Epidemiological Models and Lyapunov Functions

¹ INRIA Lorraine & Université Paul Verlaine, Metz LMAM (UMR CNRS 7122) I.S.G.M.P. Bât A, Ile du Saulcy, 57045 Metz Cedex 01, France
² Université de Saint-Louis, Sénégal
³ Université de Yaoundé, Cameroun

Corresponding author: sallet@loria.fr

Abstract

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 ≤ 1, then the disease free equilibrium is globally asymptotically stable. If > 1, there exists an unique endemic equilibrium which is asymptotically stable on the positive orthant.