We have considered a tumor growth model with the effect of tumor-immune interaction and chemotherapeutic drug. We have considered two immune components—helper (resting) T-cells which stimulate CTLs and convert them into active (hunting) CTL cells and active (hunting) CTL cells which attack, destroy, or ingest the tumor cells. In our model there are four compartments, namely, tumor cells, active CTL cells, helper T-cells, and chemotherapeutic drug. We have discussed the behaviour of the solutions of our system. The dynamical behaviour of our system by analyzing the existence and stability of the system at various equilibrium points is discussed elaborately. We have set up an optimal control problem relative to the model so as to minimize the number of tumor cells and the chemotherapeutic drug administration. Here we used a quadratic control to quantify this goal and have considered the administration of chemotherapy drug as control to reduce the spread of the disease. The important mathematical findings for the dynamical behaviour of the tumor-immune model with control are also numerically verified using MATLAB. Finally, epidemiological implications of our analytical findings are addressed critically. 1. Introduction It is well known that cancer is one of the greatest killer diseases in the world. Cancer, known medically as a malignant neoplasm, is characterized by an abnormal growth of cells. In cancer, cells divide and grow uncontrollably forming malignant tumors and invade nearby parts of the human body. The cancer may also spread to more distant parts of the body through the lymphatic system or bloodstream. Not all tumors are cancerous. The tumors which do not grow uncontrollably, do not invade neighbouring tissues, and do not spread throughout the body are not cancerous. There are over 200 different cancers including breast cancer, skin cancer, lung cancer, ovarian cancer, brain cancer, colon cancer, prostate cancer, and lymphoma and cancer. that afflict humans. Cancer symptoms vary widely based on the type of cancer. Cancer immunology is the study of interactions between the immune system and cancer cells. It is also a growing field of research that aims to discover innovative cancer immunotherapies to treat and retard progression of this disease. In the last few decades, immunotherapy has become a significant part of treating several types of cancer. Human immune system is a collection of organs, special cells, and substances that help to protect from infections and some other diseases. Immune system cells and the substances they make travel
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