Multiple stability switches and Hopf bifurcation in an epidemic model with susceptible protection period

This paper considers a delayed epidemic model with a susceptible protection period. The local asymptotic stability and the existence of Hopf bifurcation of the unique positive equilibrium of model are explored by analyzing the associated characteristic equation. The results obtained show that under a certain condition, the positive equilibrium of model is locally asymptotically stable for any nonnegative delay. However, under the other condition, there exists a sequence of critical values of delay such that the positive equilibrium of model becomes eventually unstable after undergoing finite stability switches through Hopf bifurcation. In particular, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are derived according to the center manifold theorem and the normal form method for retarded functional differential equations. To verify the theoretical predictions derived, numerical simulations are also provided at the end of the paper.