Stability and Hopf Bifurcation Analysis of an Epidemic Model with Time Delay

Epidemic models are normally used to describe the spread of infectious diseases. In this paper, we will discuss an epidemic model with time delay. Firstly, the existence of the positive fixed point is proven; and then, the stability and Hopf bifurcation are investigated by analyzing the distribution of the roots of the associated characteristic equations. Thirdly, the theory of normal form and manifold is used to drive an explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions. Finally, some simulation results are carried out to validate our theoretic analysis.