Journal of Applied Mathematics
Volume 2012 (2012), Article ID 275902, 14 pages
http://dx.doi.org/10.1155/2012/275902
Research Article

Modeling the Dynamics of an Epidemic under Vaccination in Two Interacting Populations

Ibrahim H. I. Ahmed, Peter J. Witbooi, and Kailash Patidar

Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa

Received 12 December 2011; Revised 11 April 2012; Accepted 18 April 2012

Academic Editor: Livija Cveticanin

Copyright © 2012 Ibrahim H. I. Ahmed 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

We present a model for an SIR epidemic in a population consisting of two components—locals and migrants. We identify three equilibrium points and we analyse the stability of the disease free equilibrium. Then we apply optimal control theory to find an optimal vaccination strategy for this 2-group population in a very simple form. Finally we support our analysis by numerical simulation using the fourth order Runge-Kutta method.