Ergodic property, extinction and density function of a stochastic SIR epidemic model with nonlinear incidence and general stochastic perturbations

Abstract

Focusing on the unpredictability of person-to-person contacts and the complexity of random variations in nature, this paper will formulate a stochastic SIR epidemic model with nonlinear incidence rate and general stochastic noises. First, we derive a stochastic critical value R0S related to the basic reproduction number R0. Via our new method in constructing suitable Lyapunov function types, we obtain the existence and uniqueness of an ergodic stationary distribution of the stochastic system if R0S>1. Next, via solving the corresponding Fokker-Planck equation, it is theoretically proved that the stochastic model has a log-normal probability density function when another critical value R0H>1. Then the exact expression of the density function is obtained. Moreover, we establish the sufficient condition R0C<1 for disease extinction. Finally, several numerical simulations are provided to verify our analytical results. By comparison with other existing results, our developed theories and methods will be highlighted to end this paper.