Branch Of The Infectious Disease Model

In recent years, qualitative analysis of epidemic models has become a very important part in applied mathematics. With increasingly studying of them, non-linear incidence, age and infectious age structure, together with preventing and treatment strategies have been introduced to epidemic models, which make models more perfect. In this thesis, by applying the qualitiative theory and bifurcation theory of differential equations, the dynamic behaviors of an epidemic model with nonlinear incidence rate, and an epidemic model with treatment, are studied. Especially make much investigation in the existence and stability of the equilibrium, backword bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation.This paper is composed of four chapters.In the first chapter, we introduce the signification of epidemic model exploration and the basic concepts as well as main model forms of epidemic dynamics. In addition, the main work of this paper is also simply introduced in this chapter.In Chapter 2, we present some theories and prior knowledges on this study, such as singular points of planar systems, Hopf bifurcation, Bogdanov-Takens bifurcation.In Chapter 3, an SIR epidemic model with non-linear incidence rate is investigated. It is shown that there exist some values of the model parameters such that backward bifurcation occurs for the model. Therefore, driving the basic reproduction number below the unity is not enough to eradicate the disease, and a critical value at the turning point is deduced as a new threshold. We analyse the condition which result in backward bifurcation, some sufficient conditions for the disease-free equilibrium and the endemic equilibrium being stable are also obtained.In Chapter 4, we consider an SIR epidemic model with treatment and bilinear incidence rate, to understand the effect of this treatment on the disease control. The parameter a and b in the treatment function denote different removal rate of infectives. When a1, there is an endemic equilibrium which is asymptotically stable, and the infection persists.; while a>b, the model may admits multi-equilibria, consequently the backward bifurcation will occur. By analysing the existence of equilibria, a new threshold is obtained, and it is also shown that the model undergoes a sequence of bifurcations including saddle-node bifurcation, subcritical Hopf bifurcation, and homoclinic bifurcation. Lastly, the direction of Hopf bifurcation and the stability of the bifurcating periodic solution are determined by using normal form theory and center manifold theorem. The main results are illustrated by numerical simulations.