Dynamic Behavior Of Two Kinds Of Nonlinear Incidence Infectious Disease Models With White Noise Disturbance

In recent years,mathematics as a basic subject has been applied to many fields,such as biology,physics,finance and so on.The research on the dynamics of infectious disease based on stochastic differential equation model and played an important role in the research on the prediction and control measures of the disease.The basic regeneration number,equilibrium points of asymptotic beheaviors,persistence,extinction and stationary distribution about disease in the infectious disease dynamics model play a guiding role in disease prevention and control.At the same time,the research value of stochastic differential equation theory followed by the deep study of infectious disease dynamics is more important.Dynamic models about diseases sometimes take into account more complex factors,uch as age,occupation,location.It is often necessary to establish a multi-group infectious disease dynamic model with network relationship.This is also the key point of research on stochastic epidemic models.In this paper,Lyapunov analysis and Has’minskii’s stationary distribution theory are used to study the dynamic behavior of two kinds of nonlinear incidence about epidemic models,one is the stochastic SEIS epidemic model with saturated incidence,the other is the stochastic multi-group SIS epidemic model with saturated incidence.In chapter 1,introduceing the research background and meaning of epidemic dynamics,and giving the definitions,theorems and lemmas involved in this paper.In chapter 2,the deterministic SEIS epidemic model of saturated incidence is firstly proposed.On the one hand,we calculate the basic regeneration number R0.The conditions for the existence of equilibrium points are determined and two important equilibrium points are studied,which are disease-free equilibrium point E0 and endemic equilibrium point E*.The disease-free equilibrium point is locally asymptotically stable when R0<1,and unstable when R0>1.And when R0c>R0>1,E*is locally asymptotically stable.On the other hand,for the stochastic model of saturated incidence,the existence and uniqueness of it is proved.When d>σ2/2 and R0s<1,the disease extinct in index.The existence of stationary distribution in the system are proved by Has’minskii stationary distribution theory.Then we simulated the result by matlab.In chapter 3,the deterministic multi-group SIS epidemic model with saturated incidence is proposed.For the deterministic model,using LaSalle’s Invariance Principle,P0 is globally asymptotically stable when R0≤1,and P0 is unstable and the disease is persistent in the region Γ when R0>1.The existence and uniqueness of positive solution of stochastic multi-group SIS with saturated incidence is obtained.When dk>ǒ2/2 and R0*>1 are satisfied,the disease will persist.Stationary distribution of stochastic system also obtained by Has’minskii stationary distribution theory.Finally,the result are simulated numerically.