The dynamics of an optimally controlled tumor model: A case study

We present a phase-space analysis of a mathematical model of tumor growth with an immune response and chemotherapy. We prove that all orbits are bounded and must converge to one of several possible equilibrium points. Therefore, the long-term behavior of an orbit is classified according to the basin of attraction in which it starts. The addition of a drug term to the system can move the solution trajectory into a desirable basin of attraction. We show that the solutions of the model with a time-varying drug term approach the solutions of the system without the drug once traatment has stopped. We present numerical experiments in which optimal control therapy is able to drive the system into a desirable basin of attraction, whereas traditional pulsed chemotherapy is not.