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

Abstract

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).

On the pole placement of scalar linear delay systems with two delays

“…Holomorphic functions of the form (2) are known as quasipolynomials. For a quasipolynomial under the form (2), one usually defines its degree D as D = N + N j=0 d j , where, for j ∈ {1, . .…”

“…The relations between the notions of roots of maximal multiplicity and dominance have been explored in some recent works, such as [3,5,20,21], in which it is shown that, for some particular classes of quasipolynomials, a root with maximal multiplicity is dominant. More precisely, we say that a quasipolynomial verifies the multiplicity-induced-dominancy (MID) property if it possesses a (strictly) dominant root of maximal multiplicity, and the previous references prove the MID property for some quasipolynomials of the form (2) with N = 1, corresponding thus to delay-differential equations of the form (1) with a single delay. To the best of the authors' knowledge, extensions of the MID property to the case of multiple delays have not yet been considered in the literature.…”

“…From a stability viewpoint, it is not an easy task to check when the quasipolynomial defined in ( 5) is non vanishing on one of a complex right half-plane R α of the form (3) with α < 0 and to study how the stability property depends on the five parameters a 0 , a 1 , a 2 , τ 1 , and τ 2 . A numerical investigation of the stability regions of the equation ( 4) was previously carried out in [2,18]. A qualitative analysis was proposed by [12].…”

Why Are Periodic Erythrocytic Diseases so Rare in Humans?

“…In this present work, we focus on cyclic erythropoiesis and design a model to explain this rarely observed disorder. To that purpose, we take advantage of relevant erythropoiesis mice, dogs and rabbit models (Bélair et al, 1995;Mahaffy et al, 1998;Fuertinger et al, 2013) and mathematical analysis specific to DDEs (Beretta and Kuang, 2002;Adimy et al, 2005;Crauste, 2010;Boullu et al, 2019a) Changes on parameter values accounting for the cyclic erythropoiesis have been well described: among these changes, one can note an increase of the hematopoietic stem cells apoptosis rate (Mackey, 1997) (mice and dogs) or a death rate amplification of mature red blood cells in the case of induced auto immune hemolytic anemia (Mackey, 1979) (rabbits). However, erythrocyte periodic oscillations specific to humans have been poorly investigated.…”