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 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.

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Abstract. In this paper we study the frequency and time domain behaviour of a heat exchanger network system. The system is governed by hyperbolic partial differential equations. Both the control operator and the observation operator are unbounded but admissible. Using the theory of symmetric hyperbolic systems, we prove exponential stability of the underlying semigroup for the heat exchanger network. Applying the recent theory of well-posed infinite-dimensional linear systems, we prove that the system is regular and derive various properties of its transfer functions, which are potentially useful for controller design. Our results remain valid for a wide class of processes governed by symmetric hyperbolic systems.Mathematics Subject Classification. 93D09, 93D25, 80A20, 35L50.

International audienceIn this paper we propose a malaria within-host model with k classes of age for the parasitized red blood cells and n strains for the parasite. We provide a global analysis for this model. A competitive exclusion principle holds. If R0, the basic reproduction number, satisfies R0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable. On the contrary if R0 > 1, then generically there is a unique endemic equilibrium which corresponds to the endemic stabilization of the most virulent parasite strain and to the extinction of all the other parasites strains. We prove that this equilibrium is globally asymptotically stable on the positive orthant if a mild sufficient condition is satisfied

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International audienceWe consider SIS, SIR and MSIR models with standard mass action and varying population, with $n$ different pathogen strains of an infectious disease. We also consider the same models with vertical transmission. We prove that under generic conditions a competitive exclusion principle holds. To each strain a basic reproduction ratio can be associated. It corresponds to the case where only this strain exists. The basic reproduction ratio of the complete system is the maximum of each individual basic reproduction ratio. Actually we also define an equivalent threshold for each strain. The winner of the competition is the strain with the maximum threshold. It turns out that this strain is the most virulent, i.e., this is the strain for which the endemic equilibrium gives the minimum population for the susceptible host population. This can be interpreted as a pessimization principle.On considère les modèles SIS, SIR et MSIR avec la loi de l'action de masse standard et une population non constante, avec n différentes souches de pathogènes. Nous considérons aussi les même modèles avec transmission verticale. On prouve que sous une condition générique, le principe de compétition exclusive est vérifié. Pour chaque souche, un nombre de reproduction de base est associé. Il correspond au cas où seule cette souche existe. Le nombre de reproduction de base du système complet est le maximum de tous les nombres de reproduction de base pris individuellement. Nous définissons aussi un seuil équivalent pour chaque souche. La souche qui gagne la compétition est celle qui maximise le nombre de reproduction de base. C'est aussi la souche la plus virulente, i.e., c'est la souche pour laquelle l'équilibre endémique donne le minimum des individus susceptibles dans la population hôte. C'est le principe de pessimisation

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