Global dynamics of an SEIS epidemiological model with time delay describing a latent period

In this paper, an SEIS epidemiological model with a saturation incidence rate and a time delay representing the latent period of the disease is investigated. By means of Lyapunov functional, LaSalle's invariance principle and comparison arguments, it is shown that the global dynamics is completely determined by the basic reproduction number. It is proven that the basic reproduction number is a global threshold parameter in the sense that if it is less than unity, the disease-free equilibrium is globally asymptotically stable and therefore the disease dies out; whereas if it is greater than unity, there is a unique endemic equilibrium which is globally asymptotically stable and thus the disease becomes endemic in the population. Numerical simulations are carried out to illustrate the main results.