A Mathematical Model of Tumor-Immune and Host Cells Interactions with Chemotherapy and Optimal Control

In this article, we propose the interaction of tumor cells with the immune system in the presence of chemotherapy. The existence, uniqueness, non-negativity, and boundedness of the solutions have been established. The conditions for the existence and stability of equilibrium points have been presented in both drug-free and treated systems. The local stability of the co-existing equilibrium point is proved using the Routh–Hurwitz rule, and the global stability is proved using the Lyapunov function. We have used quadratic optimal control to minimize the number of tumor cells and the side effects of chemotherapy on the immune system and healthy cells. We have demonstrated the existence of optimal control and derived the corresponding optimality system using Pontryagin’s maximum principle. The optimal system is solved using the forward-backward sweep method with fourth-order Runge–Kutta approximation. Reduction in tumor cell growth has been observed due to the increase in recruitment of immune cells activated by tumor cell antigenicity and the rate of conversion of resting immune cells into active immune cells. Additionally, the impact of administering varying chemotherapy doses on reducing tumor cell growth has been noted. Finally, a comparison between controlled and uncontrolled dynamics has been conducted to comprehend the effect of optimal control.