Dynamical Behavior of the SEIS Infectious Disease Model with White Noise Disturbance - Yuheng Song; Qixing Han (Journal of Mathematics)
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 R0. Our investigation shows that the deterministic model has two kinds of equilibria points, that is, disease-free equilibrium E0 and endemic equilibrium E∗. The conditions of asymptotic behaviors are determined by the two threshold parameters R0 and R0c. When R0<1, the disease-free equilibrium E0 is locally asymptotically stable, and it is unstable when R0>1. E∗ is locally asymptotically stable when R0c>R0>1. In addition, we show that the stochastic system exists a unique positive global solution. Conditions d>σˇ2/2 and R0s<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 R0∗>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.