The Study Of The Epidemic Models With Isolation And Vaccination

In this paper, we investigate the two types of models in mathematical biology. One is the five-dimensional difference equation model with isolation and vaccination, and the other is slightly modified on the basis of the former model, which is described by the five-dimensional delay differential equation model. We focus on bifurcations and stability from these two models.This paper is divided into four parts:In chapter 1, we review some background about epidemic dynamic models, the existed relative work, the origin of the problems we discuss and the main results we obtain.In chapter 2, we introduce some basic knowledge about difference equations and differential equation firstly, such as the Jury criterion under n-order linear constant coefficient difference equations, and some basic concepts of differential equations. We then introduce the definition of bifurcation, especially, we give Hopf bifurcation theorem of differential equations.In chapter 3, we explore a five-dimensional difference equations model with isolation and vaccination. There exists a basic reproductive number Ro for such a system. If R0<1 is true, there exists a globally asymptotically stable disease-free equilibrium. Otherwise, if R0> 1 holds, then disease-free equilbrium is unstable and there exists a endemic equilibrium. By Jury criterion, we analyse the local asymptotical stability of the endemic equilibrium. At last, we simply discuss our conclusion from the viewpoint of epidemic.In chapter 4, we study a delay differential equations model with isolation, vaccination. We assume that the susceptible is vaccinated and will become the immune after a lapse of some time. Using the time delay as a bifurcation parameter, the conditions for Hopf bifurcation to occur are derived, and some computer simulations are presented to illustrate the conclusions by MAPLE.