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

Abstract

In this paper, we investigate an SIRI epidemic model with nonlinear incidence rate and high‐order stochastic perturbation. First, we obtain a stochastic threshold scriptR0P related to the basic reproduction number scriptR0. A key contribution of our paper is to derive the existence and uniqueness of an ergodic stationary distribution of the stochastic model if scriptR0P>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 scriptR0Q<1 for disease extinction in a long term. Finally, several empirical examples and numerical simulations are provided to verify the above theoretical results.

A stochastic analysis of a SIQR epidemic model with short and long-term prophylaxis

“…In the literature, several stochastic epidemic models have been investigated such as Susceptible-Infected-Susceptible (SIS) (Wen et al, 2019), Susceptible-Infected-Removed (SIR) (Zhou et al, 2021a), Susceptible-Exposed-Infected-Removed (SEIR) (Jin and Lin, 2021), Susceptible-Infected-Removed-Susceptible (SIRS) (Lahrouz and Omari, 2013), and Susceptible-Infected-Removed-Infected (SIRI) (Zhou et al, 2022, El Fatini et al, 2018. Recent breakthroughs in SDEs allow us to include randomness into the description of biological events.…”