Stability Analysis of Equilibria for a Model of Maintenance Therapy in Acute Lymphoblastic Leukemia

Abstract

In this paper, we study two mathematical models, involving delay differential equations, which describe the processes of erythropoiesis and leukopoiesis in the case of maintenance therapy for acute lymphoblastic leukemia. All types of possible equilibrium points were determined, and their stability was analyzed. For some of the equilibrium points, conditions for parameters that imply stability were obtained. When this was not feasible, due to the complexity of the characteristic equation, we discuss the stability through numerical simulations. An important part of the stability study for each model is the examination of the critical case of a zero root of the characteristic equation. The mathematical results are accompanied by biological interpretations.

Partial Stability in a Model for Allergic Reactions Induced by Chemotherapy of Acute Lymphoblastic Leukemia

“…such that Rm = ER m and R50 = ECR 50 (see [6]). Here Rm is the maximum effect of drug on erythrocytes and R50 is the saturation constant for drug on erythrocytes.…”

“…Leukemic cells normally infiltrate the bloodstream swiftly. They can spread to other organs such the lymph nodes, liver, spleen, etc [7].…”

“…Chemotherapy's goal is that, by using drugs, to stop or decrease the growth of cancerous cells. Following the first round of chemotherapy, front line of maintenance therapy is to give the patient 6-MT (mercaptopurine) orally [7,21].…”

“…In order to mimic the ALL-immune dynamics under therapy, we use DDEs and some previous ideas (see [6], [7], [15]) to model the allergic reactions. We incorporate desensitization for mercaptopurine in our model to study the avoidance of hypersensitivity to this drug.…”

Preface to the Special Issue on “Advances in Differential Dynamical Systems with Applications to Economics and Biology”

“…Badralexi et al [8] analyzed the processes of erythropoiesis and leukopoiesis in the context of maintenance therapy for acute lymphoblastic leukemia by considering two mathematical models expressed by delay differential equations. The stability of every equilibrium point is examined either analytically or numerically.…”