On the dynamics of a class of multi-group models for vector-borne diseases

The resurgence of vector-borne diseases is an increasing public health concern, and there is a need for a better understanding of their dynamics. For a number of diseases, e.g. dengue and chikungunya, this resurgence occurs mostly in urban environments, which are naturally very heterogeneous, particularly due to population circulation. In this scenario, there is an increasing interest in both multi-patch and multi-group models for such diseases. In this work, we study the dynamics of a vector borne disease within a class of multi-group models that extends the classical Bailey-Dietz model. This class includes many of the proposed models in the literature, and it can accommodate various functional forms of the infection force. For such models, the vector-host/host-vector contact network topology gives rise to a bipartite graph which has different properties from the ones usually found in directly transmitted diseases. Under the assumption that the contact network is strongly connected, we can define the basic reproductive number $\mathcal{R}_0$ and show that this system has only two equilibria: the so called disease free equilibrium (DFE); and a unique interior equilibrium---usually termed the endemic equilibrium (EE)---that exists if, and only if, $\mathcal{R}_0>1$. We also show that, if $\mathcal{R}_0\leq1$, then the DFE equilibrium is globally asymptotically stable, while when $\mathcal{R}_0>1$, we have that the EE is globally asymptotically stable

Claude Lobry, un mathématicien militant

International audienceWe show in this communication, that Claude Lobry has always been contributing to mathematics and simultaneously promoting actions to develop a certain idea of acting in mathematicsOn montre ici que Claude Lobry, ne peut s’empêcher de faire à la fois des mathématiques et de développer des actions pour promouvoir une certaine façon de faire des mathématiques

Exponential Stabilization of Nonlinear Systems by an Estimated State Feedback

International audienceIn this paper we investigate the stabi-lizability problem of a class of multi-input multi-output nonlinear systems which linearization at the origin is controllable and observable. Under assumptions on the nonlinear part we prove : (a) the system is globally exponentially stabilizable (G.E.S) by means of linear feedback law. (b) the system can be G.E.S using a state estimation given by an observer

On the stability of nonautonomous systems

International audienceIn (Kalitine, 1982), the use of semi definite Lyapunov functions for exploring the local stability of autonomous dynamical systems has been introduced. In this paper we give an extension of the results of (Kalitine, 1982) that allows to study the local stability of nonautonomous differential systems. We give an application to the Algebraic Riccati Equation

Nonlinear stabilization by adding integrators

International audienceThe global stabilization problem of a nonlinear control system with an integrator is considered when the initial system (subsystem without integrator) is stabilizable. It is supposed that the stabilizing feedback and the Lyapunov function for the initial system satisfy the Barbashin-Krasovski\u ı asymptotic stability theorem. Explicit formulae for stabilizing feedback of a nonlinear system with an integrator are derived. The use of Lyapunov functions with derivatives of constant but not fixed signs significantly simplifies the computing of stabilizing feedback. This is confirmed by examples. The global stabilization of a nonlinear control system of the form \dot x= f(x, y), \qquad \qquad \dot y=u \tag 1 is studied, where x∈ \bbfRⁿ, y∈ \bbfR^p, u∈ \bbfR^p and f is a smooth vector field such that f (0, 0) =0. It is proved that to find a feedback stabilizer for this system we do not need to have a strict Lyapunov function for the subsystem (2) \dot x= f(x, v), where v is the input. Moreover, it is proved how to asymptotically stabilize system (1) without stabilizing system (2)

Modèles contre maladies infectieuses par Gauthier Sallet. Le paludisme fait de la résistance, entretien avec Christophe Rogier, propos recueillis par Dominique Chouchan.

National audienceComment enrayer les ravages d'une maladie telle que le paludisme ? Les modèles numériques devraient au moins offrir un outil de compréhension de sa diffusion et de l'émergence des résistances aux médicaments

Observability, Identifiability and Epidemiology -- A survey

In this document we introduce the concepts of Observability and Iden-tifiability in Mathematical Epidemiology. We show that, even for simple and well known models, these properties are not always fulfilled. We also consider the problem of practical observability and identi-fiability which are connected to sensitivity and numerical condition numbers

Multivariable boundary PI control and regulation of a fluid flow system

International audienceThe paper is concerned with the control of a fluid flow system gov-erned by nonlinear hyperbolic partial differential equations. The control and the output observation are located on the boundary. We study local stability of spatially heterogeneous equilibrium states by using Lyapunov approach. We prove that the linearized system is exponentially stable around each subcritical equilibrium state. A systematic design of proportional and integral controllers is proposed for the system based on the linearized model. Robust stabilization of the closed-loop system is proved by using a spectrum method