Survival and Stationary Distribution in a Stochastic SIS Model

Abstract

The dynamics of a stochastic SIS epidemic model is investigated. First, we show that the system admits a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: whenR0≤1, we show how the solution spirals around the disease-free equilibrium of deterministic system under some conditions; whenR0>1, we show that the stochastic model has a stationary distribution under certain parametric restrictions. In particular, we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence. Finally, numerical simulations are carried out to illustrate the theoretical results.

Synchronized stationary distribution of stochastic coupled systems based on graph theory

“…Remark In the past few decades, lots of researchers have worked on the stationary distribution of stochastic differential equations. () However, the factor of coupling effect on systems has not taken into account, since stochastic coupled systems have wide applications such as biomathematics. Moreover, when the system itself can not achieve a stationary distribution, it is necessary to introduce a response system in order to investigate the synchronized stationary distribution.…”

“…Remark There are some traditional methods to investigate the existence of stationary distribution, such as Hasmiskii's method, Lyapunov method, and so on. Zhao and Zhou et al() investigated the existence of stationary distribution of stochastic competitive models and epidemic models by using Hasmiskii's method. Huang et al used level set method and Lyapunov method to study the issue of steady states.…”

“…In previous studies,() Qian et al gave stochastic theory of nonequilibrium steady states and its applications. In Zhao et al and Zhou et al,() the authors investigated the stationary distribution of a stochastic competitive model and epidemic models. In Mao and Liu and Mao,() Mao et al explored the stationary distribution of stochastic population systems and gave a numerical method for the stationary distribution of stochastic differential equations with Markovian switching, respectively.…”

Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions

“…The objective of this paper is to study the extinction and permanence of the disease in (1). One of popular approaches to the investigation of SIS models is to introduce a reproduction value [32,34]. For example, by defining a threshold R 0 = Λβ/[µ(µ + γ + α)] for system (1) with g(S, I) = SI, Zhou et al [32] obtained the existence of its stationary distribution when R 0 > 1 and found that the disease will go to extinction when either R 0 ≤ 1 or some noise is sufficiently large, i.e., β 2 ≤ 2c 2 (γ+α+µ)+b 2 c 2 .…”

“…What we are interested in is the existence of stationary distribution and convergence of the population in system (1) in terms of a new threshold, which is different from the basic reproduction number established in [19,27,28,32,34]. For this purpose, we first obtain some estimates of solutions in probability by using some standard inequalities.…”