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 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.

Automated summary

This paper investigates a stochastic epidemic model with nonlinear incidence rate and Brownian motions, proving the permanence or extinction of the disease under different conditions. Numerical simulations illustrate the results, showing that appropriate intensities of white noises can cause fluctuations in susceptible and infected individuals.