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

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 R-S(0) related to the basic reproduction number R-0. Via our new method in constructing suitable Lyapunov function types, we obtain the exis-tence and uniqueness of an ergodic stationary distribution of the stochastic system if R-S(0) > 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 R-H(0) > 1 . Then the exact expression of the density function is obtained. Moreover, we establish the sufficient condition R-C(0) < 1 for disease extinction. Finally, several numerical simulations are provided to verify our analytical results. By com-parison with other existing results, our developed theories and methods will be highlighted to end this paper. (C) 2021 Elsevier Ltd. All rights reserved.