Table of Contents
Journal of Nonlinear Dynamics
Volume 2013, Article ID 608598, 13 pages
Research Article

Dynamical Behaviour of a Tumor-Immune System with Chemotherapy and Optimal Control

Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah 711 103, India

Received 22 April 2013; Revised 7 June 2013; Accepted 23 June 2013

Academic Editor: Giovanni P. Galdi

Copyright © 2013 Swarnali Sharma and G. P. Samanta. 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.


We have considered a tumor growth model with the effect of tumor-immune interaction and chemotherapeutic drug. We have considered two immune components—helper (resting) T-cells which stimulate CTLs and convert them into active (hunting) CTL cells and active (hunting) CTL cells which attack, destroy, or ingest the tumor cells. In our model there are four compartments, namely, tumor cells, active CTL cells, helper T-cells, and chemotherapeutic drug. We have discussed the behaviour of the solutions of our system. The dynamical behaviour of our system by analyzing the existence and stability of the system at various equilibrium points is discussed elaborately. We have set up an optimal control problem relative to the model so as to minimize the number of tumor cells and the chemotherapeutic drug administration. Here we used a quadratic control to quantify this goal and have considered the administration of chemotherapy drug as control to reduce the spread of the disease. The important mathematical findings for the dynamical behaviour of the tumor-immune model with control are also numerically verified using MATLAB. Finally, epidemiological implications of our analytical findings are addressed critically.