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Computational and Mathematical Methods in Medicine
Volume 2014 (2014), Article ID 982978, 15 pages
http://dx.doi.org/10.1155/2014/982978
Research Article

Delay Differential Model for Tumour-Immune Response with Chemoimmunotherapy and Optimal Control

1Department of Mathematical Sciences, College of Science, UAE University, P.O. Box 15551, Al-Ain, UAE
2Department of Mathematics, Faculty of Science, Helwan University, Cairo 11795, Egypt
3Zayed bin Sultan Al Nahyan Center for Health Sciences, College of Medicine and Health Sciences, UAE University, P.O. Box 17666, Al-Ain, UAE
4Institut für Angewandte Mathematik, LS III, TU Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany
5Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt

Received 13 May 2014; Revised 1 July 2014; Accepted 3 July 2014; Published 14 August 2014

Academic Editor: Loredana G. Marcu

Copyright © 2014 F. A. Rihan 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 delay differential model with optimal control that describes the interactions of the tumour cells and immune response cells with external therapy. The intracellular delay is incorporated into the model to justify the time required to stimulate the effector cells. The optimal control variables are incorporated to identify the best treatment strategy with minimum side effects by blocking the production of new tumour cells and keeping the number of normal cells above 75% of its carrying capacity. Existence of the optimal control pair and optimality system are established. Pontryagin’s maximum principle is applicable to characterize the optimal controls. The model displays a tumour-free steady state and up to three coexisting steady states. The numerical results show that the optimal treatment strategies reduce the tumour cells load and increase the effector cells after a few days of therapy. The performance of combination therapy protocol of immunochemotherapy is better than the standard protocol of chemotherapy alone.