# American Institute of Mathematical Sciences

November  2020, 13(11): 3005-3027. doi: 10.3934/dcdss.2020131

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

 1 Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France 2 Département de Mathématiques et de statistiques de l'Université de Montréal, Pavillon André-Aisenstadt, CP 6128 Succ. centre-ville, Montréal (Québec) H3C 3J7, Canada

* Corresponding author: lboullu@gmail.com

Received  December 2018 Revised  March 2019 Published  November 2020 Early access  November 2019

Fund Project: LB was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Also, LB is supported by a grant of Région Rhône-Alpes and benefited of the help of the France Canada Research Fund, of the NSERC and of a support from MITACS. JB acknowledges support from NSERC [Discovery Grant]

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: $x'(t) = -\gamma x(t) + g(x(t-\tau_1)) - g(x(t-\tau_1 - \tau_2)) e^{-\gamma \tau_2}$. 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 pseudo-phase space. Finally, the results of the local stability analysis are used to study the impact of an increase of the death rate $\gamma$ or of a decrease of the survival time $\tau_2$ 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$^{-1}$).

Citation: Loïs Boullu, Laurent Pujo-Menjouet, Jacques Bélair. Stability analysis of an equation with two delays and application to the production of platelets. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3005-3027. doi: 10.3934/dcdss.2020131
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##### References:
Region of stability for the null solution of (2.1) with $\tau = 1$. The numbers indicate the number of pairs of eigenvalues with positive real parts. The graph is the same for any positive $\tau$
Region of stability for the null solution of (2.1) with $A = 0.2$. The numbers indicate the number of pairs of eigenvalue with positive real parts
Region of stability for the null solution of (2.1) with $A = 0.55$
Region of stability for the null solution of (2.1) with A = 1
Region of stability for the null solution of (2.1) with A = 2
], Figure 3.3)">Figure 4.1.  Parametric portraits for the phase portraits near the double Hopf bifurcation (from [1], Figure 3.3)
. Once the transient dynamic is lost, a stable limit cycle appears">Figure 4.2.  Numerical simulation of (1.1) for $\tau_2 = 4.75 \times \tau_1$ in the pseudo-phase plane $(x(t), x(t-\tau_1-\tau_2))$.]{Numerical simulation of (1.1) for $\tau_2 = 4.75 \times \tau_1$ in the pseudo-phase plane $(x(t), x(t-\tau_1-\tau_2))$, corresponding to the lowest wedge of the $\mu_1<0, \mu_2>0$ quadrant of Figure 4.1. Once the transient dynamic is lost, a stable limit cycle appears
. Once the transient dynamic is lost, a stable torus appears">Figure 4.3.  Numerical simulation of (1.1) for $\tau_2 = 4.24 \times \tau_1$ in the pseudo-phase space $(x(t), x(t-\tau_1), x(t-\tau_1-\tau_2))$, corresponding to the lowest interior wedge of the $\mu_1<0$, $\mu_2>0$ quadrant of Figure 4.1. Once the transient dynamic is lost, a stable torus appears
. Once the transient dynamic is lost, a stable limit cycle appears">Figure 4.4.  Numerical simulation of (1.1) for $\tau_2 = 4.24 \times \tau_1$ in the pseudo-phase plane $(x(t), x(t-\tau_1-\tau_2))$, corresponding to the $\mu_1>0$, $\mu_2>0$ quadrant of Figure 4.1. Once the transient dynamic is lost, a stable limit cycle appears
Stability as $\tau_2$ or $\gamma$ are varied and other parameters are fixed. Blue dotted lines represent the values in healthy patients, and red dotted lines mark the limits after which the equilibrium is unstable. We see that when $\tau_2$ decreases of one day (to $\tau_2 = 7.2$), then the system loses its stability. Furthermore, if $\gamma$ is multiplied more than 12 times (to $\gamma = 0.625$) then the system also loses its stability
The evolution of the platelet count with time (blue line) for different values of $\tau_2$ and $\gamma$, after the transient phase. The green doted line represents the average platelet count of healthy patients, $20 \times 10^9$ platelets/kg, and the two red dotted lines represent the healthy range of platelet count, $11 \times 10^9$ - $32 \times 10^9$
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