Mathematical study of certain epidemiological models

Abstract

The objective of this memoire is to examine the intricacies of a reaction-diffusion (susceptib infectious-susceptible) SIS epidemic model featuring a nonlinear incidence rate, which characterizes the spread of a contagious illness among people. We demonstrate that, given a single condition, the suggested model has two steady states. We establish the local and global asymptotic stability of the non-negative constant steady states subject to the basic reproduction number being greater than unity and of the disease-free equilibrium subject to the basic reproduction number being smaller than or equal to unity in the ODE case by analyzing the eigenvalues, and using an appropriately constructed Lyapunov function. Through the application of a suitably constructed Lyapunov function, we determine the global stability condition in the PDE scenario. This is done by comparing 𝑅 with one, where in the case of 1 < 𝑅 we found that the system accepts global stability in the vicinity of the point 𝐸∗but in the case of 1 > 𝑅 the system accepts global stability in the vicinity of the point 𝐸 . Finally, we provide a few numerical examples that both illustrate and validate the analytical findings that have been made throughout the work.