Bifurcations analysis and tracking control of an epidemic model with nonlinear incidence rate

In this paper, a discrete epidemic model with nonlinear incidence rate obtained by the forward Euler method is investigated. The conditions for existence of codimension-1 bifurcations (fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation) are derived by using the center manifold theorem and bifurcation theory. Furthermore, the condition for the occurrence of codimension-2 bifurcation (fold-flip bifurcation) is presented. In order to eliminate the chaos or Neimark-Sacker bifurcation of the discrete epidemic model, a tracking controller is designed. The number of the infectives tends to zero when the number of iterations is gradually increasing, that is, the disease disappears gradually. Finally, numerical simulations not only illustrate the validity of the proposed results, but also display the interesting and complex dynamical behaviors.