Résumé:
The objective of this thesis is to introduce the field of epidemiology and its relationship with mathematics, as well as how it is modeled using partial differential equations. We specifically focus on the epidemic reaction-diffusion model for the spread of HIV, with the aim of studying the long-term stability of its solutions. We demonstrate that the model contains two types of equilibrium points for solving the proposed system, which describes the transmission of the infectious disease among individuals. The epidemic model is analyzed using the reproductive number, R0. We study both local and global stability using the Jacobian matrix and the appropriate Lyapunov function. Finally, we present numerical examples of simulation processes that illustrate the findings discussed throughout the thesis.
