Epidemiological Models and Lyapunov Functions

We give a survey of results on global stability for deterministic compartmental epidemiological models. Using Lyapunov techniques we revisit a classical result, and give a simple proof. By the same methods we also give a new result on differential susceptibility and infectivity models with mass action and an arbitrary number of compartments. These models encompass the so-called differential infectivity and staged progression models. In the two cases we prove that if the basic reproduction ratio R-0 <= 1, then the disease free equilibrium is globally asymptotically stable. If R-0 > 1, there exists an unique endemic equilibrium which is asymptotically stable on the positive orthant.