Optimal control in a mathematical model for tumor escape

Abstract

This paper deals with the medical treatment of tumors which are able to escape the immune system surveillance. A mathematical model for the tumor immune system competition is generalized by introducing a therapy which helps the immune system cells to recognize the tumor cells. The mathematical model consists into a system of two ordinary differential equations modeling the time evolution of the density of the immune system cells and the density of the tumor cells, respectively, and their competition. An optimal control strategy is introduced in order to reduce the percentage of tumor cells which are able to escape the immune system action. Specifically the control represents the percentage of effect that the therapy has on the tumor escape phenomenon production. The therapy lasts for a given period; the results, however, are not interval-dependent. The objective function to be maximized is based on the benefit related to T-cell counts less the systemic cost of the therapy. The model is mathematically analyzed and the existence of the optimal control is investigated by using the Pontryagins Maximum Principle.