Global stability of an age-structure epidemic model with imperfect vaccination and relapse

A new age-structured epidemic model with imperfect vaccination and relapse is proposed. The total population of our model is partitioned into five subclasses: susceptible class S, vaccinated class V, exposed class E, infectious class I and removed class R. Age-structures are equipped with in exposed and recovered classes. Furthermore, imperfect vaccination is also introduced in our model. The basic reproduction number R0 is defined and proved as a threshold parameter of the model. Asymptotic smoothness of solutions and uniform persistence of the system are showed via reformulating the system as a system of Volterra integral equation. Furthermore, by constructing proper Volterra-type Lyapunov functional we get when R0<1, the disease-free equilibrium is globally asymptotically stable. When R0>1, the endemic equilibrium is globally stable. Our results show that to increase the efficiency of vaccination and reduce influence of relapse are vital essential for controlling epidemic.