On the global dynamics of a chronic myelogenous leukemia model

• Dynamics of chronic myelogenous leukemia model is analyzed.
• Instability at non-lipschitzian tumor free equilibrium point is shown.
• Ultimate upper and lower bounds for state variables are obtained.
• The existence of positively invariant polytope is proved.
• The iterative localization process is described.

In this paper we analyze some features of global dynamics of a three-dimensional chronic myelogenous leukemia (CML) model with the help of the stability analysis and the localization method of compact invariant sets. The behavior of CML model is defined by concentrations of three cellpopulations circulating in the blood: naive T cells, effector T cells specific to CML and CML cancer cells. We prove that the dynamics of the CML system around the tumor-free equilibrium point is unstable. Further, we compute ultimate upper bounds for all three cell populations and provide the existence conditions of the positively invariant polytope. One ultimate lower bound is obtained as well. Moreover, we describe the iterative localization procedure for refining localization bounds; this procedure is based on cyclic using of localizing functions. Applying this procedure we obtain conditions under which the internal tumor equilibrium point is globally asymptotically stable. Our theoretical analyses are supplied by results of the numerical simulation.