Research Of Epidemic Model With The Incidence Function βI(1+vI^k-1)S

Endemic dynamics is an important branch of biological mathematics. Based on the relationship among susceptible, infected and recovered compartments of endemic sys-tems, we establish the mathematical models to describe the development quantitatively. Further, we apply the mathematical theories and methods to analyze the dynamical be-haviors of populations on quality and quantity to show and predict the evolution laws of endemic diseases, and supply the theoretical references in order to control and eradicate endemic diseases. In endemic systems, incidence rate is important. In order to study the law of endemic diseases accurately, and consider some other factors, we study the endemic models with incidence rate of nonlinear form βI(1+vIk-1)S.There are4chapters in this paper. In chapter1, we introduce the background, main work and preliminary of this paper.In chapter2, we assume that the incidence rate is the nonlinear infection function of the form βI(1+vIk-1)S, and the susceptible is vaccinated at the fixed moments. Therefore, an SEIR epidemic disease model, which is described by the impulsive d-ifferential equations, is established. The existence and the global attractivity of the infection-free periodic solution are obtained. Further, the permanence of the system is studied.In chapter3, we study multi-group endemic model with the same incidence rate. In each group, we divide people into S, E, I, R, and assume the diseases transport in each group and among groups. We use Lyapunov function to discuss the existence and asymptotically of disease free equilibrium. In chapter4, we summarize this paper and point out the shortcomings and the direction of future work.