Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure

In this paper, we deal with an SIRS epidemic model with stage structure and time delays. We perform some bifurcation analysis to the model. The delay ττ is used as the bifurcation parameter. We show that the positive equilibrium is locally asymptotically stable when the time delay is suitable small, while change of stability of positive equilibrium will cause a bifurcating periodic solution as the time delay ττ passes through a sequence of critical values. Applying the normal form theory and center manifold argument, we obtain some local bifurcation results and derive formulas for determining the bifurcation direction and the stability of the bifurcated periodic solution. In order to illustrate our theoretical analysis, some numerical simulations are also included in the end.