Stability And Bifurcation Analysis In A Delayed SIR Model

The epidemic exists extensively in modern life. Since 1920s, people had tried to study the rules of the spread of the epidemic diseases, which presented the theoretical proof for making the strategies of predicting and treating diseases. However, the ability which models predict and control disease depends greatly on the assumptions made in the modeling process. In order to gain deeper insights into the mechanism of disease transmission and evaluate therapeutic strategies, much attention has been focused on the design and analyses of mathematical models. Body has immunity ability to diseases and the immunity is not permanent. Therefore, investigating the models including of temporal delays in such models makes them more realistic. In modeling of communicable diseases, the incidence rate is considered to play a key role. In classical disease transmission models, the incidence rate is assumed to be linear. Recent clinical and theoretical studies have suggested that the non-linearity of the incidence rates is more realistic. To the study of the epidemic models, people are interested in the issue that the parameters determine the disease to die out or prevail in a population. Many models represent the transmission dynamic of differently infectious diseases by developing mathematical models using systems of ordinary differential equations, partial differential equations and function differential equations.In this paper, a function differential equation model which describe the transmission rules of the infectious diseases is investigated, that is, a delayed SIR model. Fist, we regard the recruitment rate as the bifurcation parameter. The sufficient conditions of the stability and the existence of Hopf bifurcations at the equilibrium are obtained by analyzing the distribution of the characteristic values. Furthermore, an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. At last, several numerical simulations to support our theoretically analytical conclusions are carried out using Matlab soft.