Asymptotic Behavior Of Global Positive Solution To A Stochastic SIQS Model With Isolation

The study of the dynamics of infectious diseases aims to explore the intrinsic reasons and influence factors of an epidemic, to master its transmission principles and rules, and to forecast its development trend. Such a study helps us find theoretical fundamentals that prevent even destroy the epidemic. It is well known that ordinary differential equations are frequently used to model infectious diseases. This kind of deterministic model has been investigated for many years by domestic and international researchers, and a large number of research results have been obtained. However, since deterministic epidemic models do not consider any influences from various random factors existing in the nature, it is more reasonable to apply stochastic epidemic models disturbed by white noises to describe practical situations in order to get more accurate results. In this paper, a deterministic epidemic model, that is a SIQS model, is disturbed by white noises and then transformed into a stochastic SIQS model. The main purpose is to study the dynamic behavior of such a stochastic SIQS model.Firstly, a SIQS epidemic model with isolation is introduced, then basic reproduction number ??0, disease-free equilibrium ??0 and endemic equilibrium ??? of the SIQS model are obtained. After that, the asymptotic behavior of the stochastic SIQS epidemic model is considered. The existence and the uniqueness of the global positive solution to this stochastic model is proved with the help of constructing an appropriate Lyapunov function. It has been realized that, although a deterministic SIQS model has a disease-free equilibrium and an endemic equilibrium, the corresponding stochastic SIQS model does not admit the above equilibrium points either. Therefore, the asymptotic property of the global positive solution to the stochastic SIQS model is shown with respect to the disease-free equilibrium of the deterministic SIQS model, when the basic reproductive number ??0 is less than 1. This property implies that the epidemic will vanish ultimately. For the case of ??0> 1, the asymptotic property of the global positive solution relating to the endemic equilibrium of the deterministic SIQS model is also proved.Finally, several numerical simulation examples are given to verify the correctness of main results in this thesis.