Math. Model. Nat. Phenom.
Volume 2, Number 1, 2007Epidemiology
|Page(s)||62 - 83|
|Published online||15 June 2008|
Epidemiological Models and Lyapunov Functions
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: firstname.lastname@example.org
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.
Mathematics Subject Classification: 34A34 / 34D23 / 34D40 / 92D30
Key words: nonlinear dynamical systems / global stability / Lyapunov methods / differential susceptibility models
© EDP Sciences, 2007
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