Some Systems of Ordinary Differential Equations from Cancer Modeling: Qualitative Analysis and Optimal Treatment

Abstract

In this thesis, we study four systems of ordinary differential equations, which model the interrelationships between different cell populations while tumor cells exist, and treatments, such as immunotherapy, chemotherapy, and radiotherapy, are applied.

For the first two models, we only consider the case with radiation treatment. For the first model, we consider a single general cell population with its corresponding radiated cell population. Meanwhile, two different kinds of radiation are studies separately: constant and decay; for the second model, we consider the host and tumor cell populations together with their corresponding radiated cell populations, which behave in the same way as that in the first model.

For the third and fourth models, we consider both the immunotherapy and chemotherapy. The third model includes three cell populations: host cells, tumor cells, and immune cells, as well as the drug concentration. We study the properties of its null-surfaces, equilibria, and the stability of them; in the fourth model, we not only extend the previous model into one with six populations, with the immune cells in the third model being specified into three different ones: CD8$^+$T cells, circulating lymphocytes, and IL-2. but also focus on the situation when controls are added in a linear manner. We investigate the existence of controls and find the characterization of optimal bang-bang control.