The Optimal Control of Tumor Virotherapy Treatment Model with External Supplies

This paper focuses on optimal control of a tumor virotherapy treatment model with external supplies such as chemo-drugs or viral injection. This model contains four subpopulation compartments: uninfected tumor cells by viruses, infected tumor cells, oncolytic viruses, and immune cells. The model is given by the system of ordinary differential equations. Chemo-drugs can slow down the growth of tumor cells and immune cells while the viral injection accelerate the growth of oncolytic viruses. We defined a positive control with maximum tolerated dosage (MTDs) as upper bound, that is, u₁ and u₂ represent chemotherapy drug dosage and viral injection dosage, respectively. The objective of this optimal control is to minimize tumor density and side effects of the controls. Here, we get optimal control by applying Pontryagin's Minimum Principle. Numerical simulation is implemented using the Forward-Backward Sweep Method with three strategies: u₁ only, u₂ only, and u₁ with u₂, respectively. Here, we set u₁ weight much larger than u₂ weight, that is, 50 and 1, respectively. The simulation shows that the third strategy (u₁ with u₂) has minimum side effects and density of tumors than the others controls combination. The chemo-drugs can be consumed as much as MTDs in the first quarter and need to be reduced to depletion at the end of controls. The viral injection can be consumed up to 80 percent of the MTDs at the start and needs to be reduced exponentially until the end at the period of the controls.