A stochastic epidemic model with Crowley–Martin incidence rate and Holling type III treatment

Abstract

In this work, considering the inevitable effects of environmental perturbation on disease transmission, we investigate a stochastic epidemic model with Crowley–Martin incidence rate and Holling type III treatment. Deterministic model and stochastic model are two parts in model calibration. To analyse the dynamic properties of the stochastic model, we firstly verify that there is unique positive global solution. We establish the sufficient conditions for disease extinction. The sufficient condition for disease persistence is also given. The basic reproduction number (R∗) of the deterministic system is calculated with the help of next generation matrix method. The disease will extinct in long run if the basic reproduction number R∗<1. When R∗>1, the disease will persist. We illustrate the path of the deterministic epidemic model and stochastic model using the Euler–Maruyama method to investigate numerically. The population trajectories of the stochastic model for different noise intensities are presented graphically. Our findings show the importance of considering the influence of stochasticity on the spread of epidemics, particularly in the manifestation of complex incidence mechanism and stochastic environment factors. The stochastic threshold reveals that ordinary differential equation models and white noise models underestimate the severity of disease outbreak.