Stability Analysis of a Feedback Model for the Action of the Immune System in Leukemia

Abstract

A mathematical model, coupling the dynamics of short-term stem-like cells and mature leukocytes in leukemia with that of the immune system, is investigated. The model is described by a system of seven delay differential equations with seven delays. Three equilibrium points E0, E1, E2 are highlighted. The stability and the existence of the Hopf bifurcation for the equilibrium points are investigated. In the analysis of the model, the rate of asymmetric division and the rate of symmetric division are very important.

A Complex Model of Cell Evolution in Leukemia Including Competition and the Action of the Immune System

“…Although CML is one of the most studied types of leukemia (see [13], [3], [25], [12], [19]), only relatively recently has the immune system been included in models (see [8], [21], [22], [23], [26], [27]). This is mainly due to the fact that the immune system is very complex and its mechanism is not completely understood.…”

“…A description of the activation of the immune system and the effect of the immune system on the population of malignant cells can be found in [8]. The model presented in this paper is an extension of the model from [8].…”

A stochastic individual-based model to explore the role of spatial interactions and antigen recognition in the immune response against solid tumours

“…Tumour antigen expression and the effects of epigenetic and genetic events have been modelled through differential equation models (Asatryan and Komarova, 2016;Lorenzi et al, 2016;Johnston et al, 2007;Tomasetti and Levy, 2010) and cellularautomaton (CA) models (Bouchnita et al, 2017;Manem et al, 2014). Traditionally, tumour antigen expression and recognition by the immune system have been implicitly modelled by tuning the rates of T cell recruitment, T cell proliferation or tumour cell removal (Arciero et al, 2004;Balea et al, 2014;Besse et al, 2018;De Boer et al, 1985;de Pillis et al, 2009;Köse et al, 2017;Mallet and de Pillis, 2006). More recently, these processes have been explicitly captured by mathematical models formulated in terms of either ordinary differential equations (Balachandran et al, 2017;d'Onofrio and Ciancio, 2011;Luksza et al, 2017) or integro-differential equations Kolev et al, 2013;Lorenzi et al, 2015).…”

Hybrid Models in Erythropoiesis and in Megakaryopoiesis

“…Cyclical platelet disease was a subject of mathematical modeling in Santillan et al [43] and was enriched in Apostu et al [3]. Leukemia and other blood disorders were studied in [2,4,27,28,39,45].…”