Ultimate Dynamics Analysis of the 5D Structural Leukemia Model and Partitioning of the Parameter Space

Abstract

In this paper, we study the global dynamics of the 5D structural leukemia model with 14 parameters as developed by Clapp et al. [2015]. This model describes the interaction between leukemic cell populations and the immune system. Our analysis is based on the localization method of compact invariant sets. We develop this method by introducing the notion of a partitioning of the parameter space and the notion of a localization set corresponding to this partitioning as its parameters change. Further, we obtain ultimate upper and lower bounds for all variables of a state vector without imposing additional restrictions. Local asymptotic stability conditions with respect to the leukemia-free equilibrium point (EP) are given. We deduce formulas describing inner EPs expressed in terms of positive roots of one 7D equation. Based on this equation, it is shown that the number of inner EPs cannot exceed 3 and one case of a global bifurcation of EPs is detected. Next, we prove the existence of the attracting set. Further, in two theorems, the global eradication/extinction leukemia theorems are described. The impact of using parametrically variable localization sets for a qualitative analysis of the ultimate behavior of leukemic cell populations is discussed.