ROLE OF ANTIBODIES: A PARADIGM IN MATHEMATICAL MODELING FOR CANCER TREATMENT

Abstract

SHIFERAW FEYISSA BALCHA, Role of Antibodies: A Paradigm in Mathematical Modeling for Cancer Treatment. (Under the supervision of Dr. Sandip Banerjee). In this dissertation, we propose a mathematical model using a system of non-linear ordinary differential equations (ODEs), delay differential equations (DDEs) and partial differential equations (PDEs) -for the qualitative and quantitative analy-sis of cancer immune interaction, considering the role of humoral (antibody) mediated immune responses. The model is based on the clinical evidence, which states that antibodies can directly kill cancerous cells. We use Sotomayor theorem to show the existence of transcritical and saddle node bifurcations, which provide strong biological implications. Through numerical simulations, we illustrate that under certain therapy (like monoclonal antibody therapy), which is capable of altering system parameters, cancer free state can be achieved. Next, we add two discrete time delays, namely, maturation and interaction delays, to assess its effect on the interaction between can-cerous cells and the antibodies. After categorizing the parametric plane into different regions based on the existence of equilibria, we investigate both analytically and through simulations, the stability of equilibria and the onset of sustained oscillations through Hopf bifurcations. We observe that both the delays play an important role in stability switching. Appropriate therapy with proper choice of system parameters are suggested to obtain cancer free equilibrium point. We have also considered the spatiotemporal aspect of the model. In view of that, from the stability analysis of the it system, we observe that the temporally stable positive interior equilibrium point of the system also remain stable under the spatio-temporal perturbations. Numerically, it is shown that for a particular choice of the system parameters, the model was able to simulate the phenomenon of cancer dormancy