Research On The Stability Of Several Types Of Epidemic Models

In recent years,the mathematical model has become a common tool to study the laws of disease epidemic,predict the trend of development and prevent and control the infectious diseases.Based on the study of the various factors in the process of the spread of the disease,we established the corresponding infectious disease model to study the stability of the epidemic.Applying the next generation matrix approach to determine the basic reproduction number,using the theory of characteristic value and the RouthHurwitz criterion to study the local stability of equilibrium.According to compound matrix theory,we use a geometric approach to investigate the global stability of the endemic equilibrium.By analyzing the stability of several infectious disease models,some conclusions are obtained.According to the content,this paper can be divided into the following four chapters:Chapter 1,this chapter briefly introduce the background and significance of epidemic models.Chapter 2,this chapter study the stability of a generalized SIRS epidemic model with transfer from infectious to susceptiblewith nonlinear incidence and temporary immunity.We use the method od the next generation matrix,calculate the basic reproduction number.According to the eigenvalue theory and Routh-Hurwitz criterion respectively discuss the local stability of the disease-free equilibrium and the endemic equilibrium point,the global stability of geometric method to study the endemic equilibrium point.Chapter 3,this chapter study the stability analysis of an SEIR epidemic model with Beddington-De Angelis function responsewith Bedding-De Angelis functional response function.The existence and uniqueness of model equilibrium are discussed.The local stability of equilibrium points is studied by using eigenvalue theory and Routh-Hurwitz criterion.Finally,the four order differential equation is reduced to three order,and the global stability of endemic equilibrium is discussed according to the geometric method of three order differential equation.Chapter 4,this chapter study the stability analysis of an SEIR epidemic model with relapse and general nonlinear incidence ratewith replase and nonlinear incidence.The existence and positive of solutions of four order differential equations are analyzed.The local stability of the two solutions is studied by using eigenvalue theory.The global stability of endemic equilibrium is discussed on the basis of the four differential equation geometry method.