Analysis of the Dynamics of a Tumor-Immune System with Chemotherapy and Immunotherapy and Quadratic Optimal Control

In this paper, we consider a tumor growth model with the effect of tumor-immune interactions and chemotherapeutic as well as immunotherapeutic drugs. In our model there are four compartments, namely tumor cells, immune cells, chemotherapeutic drug concentration and immunotherapeutic drug concentration. The dynamical behaviour of our system by analyzing the existence and stability of our system at various equilibria is discussed elaborately. We set up an optimal control problem relative to the model so as to minimize the number of tumor cells and the chemotherapeutic and immunotherapeutic drugs administration. Here we use a quadratic control to quantify this goal and consider the administration of chemotherapy and immunotherapeutic drugs as controls to reduce the spread of the tumor growth. The important mathematical findings for the dynamical behaviour of the tumor-immune model are also numerically verified using MATLAB. Finally epidemiological implications of our analytical findings are addressed critically.