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doi: 10.3934/dcdss.2020131

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

Loïs Boullu 1,, , Laurent Pujo-Menjouet 1, and  Jacques Bélair 2,

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 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} $).

Keywords: Platelets, oscillations, stability, two delays, D-decomposition, centre manifold analysis.
Mathematics Subject Classification: Primary: 34K13, 34K18; Secondary: 92D25.
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, doi: 10.3934/dcdss.2020131
References:
[1]

J. Bélair and S. A. Campbell, Stability and bifurcations of equilibriain a multiple-delayed differential equation, SIAM Journal on Applied Mathematics, 54 (1994), 1402-1424.  doi: 10.1137/S0036139993248853.  Google Scholar

[2]

J. Bélair and M. C. Mackey, A model for the regulation of mammalian platelet production, Annals of the New York Academy of Sciences, 504 (1987), 280-282.  doi: 10.1111/j.1749-6632.1987.tb48740.x.  Google Scholar

[3]

J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two delay models for erythropoiesis, Math. Biosciences, 128 (1995), 317-346.  doi: 10.1016/0025-5564(94)00078-E.  Google Scholar

[4]

A. Besse, Modélisation Mathématique de La Leucémie Myéloïde Chronique, Ph.D thesis, Université Claude Bernard Lyon 1, 2017. Google Scholar

[5]

L. Boullu, M. Adimy, F. Crauste and L. Pujo-Menjouet, Oscillations and asymptotic convergence for a delay differential equation modeling platelet production, Discrete and Continuous Dynamical Systems Series B, 24 (2019), 2417-2442.  doi: 10.3934/dcdsb.2018259.  Google Scholar

[6]

L. Boullu, L. Pujo-Menjouet and J. H. Wu, A model for megakaryopoiesis with state-dependent delay, SIAM J. Appl. Math., 79 (2019), 1218-1243.  doi: 10.1137/18M1201020.  Google Scholar

[7]

T. C. Busken and J. M. Mahaffy, Regions of stability for a linear differential equation with two rationally dependent delays, Discrete and Continuous Dynamical Systems, 35 (2015), 4955-4986.  doi: 10.3934/dcds.2015.35.4955.  Google Scholar

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S. A. Campbell, Calculating centre manifolds for delay differential equations using MapleTM, Delay Differential Equations, Springer, New York, (2009), 221-244. doi: 10.1007/978-0-387-85595-0_8.  Google Scholar

[9]

F. J. de Sauvage, K. Carver-Moore, S. M. Luoh, A. Ryan, M. Dowd, D. L. Eaton and M. W. Moore, Physiological regulation of early and late stages of megakaryocytopoiesis by thrombopoietin, The Journal of Experimental Medicine, 183 (1996), 651-656.  doi: 10.1084/jem.183.2.651.  Google Scholar

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H. A. El-Morshedy, G. Röst and A. Ruiz-Herrera, Global dynamics of delay recruitment models with maximized lifespan, Zeitschrift für angewandte Mathematik und Physik, 67 (2016), Art. 56, 15 pp. doi: 10.1007/s00033-016-0644-0.  Google Scholar

[11]

R. S. Go, Idiopathic cyclic thrombocytopenia, Blood Reviews, 19 (2005), 53-59.  doi: 10.1016/j.blre.2004.05.001.  Google Scholar

[12]

R. Grozovsky, A. J. Begonja, K. F. Liu, G. Visner, J. H. Hartwig, H. Falet and K. M. Hoffmeister, The Ashwell-Morell receptor regulates hepatic thrombopoietin production via JAK2-STAT3 signaling, Nature Medicine, 21 (2015), 47-54.  doi: 10.1038/nm.3770.  Google Scholar

[13]

E. N. Gryazina, The D-decomposition theory, Automation and Remote Control, 65 (2004), 1872-1884.  doi: 10.1023/B:AURC.0000049874.93222.2c.  Google Scholar

[14]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[15]

K. Kaushansky, Megakaryopoiesis and thrombopoiesis, in Williams Hematology, McGraw-Hill, 9th edition, (2016), 1815-1828. Google Scholar

[16]

