Journal of Applied Mathematics
Volume 2012 (2012), Article ID 946504, 17 pages
http://dx.doi.org/10.1155/2012/946504
Research Article

Presentation of Malaria Epidemics Using Multiple Optimal Controls

Abid Ali Lashari,1 Shaban Aly,2,3 Khalid Hattaf,4 Gul Zaman,5 Il Hyo Jung,6 and Xue-Zhi Li7

1Center for Advanced Mathematics and Physics, National University of Sciences and Technology, Islamabad 44000, Pakistan
2Department of Mathematics, Faculty of Science, King Khalid University, Abha 9004, Saudi Arabia
3Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71511, Egypt
4Department of Mathematics and Computer Science, Faculty of Sciences Ben M'sik, Hassan II University, Casablanca, Morocco
5Department of Mathematics, University of Malakand, Chakdara Dir (Lower), Khyber Pukhtunkhwa, Pakistan
6Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
7Department of Mathematics, Xinyang Normal University, Xinyang 64000, China

Received 9 March 2012; Revised 4 April 2012; Accepted 6 April 2012

Academic Editor: Junjie Wei

Copyright © 2012 Abid Ali Lashari 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

An existing model is extended to assess the impact of some antimalaria control measures, by re-formulating the model as an optimal control problem. This paper investigates the fundamental role of three type of controls, personal protection, treatment, and mosquito reduction strategies in controlling the malaria. We work in the nonlinear optimal control framework. The existence and the uniqueness results of the solution are discussed. A characterization of the optimal control via adjoint variables is established. The optimality system is solved numerically by a competitive Gauss-Seidel-like implicit difference method. Finally, numerical simulations of the optimal control problem, using a set of reasonable parameter values, are carried out to investigate the effectiveness of the proposed control measures.