Stability and instability induced by time delay in an erythropoiesis model (original) (raw)
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SIAM Journal on Applied Mathematics, 2010
We propose a mathematical model describing the dynamics of a hematopoietic stem cell population. The method of characteristics reduces the age-structured model to a system of differential equations with a state-dependent delay. A detailed stability analysis is performed. A sufficient condition for the global asymptotic stability of the trivial steady state is obtained using a Lyapunov-Razumikhin function. A unique positive steady state is shown to appear through a transcritical bifurcation of the trivial steady state. The analysis of the positive steady state behavior, through the study of a first order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation and gives criteria for stability switches. A numerical analysis confirms the results and stresses the role of each parameter involved in the system on the stability of the positive steady state.
Acta Biotheoretica, 2019
We present an age-structured model for erythropoiesis in which the mortality of mature cells is described empirically by a physiologically realistic probability distribution of survival times. Under some assumptions, the model can be transformed into a system of delay differential equations with both constant and distributed delays. The stability of the equilibrium of this system and possible Hopf bifurcations are described for a number of probability distributions. Physiological motivation and interpretation of our results are provided.
Journal of Theoretical Biology, 1998
An age-structured model for erythropoiesis is extended to include the active destruction of the oldest mature cells and possible control by apoptosis. The former condition, which is applicable to other population models where the predator satiates, becomes a constant flux boundary condition and results in a moving boundary condition. The method of characteristics reduces the age-structured model to a system of threshold type differential delay equations. Under certain assumptions, this model can be reduced to a system of delay differential equations with a state dependent delay in an uncoupled differential equation for the moving boundary condition. Analysis of the characteristic equation for the linearized model demonstrates the existence of a Hopf bifurcation when the destruction rate of erythrocytes is modified. The parameters in the system are estimated from experimental data, and the model is simulated for a normal human subject following a loss of blood typical of a blood donation. Numerical studies for a rabbit with an induced auto-immune hemolytic anemia are performed and compared with experimental data.
Periodic oscillations in leukopoiesis models with two delays
Journal of Theoretical Biology, 2006
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.
Mathematical Modelling of Natural Phenomena, 2014
A two dimensional two-delays differential system modeling the dynamics of stemlike cells and white-blood cells in Chronic Myelogenous Leukemia is considered. All three types of stem cell division (asymmetric division, symmetric renewal and symmetric differentiation) are present in the model. Stability of equilibria is investigated and emergence of periodic solutions of limit cycle type, as a result of a Hopf bifurcation, is eventually shown. The stability of these limit cycles is studied using the first Lyapunov coefficient.
2009
Hematopoiesis is a complex biological process that leads to the production and regulation of blood cells. It is based upon differentiation of stem cells under the action of growth factors. A mathematical approach of this process is proposed to carry out explanation on some blood diseases, characterized by oscillations in circulating blood cells. A system of three differential equations with delay, corresponding to the cell cycle duration, is analyzed. The existence of a Hopf bifurcation for a positive steady-state is obtained through the study of an exponential polynomial characteristic equation with delay-dependent coefficients. Numerical simulations show that long period oscillations can be obtained in this model, corresponding to a destabilization of the feedback regulation between blood cells and growth factors. This stresses the localization of periodic hematological diseases in the feedback loop.
Stability of equilibrium and periodic solutions of a delay equation modeling leukemia
2010
We consider a delay differential equation that occurs in the study of chronic myelogenous leukemia. After shortly reminding some previous results concerning the stability of equilibrium solutions, we concentrate on the study of stability of periodic solutions emerged by Hopf bifurcation from a certain equilibrium point. We give the algorithm for approximating a center manifold at a typical point (in the parameter space) of Hopf bifurcation (and an unstable manifold in the vicinity of such a point, where such a manifold exists). Then we find the normal form of the equation restricted to the center manifold, by computing the first Lyapunov coefficient. The normal form allows us to establish the stability properties of the periodic solutions occurred by Hopf bifurcation.
A delay differential-difference system of hematopoietic stem cell dynamics
Comptes Rendus Mathematique, 2015
All functionally blood cells are generated in the bone marrow through hematopoiesis from a small population of cells called hematopoietic stem cells (HSCs). HSCs have the capacity to self-renew and also the capacity to differentiate into any types of blood cells. We consider a system of two age-structured partial differential equations, describing the evolution of HSC population. By integrating this system over the age and using the characteristics method, we reduce it to a system composed with a differential equation and a delay difference equation. We investigate the asymptotic stability of steady states and the existence of a Hopf bifurcation. We conclude our work by numerical simulations. Résumé Equations différentielles et aux différencesà retard pour des modèles de dynamique des cellules souches hématopoïétiques. Toutes les cellules sanguines sont produites dans la moelle osseuse lors de l'hématopoïèseà partir d'une petite population de cellules appelées cellules souches hématopoïétiques (CSHs). Les CSHs ont la capacité de s'auto-renouveler etégalement de se différencier en tous types de cellules sanguines. Le sytème mathématique que nous considérons pour modéliser ces populations de CSHs est un système de deux equations aux dérivées partielles structurées enâge. Par intégration suivant les caractéristiques, le modèle est réduit a un système composé d'uneéquation différentielle et d'uneéquation aux différencesà retard. Nousétudions le comportement asymptotique desétats d'équilibre et l'existence d'une bifurcation de Hopf. Nous concluons notre travail par des simulations numériques.
Oscillation and global attractivity in a periodic delay hematopoiesis model
Journal of Applied Mathematics and Computing, 2003
In this paper we shall consider the nonlinear delay differential equation p (t) = β(t) 1 + p n (t − mω) − δ(t)p(t), (*) where m is a positive integer, β(t) and δ(t) are positive periodic functions of period ω. In the nondelay case we shall show that (*) has a unique positive periodic solution p(t), and show that p(t) is a global attractor all other positive solutions. In the delay case we shall present sufficient conditions for the oscillation of all positive solutions of (*) about p(t), and establish sufficient conditions for the global attractivity of p(t). Our results extend and improve the well known results in the autonomous case.