17. Moreover, we see that if σ i = 0 (i = 1, 2, 3, 4), the above condition is reduced to R 0 > 1, which is the condition for globally asymptotically stable of endemic equilibrium P* of system (1.1). And R s 0 is smaller than the basic reproduction number R 0 of system (1.1).