Ergodic property, extinction, and density function of an SIRI epidemic model with nonlinear incidence rate and high-order stochastic perturbations

In this paper, we investigate an SIRI epidemic model with nonlinear incidence rate and high-order stochastic perturbation. First, we obtain a stochastic threshold (Formula presented.) related to the basic reproduction number (Formula presented.). A key contribution of our paper is to derive the existence and uniqueness of an ergodic stationary distribution of the stochastic model if (Formula presented.). Next, by solving the corresponding Fokker-Planck equation, the exact expression of probability density function of the stochastic model is obtained. Moreover, we establish the sufficient condition (Formula presented.) for disease extinction in a long term. Finally, several empirical examples and numerical simulations are provided to verify the above theoretical results.