Dynamic Behaviors For Two Classes Of Epidemic Models With Non-linear Incidence Rate

As we all know,the prevalence of the infectious disease is bound to bring devas-tating damage to the development of the human society.In order to control the spread of epidemic,mathematicians always establish mathematical dynamic models which can accurately describe the characteristics of epidemic transmission to provide theoretical basis.Then they use various mathematical theories to study the laws of the disease transmission.This approach plays an important role in both theory and practice.In the environment,there are full of random disturbance.So,it is more practical to consider random interference in the study of epidemic models.In recent years,the introduction of stochastic perturbation into the deterministic epidemic dynamics models and the study of the behavior of stochastic epidemic models have become more and more hot in the study of epidemic dynamics models.In this article,we use a general incidence rate,and introduce the random dis-turbance,then we establish the SIRS and SIVS epidemic models.Subsequently,we apply the basic theories of stochastic differential equation to discuss the behavior of the system and analysis the influence of the stochastic perturbation.In chapter one,we describe the significance and the current situation of epidemic model research briefly.Then we give some basic definitions and preliminaries of the stochastic differential equation relating to this paper.In chapter two,we study the dynamics behavior of the stochastic SIRS epidemic models.Firstly,We introduce random disturbance around the disease-free equilibrium and the endemic equilibrium of deterministic SIRS epidemic model to get the corre-sponding stochastic SIRS epidemic models.Then we give the sufficient conditions for the asymptotic stability of the equilibrium points in the stochastic SIRS models through constructing suitable Lyapunov function and using Ito formula.Finally,the numerical simulation is used to prove the correctness of the conclusion.In chapter three,we study the SIVS epidemic models.Firstly,we discuss the existence and stability of the disease-free equilibrium and the endemic equilibrium of the deterministic system.Then we give the expression of the basic reproductive number R0,and obtain the sufficient condition for the stability of the endemic equilibrium.Secondly,we introduce random disturbance and study the corresponding stochastic SIVS epidemic model.Then we obtain the sufficient conditions for the extinction and prevalence of the disease through constructing suitable Lyapunov function and using Ito formula.Finally,we prove the conclusion is correct by numerical simulation.In chapter four,we summarize the full text,and point out the deficiencies.