Stability and Hopf bifurcation analysis of an SEIR model with nonlinear incidence rate and relapse

Abstract

In this paper, a delayed SEIR dynamic model with relapse and the nonlinear incidence rate is considered. The basic reproductive rate, R0, is derived and the existence of equilibria is established. It has been shown that if R0 < 1, the disease-free equilibrium is locally asymptotically stable and the disease dies out. The effect of the time delay on stabilities of the endemic equilibrium has been studied. If R0 > 1, the unique endemic equilibrium is local asymptotical stability when time delays are small enough, and when increasing of time delay, it has been shown that the existence of Hopf bifurcations by analyzing the root distribution of the transcendental characteristic equation. Moreover, a sufficient condition has been obtained to establish the local stability properties of the endemic equilibrium. At the end of the paper, some numerical simulations have been presented to confirm the correctness of theoretical analyses.