Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay

Abstract

Under the action of growth factors, proliferating andnonproliferating hematopoietic stem cells differentiate anddivide, so as to produce blood cells. Growth factors act atdifferent levels in the differentiation process, and we considertheir action on the mortality rate (apoptosis) of theproliferating cell population. We propose a mathematical modeldescribing the evolution of a hematopoietic stem cell populationunder the action of growth factors. It consists of a system of twoage-structured evolution equations modeling the dynamics of thestem cell population coupled with a delay differential equationdescribing the evolution of the growth factor concentration. Wefirst reduce our system of three differential equations to asystem of two nonlinear differential equations with two delays anda distributed delay. We investigate some positivity andboundedness properties of the solutions, as well as the existenceof steady states. We then analyze the asymptotic stability of thetwo steady states by studying the characteristic equation withdelay-dependent coefficients obtained while linearizing oursystem. We obtain necessary and sufficient conditions for theglobal stability of the steady state describing the cellpopulation's dying out, using a Lyapunov function, and we provethe existence of periodic solutions about the other steady statethrough the existence of a Hopf bifurcation.