Bifurcation analysis of a discrete ${SIRS}$ epidemic model with standard incidence rate.

Discrete epidemic models are popularly used to detect the pathogenesis, spreading, and controlling of the diseases. The three-dimensional discrete ${SIRS}$ epidemic models are more suitable than the two-dimensional discrete models to describe the spreading characters of the diseases. In this paper, the complex dynamical behaviors of a three-dimensional discrete ${SIRS}$ epidemic model with standard incidence rate are discussed. We choose the time step size parameter as a bifurcation parameter, the existence, stability, and direction of Hopf bifurcation are proved by using the normal form theorem and bifurcation theory. Moreover, the numerical simulations not only illustrate our results, but they also exhibit the complex dynamical behaviors, such as the invariant cycle, period-7 orbits and period-12 orbits with more than one attractors and chaotic sets. The flip bifurcation caused by the step size parameter is also obtained by a numerical simulation. Most importantly, when the adequate contact rate and the death rate of the infective individuals are chosen as the bifurcation parameters, there also exist a Hopf bifurcation, a flip bifurcation, chaos, and strange attractors. These results provide significant information for the disease controlling when there appear complex dynamical behaviors in the epidemic model. [ABSTRACT FROM AUTHOR]