Rui Xu

A mathematical model describing the transmission dynamics of an infectious disease with an exposed (latent) period, relapse and a saturation incidence rate is investigated. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is established. By using suitable Lyapunov functionals and LaSalle’s invariance principle, it is proven that if the basic reproduction number is less than unity, the diseasefree equilibrium is globally asymptotically stable and therefore the disease fades out; and if the basic reproduction number is greater than unity, the endemic equilibrium is globally asymptotically stable and the disease becomes endemic.

In this paper, a predator-prey system with Holling type II functional response and stage structure is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is studied. The existence of the orbitally asymptotically stable periodic solution is established. By using suitable Lyapunov functions and the LaSalle invariance principle, it is proven that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and sufficient conditions are derived for the global stability of the coexistence equilibrium.

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.

A hepatitis B virus infection model with time delay is discussed. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the model is studied. By using comparison arguments, it is proved that if the basic reproduction ratio is less than unity, the infection‐free equilibrium is globally asymptotically stable. If the basic reproduction ratio is greater than unity, by means of an iteration technique, sufficient conditions are derived for the global asymptotic stability of the virus‐infected equilibrium. Numerical simulations are carried out to illustrate the theoretical results.