With The Analysis Of The Saturated Treatment Function And Saturation Incidence Of Infectious Disease Model

In this thesis, by applying the qualitative theory and bifurcation theory of differentialequations, the dynamic behaviors of a SIS epidemic model with density dependent birth,death rates and saturated treatment function and a SIV epidemic model with saturatedincidence rate are studied. The whole thesis consists of four chapters.In the first and second chapters, we introduce the background,the relational works ofepidemic dynamics and the basic concepts and theorems related to this thesis. At the sametime, the main work of this thesis are given.In the third chapter, an epidemic model with saturated treatment function and densitydependent birth and death rates is studied. Here the total number of the population is governedby Logistic equation and the treatment function adopts a continuous and differentiablefunction which can describe the effect of delayed treatment when the medical condition islimited. However, it is shown that a backward bifurcation will take place when this delayedeffect for treatment is strong. Therefore, the basic reproduction number below the unityis not enough to eradicate the disease and a critical value .... is deduced as a new threshold.Some sufficient conditions for the disease-free equilibrium and the endemic equilibriumbeing globally asymptotically stable are also obtained. The main results are illustrated bynumerical simulations.In the fourth chapter, we consider an epidemic model with saturated incidence ratein its stable invariant manifold．The disease-free equilibrium is proved to be the globalasymptotically. By carrying out the qualitative analysis of the model,it is shown that themodel admits rich dynamical behaviors．It has saddle-node bifurcation, subcritical Hopfbifurcation, and homoclinic bifurcation．We have also found that the model admits bistablesteady states under certain conditions. Lastly, the direction of Hopf bifurcation and thestability of the bifurcating periodic solution are determined by using normal form theoryand center manifold theorem.The main results are illustrated by numerical simulations.