Global stability of an epidemic model with age-dependent vaccination, latent and relapse

The vaccination, latent and relapse period are three important factors affecting the whole disease development. In this paper, we propose an SVEIR epidemic model with continuous age-dependent vaccination, latency and relapse, at the same time, the nonlinear incidence rate is also considered. Uniform persistence of the model is proved by reformulating it as the so called Volterra integral equations. The basic reproduction number R0, which completely determines the global dynamics of the model, is derived. By using Lyapunov functionals, the global stability of the equilibria is obtained. Namely, the disease-free equilibrium is globally asymptotically stable if R0<1, while if R0>1 the endemic equilibrium is globally asymptotically stable. Finally, two numerical examples support our main analytical results.