Seeking bang-bang solutions of mixed immuno-chemotherapy of tumors

It is known that a beneficial cancer treatment approach for a single patient often involves the administration of more than one type of therapy. The question of how best to combine multiple cancer therapies, however, is still open. In this study, we investigate the theoretical interaction of three treatment types (two biological therapies and one chemotherapy) with a growing cancer, and present an analysis of an optimal control strategy for administering all three therapies in combination. In the situations with controls introduced linearly, we find that there are conditions on which the controls exist singularly. Although bang-bang controls (on-off) reflect the drug treatment approach that is often implemented clinically, we have demonstrated, in the context of our mathematical model, that there can exist regions on which this may not be the best strategy for minimizing a tumor burden. We characterize the controls in singular regions by taking time derivatives of the switching functions. We will examine these representations and the conditions necessary for the controls to be minimizing in the singular region. We begin by assuming only one of the controls is singular on a given interval. Then we analyze the conditions on which a pair and then all three controls are singular