Oscillations and asymptotic convergence for a delay differential equation modeling platelet production

Boullu, Loïs; Adimy, Mostafa; Crauste, Fabien; Praly, Laurent

doi:10.3934/dcdsb.2018259

Cited by 3 publications

(2 citation statements)

References 34 publications

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“…A similar phenomenon was obtained later by Mahaffy et al [20] in a second paper on a model of erythropoiesis with a state-dependent delay. Finally, we notice that equation (1.1) can also be obtained by adding a survival time for platelets to a model of megakaryopoiesis whose stability was recently analyzed [5]. In the case of this model, we have τ 1 > τ 2 , which according to preliminary numerical explorations (not shown) seems to imply that stability is not affected of the value of γ.…”

Section: Discussionmentioning

confidence: 62%

Stability analysis of an equation with two delays and application to the production of platelets

Boullu¹,

Praly²,

Bélair³

2020

Discrete &Amp; Continuous Dynamical Systems - S

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We analyze the stability of a differential equation with two delays originating from a model for a population divided into two subpopulations, immature and mature, and we apply this analysis to a model for platelet production. The dynamics of mature individuals is described by the following nonlinear differential equation with two delays: x (t) = −γx(t) + g(x(t − τ 1)) − g(x(t − τ 1 − τ 2))e −γτ 2. The method of D-decomposition is used to compute the stability regions for a given equilibrium. The centre manifold theory is used to investigate the steady-state bifurcation and the Hopf bifurcation. Similarly, analysis of the centre manifold associated with a double bifurcation is used to identify a set of parameters such that the solution is a torus in the pseudophase space. Finally, the results of the local stability analysis are used to study the impact of an increase of the death rate γ or of a decrease of the survival time τ 2 of platelets on the onset of oscillations. We show that the stability is lost through a small decrease of survival time (from 8.4 to 7 days), or through an important increase of the death rate (from 0.05 to 0.625 days −1).

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Section: Discussionmentioning

confidence: 62%

Stability analysis of an equation with two delays and application to the production of platelets

Boullu¹,

Praly²,

Bélair³

2020

Discrete &Amp; Continuous Dynamical Systems - S

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“…Indeed, our choice of a TPO-dependent speed of cell division (rather than overall volume expansion as in [34]) together with the necessity of an A >> 3000 implies that a progenitor divides multiple times as it goes from one end of the compartment M (see Figure 2.1) to the other. Taking this feature into account in the computation of the concentration on c-Mpl receptors (instead of α M M (t) + α P P (t) currently) could be enough to reproduce the clinical features of CT. An example of a model for megakaryopoiesis with multiple compartments, although with a fixed division time τ , is found in [8].…”

Section: Application To a System With A Third-degree Transcendental Ementioning

confidence: 99%

A Model for Megakaryopoiesis with State-Dependent Delay

Boullu¹,

Praly²,

Wu³

2019

SIAM J. Appl. Math.

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We analyze the stability of a system of differential equations with a threshold defined delay arising from a model for platelet production. We consider a maturity-structured population of megakaryocyte progenitors and an age-structured population of platelets, where the cytokine thrombopoietin (TPO) increases the maturation rate of progenitors. Using the quasi-steady-state approximation for TPO dynamics and the method of characteristics, partial differential equations are reduced to a system of two differential equations with a state-dependent delay accounting for the variable maturation rate. We start by introducing the model and proving the positivity and boundedness of the solutions. Then we use a change of variables to obtain an equivalent system of two differential equations with a constant delay, from which we prove existence and uniqueness of the solution. As linearization around the unique positive steady state yields a transcendental characteristic equation of third degree, we introduce the main result, a new framework for stability analysis on models with fixed delays. This framework is then used to describe the stability of the megakaryopoiesis with respect to its parameters. Finally, with parameters being obtained and estimated from data, we give an example in which oscillations appear when the death rate of progenitors is increased 10-fold.

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Periodic Solutions in a Differential Delay Equation Modeling Megakaryopoiesis

Ivanov,

Lani-Wayda

2023

Trends in Mathematics

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Oscillations and asymptotic convergence for a delay differential equation modeling platelet production

Cited by 3 publications

References 34 publications

Stability analysis of an equation with two delays and application to the production of platelets

Stability analysis of an equation with two delays and application to the production of platelets

A Model for Megakaryopoiesis with State-Dependent Delay

Periodic Solutions in a Differential Delay Equation Modeling Megakaryopoiesis

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