Dynamics, stationary distribution and application of a stochastic SIRS model with Stratonovich perturbation

Abstract

In this paper, we investigate the Stratonovich stochastic SIRS model with the nonlinear incidence rate. Firstly, the existence and uniqueness of the globally positive solution to the stochastic model are obtained by constructing suitable Lyapunov function. Then it’s worth noting that the basic reproduction number of Stratonovich model \({R}_0\) is the same as that of the deterministic model. And when \({R}_0>1\), we obtain the low bound of the number of infected people in mean, which is less than that of It\({\hat{o}}\) model. Moreover, there is a unique stationary distribution to the model when \({{\hat{R}}}_0\) is greater than one. Finally, in numerical simulations, all the theoretical results are verified by Milstein’s higher-order method, where the dynamical behaviors, the sufficient conditions for the existence of the stationary distribution and changes of kernel densities over time are depicted or calculated in detail. In addition, we obtain and fit the real data of COVID-19 in Australia, which further demonstrates the practicability of the model in the real world.