Table of Contents
International Journal of Partial Differential Equations
Volume 2014 (2014), Article ID 186437, 6 pages
http://dx.doi.org/10.1155/2014/186437
Research Article

Partial Differential Equations of an Epidemic Model with Spatial Diffusion

1Department of Mathematics and Computer Science, Faculty of Sciences Ben M’sik, Hassan II University, P.O. Box 7955, Sidi Othman, Casablanca, Morocco
2Centre Régional des Métiers de l’Education et de la Formation, 20340 Derb Ghalef, Casablanca, Morocco

Received 30 August 2013; Revised 6 December 2013; Accepted 20 December 2013; Published 10 February 2014

Academic Editor: William E. Fitzgibbon

Copyright © 2014 El Mehdi Lotfi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.