Global stability for an special SEIR epidemic model with nonlinear incidence rates

A SEIR epidemic model with nonlinear incidence rates, constant recruitment and disease-caused death in epidemiology is considered. It is shown that the global dynamics is completely determined by the contact number R0. If R0 ⩽ 1, the disease-free equilibrium is globally stable and the disease dies out. If R0 > 1, the unique endemic equilibrium is globally stable in the interior of the feasible region by using the methods established in Butler GJ, Freedman HI, Waltman P. Uniformly persistent systems, Proc Am Math Soc 1986;96:425–30, and the disease persists at the endemic equilibrium.