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

 School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

* Corresponding author: Shangjiang Guo

Received  January 2020 Revised  April 2020 Published  June 2020

Fund Project: The second author is supported by NSFC (Grant No. 11671123)

This paper is devoted to investigate the dynamics of a stochastic susceptible-infected-susceptible epidemic model with nonlinear incidence rate and three independent Brownian motions. By defining a threshold $\lambda$, it is proved that if $\lambda>0$, the disease is permanent and there is a stationary distribution. And when $\lambda<0$, we show that the disease goes to extinction and the susceptible population weakly converges to a boundary distribution. Moreover, the existence of the stationary distribution is obtained and some numerical simulations are performed to illustrate our results. As a result, appropriate intensities of white noises make the susceptible and infected individuals fluctuate around their deterministic steady–state values; the larger the intensities of the white noises are, the larger amplitude of their fluctuations; but too large intensities of white noises may make both of the susceptible and infected individuals go to extinction.

Citation: Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020201
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##### References:
Trajectories of solutions of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = b = c = 0.1$
Stationary distribution of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = b = c = 0.1$: (a) the graph of the relative frequency densities of $S$ and $I$; (b) the joint density distribution of solution $(S,I)$
Trajectories of solutions of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = 1$, $b = 0.1$, $c = 0.1$
Trajectories of solutions of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = 2$, $b = 0.1$, $c = 0.1$
Trajectories of solutions of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = 0.1$, $b = 1$, $c = 0.1$
Trajectories of solutions of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = 0.1$, $b = 0.1$, $c = 0.31$
Numerical simulations of model (26) with initial value $(S(0),I(0)) = (1.9,1.9)$ and parameters $a = 0.1$, $b = 0.1$, $c = 0.31$: (a) Trajectories of the solution $S(t)$ of (26) and the solution $\varphi$ of (10) shows the convergence of $S$ to a boundary distribution; (b) The dynamics of $S(t)$ and $I(t)$ in time average
Trajectories of solution $(S(t),I(t))$ of model (27) with initial value $(S(0),I(0)) = (1,1)$ and parameters $a = 0.1$, $b = 0.1$, $c = 0.1$
Trajectories of solution $(S(t),I(t))$ of model (27) with initial value $(S(0),I(0)) = (1,1)$ and parameters $a = 1$, $b = 0.1$, $c = 0.1$
Trajectories of solution $(S(t),I(t))$ of model (27) with initial value $(S(0),I(0)) = (1,1)$ and parameters $a = 0.1$, $b = 1$, $c = 0.1$
Trajectories of solution $(S(t),I(t))$ of model (27) with initial value $(S(0),I(0)) = (1,1)$ and parameters $a = 0.1$, $b = 0.1$, $c = 1$
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