| An SVIR model with different time delay and incidence rate is considered. By choosing time delay as the bifurcation parameter and analyzing the corresponding characteristic equation, the local stability of the positive equilibrium is investigated and the existence of Hopf bifurcations is established. Formulas are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center mainfold theorem. So the local stability of the positive equilibrium is effected by time delay and incidence rate.In chapter 1, introduces the research background of the mathematical model of infectious diseases, as well as its theoretical results have important practical significance for the spread of infectious diseases control and prevention. By analyzing the research status at home and abroad, thus lead to the formation process of SVIR infectious disease model.In chapter 2, we give the SVIR model with a bilinear incidence rate and by incorporating a time delay to express the latent period. the existence of the positive equilibrium is discussed when basic reproductive number is more than 1.the local stability of the positive equilibrium is obtained and Hopf bifurcation occurs when the delay passed through some critical values. Formulas are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main theoretical results.In chapter 3, we discuss the SVIR model with a nonlinear incidence rate and by employing a time delay to describe the immune period. Then the existence of the positive equilibrium is investigated, and it is locally stable. In the same time,considering the delay as bifurcation parameter, it is found that Hopf bifurcation occurs when the delay passed through some critical values.In chapter 4, Summarizes the full text, and prospects for future research on SVIR infectious disease model. |
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