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 R0P related to the basic reproduction number R0. A key contribution of our paper is to derive the existence and uniqueness of an ergodic stationary distribution of the stochastic model if R0P>1. 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 R0Q<1 for disease extinction in a long term. Finally, several empirical examples and numerical simulations are provided to verify the above theoretical results.

Automated summary

This paper investigates an SIRI epidemic model with nonlinear incidence rate and high-order stochastic perturbation. The existence and uniqueness of an ergodic stationary distribution, probability density function, and conditions for disease extinction are discussed. The theoretical results are verified through empirical examples and numerical simulations.