The asymptotic behavior of a logistic SIR epidemic model with stochastic perturbation

A asymptotic behavior of a stochastic logistic SIR epidemic model was studied, whose natural death rates are subject to the environmental white noise. First, it was demonstrated that the model possesses non-negative solutions with probability one. Then, the stochastically asymptotical constancy of the equilibrium was obtained by means of the stochastic Lyapunov functional technique, when R0≤1. Additionally, when R0>1, some asymptotic outcomes regarding large time behavior were given. When the noise is small and the diseased death rate is limited, the solution will oscillate around the endemic equilibrium of the deterministic model for a long time, and the fluctuation decreases with the decrease of white noise, which reflects the prevalence of the disease.