A new way of constructing Lyapunov functions with application to an SI epidemic model

Abstract

We reconsider an SI epidemic model studied by Li et al. (2007). The model is shown to have complicated dynamics including backward bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation. But for the case where the basic reproductive number is larger than one, the global stability of the unique endemic equilibrium is not investigated. In this paper, using a novel function, we construct a new type of Lyapunov functions to derive a sufficient condition on the global stability of the endemic equilibrium, which is weaker than the one obtained by using the traditional way of constructing Lyapunov functions. We also propose some other ways of constructing Lyapunov functions in the discussion.