Local dynamics, chaos control and flip bifurcation analysis of a discrete epidemic model with vital dynamics and vaccination

Abstract

We study the local dynamics and existence of bifurcation sets at equilibrium states, bifurcation analysis and chaos of the epidemic model with vital dynamics and vaccination in R+2={(I,S): I,S≥0}. More specifically, it is proved that discrete epidemic model has disease-free and endemic equilibrium states under model’s parameters restriction(s), and we have studied local dynamical properties at equilibrium states by the theory of linear stability. Furthermore, first we have pointed out the bifurcations sets at equilibrium states, and then proved that at disease-free equilibrium state discrete epidemic model does not undergo flip bifurcation but it undergoes only flip bifurcation at endemic equilibrium state by center manifold theorem and bifurcation theory. Additional, hybrid control strategy is utilized to control chaos in the epidemic model due to the occurrence of flip bifurcations. Finally, numerical simulations are given to verify theoretical results.