A Fitted Operator Method for a System of Delay Model of Tumor Cells Dynamics within their Micro-Environment
This paper deals with the extension of the dynamics modeling the interaction among transformed epithelial cells (TECs), fibroblasts, myofibroblasts, transformed growth factor (TGF−β), and epithelial growth factor (EGF), in silico, in a setup mimicking experiments in a tumor chamber invasion assay. In the sequel of establishing solution continuously depends on the data and existence of unique solution, we were able to extend the Gronwall’s inequality for linear ordinary differential equations, to the Gronwall’s inequality for linear, delay ordinary differential equations. The method of upper and lower solutions is utilized to present that the equilibrium points are globally stable, whereas, equilibrium points are analyzed and the conditions for the existence of Hopf bifurcation are also established. Since it is not possible to solve the extended dynamics, nor the original dynamics, we derive, analyze, implement a fitted operator method and present our results. Analysis of the basic properties of the fitted operator method presents that it is consistent, stable and convergent. Since our numerical results are in agreement with our findings, we thus believe that our findings in this study, can indeed contribute more toward the design of the drug which can slow and/or confine tumor invasion.