A Cancer Model for the Angiogenic Switch and Immunotherapy: Tumor Eradication in Analysis of Ultimate Dynamics

In this paper, we study ultimate dynamics and derive tumor eradication conditions for the angiogenic switch model developed by Viger et al. This model describes the behavior and interactions between host (x); effector (y); tumor (z); endothelial (w) cell populations. Our approach is based on using the localization method of compact invariant sets and the LaSalle theorem. The ultimate upper bound for each cell population and ultimate lower bound for the effector cell population are found. These bounds describe a location of all bounded dynamics. We construct the domain bounded in x- and z-variables which contains the attracting set of the system. Further, we derive conditions imposed on the model parameters for the location of omega-limit sets in the plane w = 0 (the case of a localized tumor). Next, we present conditions imposed on the model and treatment parameters for the location of omega-limit sets in the plane z = 0 (the case of global tumor eradication). Various types of dynamics including the chaotic attractor and convergence dynamics are described. Numerical simulation illustrating tumor eradication theorems is fulfilled as well.