Analysis On The Dynamics Of A Class Of Epidemic Model

The study on epidemic model is always one of the topics of interest tomathematicians and scientists of infectious diseases in recent years. Purpose is tocharacterize the spread of infectious diseases mechanism through a mathematicalmodel, and then on the basis of the study on the dynamic properties of thecorresponding mathematical model, it can help human beings to understand thetrends and propagation of infectious diseases more clearly, propose a reasonablecontrol and treatment programs, and achieve the prevention of infectious diseasesoccur or control the spread of infectious diseases, thereby improving the quality ofhuman life. In this article, on the basis of previous work, we construct a time delayepidemic model with with sub-optimal immunity and any recovery rate function tostudy the impact of recovery rates for different forms. The model suggests that thenon-linear recovery rate and time delay can lead to a rich dynamical behavior in it,for example, non-linear recovery rate can cause the branch of the equilibrium point,time delay can lead to the generation of the Hopf bifurcation, that is caused byperiodic oscillations. Further, we also found that there is a special function of therecovery rate with very similar dynamic properties in SIS and SIR models.By selecting the time delay branch parameters, we obtain a sufficient conditionfor the stability of equilibrium point and existence of Hopf bifurcation of the model,revealing that how a time delay can affect the dynamics of epidemic model.Furthermore, by using canonical form method and center manifold theory offunctional differential equations which is proposed by Hassard etc, the calculationof several important amount about determining the nature of the Hopf bifurcation ofthe model is given. In the last, we do numerical simulation by two sets of concreteexperimental data in the literature to illustrate the obtained theoretical results.