Study On Dynamical Models Of Epidemic Diseases

Epidemie dynamical models are inportant parts of mathematieal models in biology. Recently, they have got much attention and been studied by lots of scholars. We have got many great results in this field. In this paper, we construct several epidemic dynamical models by the method of compartment modeling and research their dynamical behaviors.First, for convenience, we introduce the background, basic concepts and some theories in epidemic models research.Second, we construct SIRS models with nonlinear incidence rate, define the reproduction number, and analyze the existence and stability of the disease-free equilibrium and the endemic equilibrium. And then we show them numerically; based on the model, we replace the nonlinear incidence rate with a general form, and consider continuous vaccination on S class. Then we analyze the model in a similar way. At last, we compare the two models, and give some measures to prevent and control these diseases.In the third part, we construct an epidemic model with constant input, exponential death and bilinear incidence rate. And the model is combined by SIRS and SIS models. We define the basic reproduction number, analyze the existence and stability of the disease-free equilibrium and the endemic equilibrium. Then, we compare the models with different parameter values numerically. At last, we analyze how to prevent and control the epidemic disease.In the fourth part, we consider a discrete-time SIRS epidemic model, study its dynamical behaviors. We analyze the existence and stability of the disease-free equilibrium and the endemic equilibrium. Then we give the condition of extinct of the disease.In the last part, we conclude this work, and show the work we will do next.