Global dynamics of an SEIRI epidemiological model with time delay

In this paper, an epidemiological model with disease relapse, nonlinear incidence rate and a time delay representing an exposed (latent) period is investigated. The basic reproduction number is identified. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is completely established. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is proven that if the basic reproduction number is greater than unity, the endemic equilibrium is globally asymptotically stable and the disease becomes endemic; if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable and therefore the disease fades out.