Global stability of vaccine-age/staged-structured epidemic models with nonlinear incidence

We consider two classes of infinitely dimensional epidemic models with non- linear incidence, where one assumes that the rate of a vaccinated individual losing im- munity depends on the vaccine-age and another assumes that, before the vaccine begins to wane, there is a period during which the vaccinated individuals have complete im- munity against the infection. The first model is given by a coupled ordinary-hyperbolic differential system and the second class is described by a delay differential system. We calculate their respective basic reproduction numbers, and show they characterize the global dynamics by constructing the appropriate Lyapunov functionals.