In this thesis, our interest is on the aspect in space of the establishment of a spatial model in epidemiology and the conditions leading to the stability of the systems that we present, in epidemiology, from the classical models by Ross and Mckendrick. Firstly, we intend to examine the eects of the Normalized DiFerence Vegetation Index(NDVI) in a model of contamination of malaria in Bankoumana, a region in Mali. From the system obtained, we willnd the basic reproduction rate. Then we deduce two point of equilibrium, among which one point of equilibrium without the disease and another one with an endemic point. The latter with the basic reproduction rate vary according to the indices of normalized vegetation. Then, we will build a model having equations delay, containing the NDVI. The rate of basic reproduction and the two points of equilibrium that come from our system depend upon the delay introduced. We will show that the stability of our points of equilibrium is not only dependent upon the basic reproduction rate, but also closely related to the delays introduced. In another way, we will divide the region of study in areas where we will set hypotheses that the rate of contamination brought about by individuals in an area of study on the others, can be dierent. It will permit us to obtain a system in which we will determine the points of equilibrium and the conditions that will lead us to obtain the stability according to Lyapunov. Then, we will disturb the previous system at the level of its unique endemic point of equilibrium, with the introduction of an additional noise. The conditions leading to stability according to Lyapunov, on the new system obtained, are generally deduced here. In a similar framework, we will elaborate a multigroups model, in which we will introduce spatial coordinates. The groups are formed according to a closeness depending to a radius of a circle at random. Here, the rate of contamination is supposed to be uniform in the groups. After having determined the point of equilibrium and the rate of basic reproduction, we will nd the conditions facilitating stability in as by Lyapunov in a global framework. In the order1, it means that supposing that we have only one group, the conditions of stability are obtained according to the Routh-Hurvitz criteria.
Spatial aspect in the epidemiological modeling