Stability And Hopf Bifurcation For A SIR Epidemiological Model With Vaccinal Immunity And Delay System

In this paper, stability and Hopf bifurcation of a SIR epidemic model with vaccinal immunity and delay will be considered.The letter divides into two parts:the progress of modeling firstly; then analysis of the stabilities and Hopf bifurcations of the disease-free equilibrium point and the endemic equilibrium pointWhen we build the model, definite population increase, the spread of disease among them, vaccination, natural causes, death by disease, cure condition and the incubation period for the disease will be considered. In this way, the model can be more close to the reality we concerned.The model we established is as follows:Then the full text cares about how incubation period effects the stability and Hopf bifurcation of the model. In the article we make use of the stability and bifurcation theory, eigenvalue method and normal form theoryAt last, through the strict reasoning, we have(1) Existence conditions of the disease-free equilibrium point and the endemic equilibrium point;(2) Stability and Hopf bifurcation of the disease-free equilibrium point;(3) Stability and Hopf bifurcation of the endemic equilibrium point;(4) The computational formula of the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation.