Stability And Hopf Bifurcation Of Delayed Epidemic Models

In the last recent20years, the stability and Hopf bifurcation of the delay differentialequation have received extensive attention of many scholars. Specially, the existence of theperiodical solution caused by the occurrence of Hopf bifurcation is one of an importantsubject focused by the experts at home and abroad. This thesis researches three differentialdelayed epidemic models by using the bifurcation method of ordinary differential equations,and the stable conditions of the endemic equilibrium and the ones under which the modeloccurs Hopf bifurcation are obtained respectively. In the same time, stability and direction ofthe Hopf bifurcation are discussed by the normal form method and center manifold theorem.In chapter two, a delayed SEIS model is formulated by incorporating a time delay toexpress the latent period. By means of the Hopf bifurcation theorem and considering thedelay as a bifurcation parameter, the local stability of equilibrium and the existence of Hopfbifurcation are discussed. The endemic equilibrium is locally stable when is small enoughand Hopf bifurcation occurs when passed through some critical values. Furthermore, theformula expressing the stability and the direction of the Hopf bifurcation are computed by thenormal form method and center manifold theorem. Finally, Matlab is employed to carry outnumerical simulation to verify the theoretical results.In chapter three, an SEIS epidemic model is formulated by employing a time delay todescribe the immune period. Then, the stability of equilibrium is investigated by the stabilitytheory of the differential equation, and sufficient conditions are obtained under which theendemic equilibrium is locally stable. The endemic equilibrium is locally stable when0.In the same time, considering as bifurcation bifurcation, it is found that Hopf bifurcationoccurs when passes through some critical values. Finally, numerical simulation is carriedto illustrate the theoretical results.Based on chapters two and three, chapter four formulates an SEIRS epidemic model with two different delays to investigate the effect of the latent period1and the immune period2on the transmission of disease. Then, viewing the delays1and2as the bifurcationparameters respectively, some sufficient conditions under which the model occurs Hopfbifurcation are obtained by the Hopf bifurcation theory.