Periodic Oscillations in Leukopoiesis Models with Two Delays

Mostafa Adimy (1, 2) , Fabien Crauste (1, 2) , Shigui Ruan (3)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 ANUBIS - Tools of automatic control for scientific computing, Models and Methods in Biomathematics
3 Department of Mathematics [Miami]

The term leukopoiesis describes processes leading to the production and regulation of white blood cells. It is based on stem cells differentiation and may exhibit abnormalities resulting in severe diseases, such as cyclical neutropenia and leukemias. We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analyzing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of a Hopf bifurcation which leads to periodic solutions. Numerical simulations of the model with parameter values reported in the literature demonstrate that periodic oscillations (with short and long periods) agree with observations of cyclical neutropenia in patients.

Mots clés: Leukopoiesis; hematopoietic stem cells; delay differential equations; stability; Hopf bifurcation

Domaines: Analyse classique [math.CA]; Systèmes dynamiques [math.DS]

Mostafa Adimy, Fabien Crauste, Shigui Ruan. Periodic Oscillations in Leukopoiesis Models with Two Delays. Journal of Theoretical Biology, 2006, 242, pp.288-299. ⟨10.1016/j.jtbi.2006.02.020⟩. ⟨hal-00258393⟩