Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model

In this paper, an SEIRS system with two delays and the general nonlinear incidence rate is considered. The positivity and boundedness of solutions are investigated. The basic reproductive number, R0R0, is derived. If R0≤1R0≤1, then the disease-free equilibrium is globally asymptotically stable and the disease dies out. If R0>1R0>1, then there exists a unique endemic equilibrium whose locally asymptotical stability and the existence of local Hopf bifurcations are established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions is derived by using the center manifold and the normal form theory. Furthermore, there exists at least one positive periodic solution as the delay varies in some regions by using the global Hopf bifurcation result of Wu for functional differential equations. If R0>1R0>1, then the sufficient conditions of the permanence of the system are obtained, i.e., the disease eventually persists in the population. Especially, the upper and lower boundaries that each population can coexist are given exactly. Some numerical simulations are performed to confirm the correctness of theoretical analyses.