Dynamical Behavior of the SEIS Infectious Disease Model with White Noise Disturbance

Abstract

Mathematical model plays an important role in understanding the disease dynamics and designing strategies to control the spread of infectious diseases. In this paper, we consider a deterministic SEIS model with a saturation incidence rate and its stochastic version. To begin with, we propose the deterministic SEIS epidemic model with a saturation incidence rate and obtain a basic reproduction number $R_0$ . Our investigation shows that the deterministic model has two kinds of equilibria points, that is, disease-free equilibrium $E_0$ and endemic equilibrium $E_{\ast }$ . The conditions of asymptotic behaviors are determined by the two threshold parameters $R_0$ and $R_0^c$ . When $R_0<1$ , the disease-free equilibrium $E_0$ is locally asymptotically stable, and it is unstable when $R_0>1$ . $E_{\ast }$ is locally asymptotically stable when $R_0^c>R_0>1$ . In addition, we show that the stochastic system exists a unique positive global solution. Conditions $d>{\check{\sigma }}^2/2$ and $R_0^s<1$ are used to show extinction of the disease in the exponent. Finally, SEIS with a stochastic version has stationary distribution and the ergodicity holds when $R_0^{\ast }>1$ by constructing appropriate Lyapunov function. Our theoretical finding is supported by numerical simulations. The aim of our analysis is to assist the policy-maker in prevention and control of disease for maximum effectiveness.