Polynomial ergodicity of an SIRS epidemic model with density-dependent demographics

In this paper, a stochastic susceptible-infective-recovered-susceptible (SIRS) model with density-dependent demographics is proposed to study the dynamics of transmission of infectious diseases under stochastic environmental fluctuations. We demonstrate that the position of the basic reproduction number R0s$R_0<^>s$ with respect to 1 is the threshold between extinction and persistence of the disease under mild extra conditions. That is, under mild extra conditions, when R-0(s) < 1, the disease is eradicated with probability 1; when R-0(s) < 1, the disease is persistent almost surely and the Markov process has a unique stationary distribution and is polynomial ergodic. As an application, we use the 2017 influenza A data from Western Asia to estimate the parameter values of the model and based on that investigate the effect of random noises on the dynamics of the model. Our study reveals that the basic reproduction number R-0(s) is negatively correlated with the noise intensity for the infected but positively correlated with that for the susceptible population, which are different from the findings obtained in the existing literature.