Delay differential model for tumour-immune response with chemoimmunotherapy and optimal control

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.