Global dynamics of a vector disease model with saturation incidence and time delay

In this paper, we study a vector disease model with a saturation incidence rate and a time delay describing a fixed time during which the infectious agents develop in the vector and it is only after that time that the infected vector can infect a susceptible human. By analysing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is discussed. The existence of Hopf bifurcations is established. By using the persistence theory for infinite-dimensional dynamic systems, it is proved that when the basic reproduction number is greater than unity, the disease is permanent. By comparison arguments, it is verified that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium by means of an iteration scheme.