MATHEMATICAL MODELLING OF THE INTERACTION BETWEEN CANCER AND THE IMMUNE SYSTEM WITH TREATMENT (THERAPY)

Abstract

Cancer is a disease caused by the accumulation of phenotype-altering genetic mutations in somatic cells. A mathematical model describes the impact of the immunodeficiency virus on the cellular population of tumor-immune interaction cultured with immunotherapy and interleukin IL-2 in terms of mathematical equations and these equations represent the relations between relevant properties of the compartments. The aim of this study is to understand the dynamics of the interaction between cancer and the immune system with immunotherapy treatment, and try to identify possible intervention mechanisms to cure the disease by applying mathematical modelling. The dynamic model is developed based on basic mathematical modelling techniques leading to a system of ordinary differential equations (ODEs). Qualitative analysis of the model applies dimensional analysis, scaling and bifurcation analysis in addition to stability theory for ODE systems. We also derive the equilibrium points of the model and investigate their stability both locally and globally. Our results show that, there are two threshold values of the burst size: below the first threshold the tumor always grows to its maximum size; while passing this threshold, there is a locally stable positive equilibrium solution appearing through transcritical bifurcation: while at or above the second threshold, there exist one or three families of periodic solutions arising from Hopf bifurcations. Our study suggests that the tumor lode can drop to a undetectable level either during the oscillation or when the burst size is large enough. Numerical simulations are carried out applying the numerical software Matlab. These simulations show the behavior of the populations in time and the stability of equilibrium points. The numerical simulation with its estimated and the current medical literature parameters indicates the highlight key values of the burst size of a virus in immunodeficiency virus treatment. When the burst size is smaller than the first threshold value, immunotherapy always fail and it is in between of two threshold values, we have partial success of immunotherapy represented by the stable positive equilibrium solutions.