Epidemiological models and Lyapunov functions

A. A. Fall (1, 2, 3) , Abderrahman Iggidr (2, 3) , Gauthier Sallet (2, 3) , Jean-Jules Tewa (2, 3, 4)

1 LANI - Laboratoire d'Analyse Numérique et Informatique [Sénégal]
2 LMAM - Laboratoire de Mathématiques et Applications de Metz
3 MASAIE - Tools and models of nonlinear control theory for epidemiology and immunology
4 Département de Mathématiques [Yaoundé I] = Department of Mathematics [Yaoundé, Cameroon]

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 R0 \leq 1, then the disease free equilibrium is globally asymptotically stable. If R0 > 1, there exists an unique endemic equilibrium which is asymptotically stable on the positive orthant.

Mots clés: Nonlinear dynamical systems; Global stability; Lyapunov methods; Differential susceptibility models.; Differential susceptibility models

Domaines: Systèmes dynamiques [math.DS]; Santé publique et épidémiologie

A. A. Fall, Abderrahman Iggidr, Gauthier Sallet, Jean-Jules Tewa. Epidemiological models and Lyapunov functions. Mathematical Modelling of Natural Phenomena, 2007, 2 (1), pp.62 - 83. ⟨10.1051/mmnp:2008011⟩. ⟨inria-00596236⟩