Optimal control of tumour-immune model with time-delay and immuno-chemotherapy

Herein, we study an optimal control problem of delay differential model to describe the dynamics of tumour-immune interactions in presence of immuno-chemotherapy. The model includes constant delays in the mitotic phase to justify the time required to stimulate the effector cells and for the effector cells to develop a suitable response to the tumour cells. By applying optimal control theory, we seek to minimize the cost associated with the immuno-chemotherapy and to reduce load of of tumour cells. Non-Negativity of the solutions of the model and existence of an optimal control has also been proven. Optimality conditions and characterization of the control are also discussed. We numerically approximate the solution of the optimal control problem by solving the state system forward and adjoint system backward in time. The numerical simulations show that the combination of immuno-chemotherapy protocol reduces the tumour load in few months of therapy.