Research Epidemic Model Types Have Saturated The Incidence Of Lag

Epidemic dynamic models ia an important part of mathematical models in biology.On the basis of previous research, two classes delayed SIR epidemic models with saturatedincidence rate have been investigated in this paper. One is a delayed SIRI model withrecovery, the other is the one with exponential birth. By analyzing the dynamics propertiesof these models, we obtain some conclusions.The organization of this article is as follows:In chapter one, we introduces the historical background of this issue and the signifi-cance of this research subject. And we analyze the present research in both domestic andinternational, and propose the structure of this article.In chapter two, a delayed SIRI model with saturation incidence incidence has been inves-tigated. We established the local asymptotically of a disease-free and an endemic equilibriumby characteristic equations. When R0<1, the disease-free equilibrium is local asymptoti-cally stable; and when R0>1the endemic equilibrium is local asymptotically stable. Then,by constructing proper Liapunov function, the global asymptotic stability of the equilibriumpoint are obtained. Numerical simulations are carried out to illustrate the main theoreticalresults.In chapter three, a delayed SIR model with exponential demographic structure is for-mulated. We analyze the stability of the endemic equilibrium with the incidence rates（βS（t-τ）I（t-τ））/（1+α₁S（t-τ）+α₂I（t-τ））e（-μτ）. when τ=0, the endemic equilibrium is locally stable; When τ>0,endemic equilibriums are local stable with suitably small. Their behavior from stable tounstable with appropriate critical value τ. Then, we analyzed the dynamic behavior ofthe model when α1=0and α₁=α₂=0. Compare with the model with α1, α2=0,Some numerical simulations in the last section of this paper are given to illustrate the maintheoretical results.