Global stability of a multi-group SVIR epidemic model

In this paper, we investigate the global stability of a multi-group SVIR epidemic model, with which we can consider the heterogeneity of population and the effect of immunity induced by vaccination. We prove that the basic reproduction number ℛ0ℛ0 plays the role of a threshold for the long-time behavior of the model, that is, the disease-free equilibrium E0E0 of the model is globally asymptotically stable if ℛ0≤1ℛ0≤1 while an endemic equilibrium E∗E∗ exists uniquely and is globally asymptotically stable if ℛ0>1ℛ0>1. For the proofs we use the classical method of Lyapunov, a recently developed graph-theoretic approach and an approach of max functions.