Modélisation et étude mathématique de la propagation d’une maladie vectorielle (paludisme) au sein d’une population.

Abstract

The main purpose of this thesis is to study a class of mathematical models describing some problems related to the infection by the Plasmodium falciparum parasite which causes malaria and whose vector is the female mosquito of the species Anopheles. The work is divided into three main parts, the first part is related to the analysis of the spread of malaria in an isolated population. The global stability of the disease-free equilibrium is studied according to the different epidemiological parameters when the basic reproduction number is lower than one. When this number is higher than one, the existence of a unique endemic equilibrium is proved. Inspired by the geometric approach introduced by Li and Muldowney, we provided a sufficient condition for this endemic equilibrium to be globally asymptotically stable. A state estimator was constructed to estimate the size of human populations based on the measurement of the number of newly infected humans per unit time. We also proposed two control strategies to eradicate the disease. Finally, to better understand the dynamics of the spread of the disease and to identify the most influential parameters, we have studied the local sensitivity of the number of basic reproduction with respect to each parameter. The second part is about the study of a model that describes the interaction and the spread of the disease within a human population that is divided into two subpopulations, local and non-local. The first subpopulation follows a linear growth while the non-local population follows a logistic growth among the first. We choose to study the impact of the migration of people from an endemic country to another country declared free of the disease or towards the eradication of the disease. Our analysis yielded conditions of the persistence of the disease, we studied the possibility of controlling the disease in a first step through the control of the carrying capacity, then we developed a method based on a matrix called matrix of vectorial transmission which was used to determine the link between the two subpopulations and the population of mosquitoes, according to the values of this matrix entries in order to ensure the control of the disease spread. In addition, a local and global sensitivity study of the level of local and non-local infection was performed to determine the most influential model input parameters. The last part is devoted to the study of the global dynamics of models with multiple subpopulations that are assumed to be weakly interconnected. Our work highlights a process that allows us to perform a complete analysis of many dynamical systems modeling the spread of a disease that involves different populations. The objective is to be able to determine the global stability of the disease-free equilibrium when the basic reproduction number is less than one as well as the global stability of the different types (interior or frontier) of endemic equilibria as a function of the different local basic reproduction numbers and the nature of the interconnections between the network components.