# American Institute of Mathematical Sciences

2005, 2(3): 499-510. doi: 10.3934/mbe.2005.2.499

## Immunotherapy: An Optimal Control Theory Approach

 1 Department of Mathematics and Statistics, Murray State University, 6C Faculty Hall, Murray, KY 42071, United States

Received  January 2005 Revised  May 2005 Published  August 2005

We investigate mathematical models for the dynamics between tumor cells, immune-effector cells, and cytokine interleukin-2 (IL-2). To better determine under what circumstances the tumor can be eliminated, we implement optimal control theory. We design two control functionals, the first functional having one control and the second having two controls, to maximize the effector cells and interleukin-2 concentration and to minimize the tumor cells. Next, we show that bang-bang optimal controls exist for each problem. Then, we characterize our optimal controls in terms of the solutions to the optimality system, which is the state system coupled with the adjoint system. Finally, we analyze the various optimal controls and optimality systems using numerical techniques.
Citation: K. Renee Fister, Jennifer Hughes Donnelly. Immunotherapy: An Optimal Control Theory Approach. Mathematical Biosciences & Engineering, 2005, 2 (3) : 499-510. doi: 10.3934/mbe.2005.2.499
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