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

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 \begin{document}$ \lambda $\end{document} , it is proved that if \begin{document}$ \lambda>0 $\end{document} , the disease is permanent and there is a stationary distribution. And when \begin{document}$ \lambda , 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.