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Therapy strategy in tumour cells and immune system interaction mathematical model

In this paper, we propose a method to optimise a theoretical control on a mathematical model of interactions between cancer cells and the immune system. The model consists of five ordinary differential equations describing the behaviour of a tumour cell population interacting with populations of immune cells, all cells being submitted to the effects of chemotherapy. Two constraints were added to reflect the biological specificity of the problem. The first one asserts that the number of cytotoxic T lymphocytes must remain above a given threshold at the terminal point in time. The second one keeps the drug concentration in the bloodstream below a threshold to reduce toxic side effects resulting from the damages of chemotherapy to healthy cells. Using Pontryagin’s Maximum Principle and approximation methods for optimal control, we obtain bang–bang solutions for different scenarios.