Analyse asymptotique d’un modèle épidémique de la dynamique du Choléra.

Abstract

Cholera is an infectious desease caused by the pathogen Vibrio Cholerae. Many mathematical models have been proposed to study the dynamic of this desease, like J. Wang and S. Liao model (Wang et Liao 2010). This model is make up of four compartments : S the Susceptibles, I the infected populations, R the recovered populations and B the concentration of Vibrio Cholerae in water resource. The specificity of this model is that, it resumes the multiple transmission pathways of the disease by the function f(I,B), which is also termed the incidence function, and the growth of the concentration of Vibrio Cholerae is characterised by the function h(I,B). In this work, we are going to analyse the local and global asymptotical stability of the points of equilibrium of this model in two cases which are : the case where the incidence function depends only on B and the other case where it depends linearly on I and non-linearly on B. In order to achive this, we will make use of the Lyapunov function, the theory of monotone dynamical systems, and geometric approach. Finally we are going to perform certain numerical simulations with the aid of the MATLAB program in order to visualize the stability of these points of equilibrium. The analysis presented here permits us to have a deeper understanding of the fundamental mechanism in cholera dynamics.