Dynamics Analysis Of Two Classes Of Epidemic Models

In this thesis, by applying the qualitiative theory and bifurcation theory ofdi?erential equations, we study the dynamic behaviors of two classes of epidemicmodels, including stability and bifurcations. Bifurcating in epidemic models hasimportant significance, specially, the biologic of the bifurcating periodic solutionsis that the epidemic oscillates at a certain conditions.This paper is composed of four chapters.In Chapter 1, the background and the relational works of epidemic dynamicsare introduced. At the same time , the main work of this paper are gived.In Chapter 2, we present some theories and prior knowledges on this study,such as Hopf bifurcation, Bogdanov-Takens bifurcation, the local structure of theequilibrium point of ordinary di?erential equation, some knowledges about func-tional di?erential equation and Hurwitz law.In Chapter 3, we consider a SIRS epidemic model with constant vaccinationrate and nonlinear infection rate. Firstly, the existence of endemic equilibrium pointis analysised, then by applying the Jacbian matrix of the model at the endemicequilibrium point, we obtain the local asymptotically stable at this equilibriumpoint. Next, choosing the suitable bifurcation parameter, the existence of Hopfbifurcation and Bogdanov-Takens bifurcation are investigated.In Chapter 4, stability and Hopf bifurcation of a delayed SIR epidemic modelwith stage structure are studied. We obtain the stability of the endemic equilibriumpoint. By regarding the delay as the bifurcation parameter, Hopf bifurcation of thismodel is investigated. Following, the direction of Hopf bifurcation and the stabilityof the bifurcating periodic solution are determined by using normal form theoryand center manifold theorem.