Analysis of an epidemic model with density-dependent birth rate, birth pulses

In this paper, we propose an SIS epidemic model for which population births occur during a single period of the year. Using the discrete map, we obtain exact periodic solutions of system which is with Ricker function. The existence and stability of the infection-free periodic solution and the positive periodic solution are investigated. The Poincaré map, the center manifold theorem and the bifurcation theorem are used to discuss flip bifurcation and bifurcation of the positive periodic solution. Numerical results imply that the dynamical behaviors of the epidemic model with birth pulses are very complex, including small-amplitude periodic 1 solution, large-amplitude multi-periodic cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allow for a period-doubling route to chaos.