D. J. Kuter, The biology of thrombopoietin and thrombopoietin receptor agonists, International Journal of Hematology, 98 (2013), 10-23.  doi: 10.1007/s12185-013-1382-0.  Google Scholar

[17]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998. doi: 10.1007/b98848.  Google Scholar

[18]

G. P. Langlois, M. Craig, A. R. Humphries, M. C. Mackey, J. M. Mahaffy, J. Bélair, T. Moulin, S. R. Sinclair and L. L. Wang, Normal and pathological dynamics of platelets in humans, Journal of Mathematical Biology, 75 (2017), 1411-1462.  doi: 10.1007/s00285-017-1125-6.  Google Scholar

[19]

J. Li, D. E. van der Wal, G. H. Zhu, M. Xu, I. Yougbare, L. Ma, B. Vadasz, N. Carrim, R. Grozovsky, M. Ruan, L. Y. Zhu, Q. S. Zeng, L. L. Tao, Z.-M. Zhai, J. Peng, M. Hou, V. Leytin, J. Freedman, K. M. Hoffmeister and H. Y. Ni, Desialylation is a mechanism of Fc-independent platelet clearance and a therapeutic target in immune thrombocytopenia, Nature Communications, 6 (2015), 7737. doi: 10.1038/ncomms8737.  Google Scholar

[20]

J. M. Mahaffy, J. Bélair and M. C. Mackey, Hematopoietic model with moving boundary condition and state dependent delay: Applications in erythropoiesis, Journal of Theoretical Biology, 190 (1998), 135-146.  doi: 10.1006/jtbi.1997.0537.  Google Scholar

[21]

J. M. Mahaffy, K. M. Joiner and P. J. Zak, A geometric analysis of stability regions for a linear differential equation with two delays, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 779-796.  doi: 10.1142/S0218127495000570.  Google Scholar

[22]

S. E. McKenzie, S. M. Taylor, P. Malladi, H. Yuhan, D. L. Cassel, P. Chien, E. Schwartz, A. D. Schreiber, S. Surrey and M. P. Reilly, The role of the human Fc receptor FcgRIIA in the immune clearance of platelets: A transgenic mouse model, The Journal of Immunology, 162 (1999), 4311-4318.   Google Scholar

[23]

L. Pitcher, K. Taylor, J. Nichol, D. Selsi, R. Rodwell, J. Marty, D. Taylor, S. Wright, D. Moore, C. Kelly and A. Rentoul, Thrombopoietin measurement in thrombocytosis: Dysregulation and lack of feedback inhibition in essential thrombocythaemia, British Journal of Haematology, 99 (1997), 929-932.  doi: 10.1046/j.1365-2141.1997.4633267.x.  Google Scholar

[24]

H. Y. Shu, L. Wang and J. H. Wu, Global dynamics of Nicholson's blowflies equation revisited: Onset and termination of nonlinear oscillations, J. Differential Equations, 255 (2013), 2565-2586.  doi: 10.1016/j.jde.2013.06.020.  Google Scholar

[25]

H. Y. Shu, L. Wang and J. H. Wu, Bounded global Hopf branches for stage-structured differential equations with unimodal feedback, Nonlinearity, 30 (2017), 943-964.  doi: 10.1088/1361-6544/aa5497.  Google Scholar

[26]

Y. L. Song and J. Jiang, Steady-state, Hopf and steady-state-hopf bifurcations in delay differential equations with applications to a damped harmonic oscillator with delay feedback, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250286, 31 pp. doi: 10.1142/S0218127412502860.  Google Scholar

[27]

M.-F. Tsan, Kinetics and distribution of platelets in man, American Journal of Hematology, 17 (1984), 97-104.  doi: 10.1002/ajh.2830170114.  Google Scholar

show all references

References:
[1]

J. Bélair and S. A. Campbell, Stability and bifurcations of equilibriain a multiple-delayed differential equation, SIAM Journal on Applied Mathematics, 54 (1994), 1402-1424.  doi: 10.1137/S0036139993248853.  Google Scholar

[2]

J. Bélair and M. C. Mackey, A model for the regulation of mammalian platelet production, Annals of the New York Academy of Sciences, 504 (1987), 280-282.  doi: 10.1111/j.1749-6632.1987.tb48740.x.  Google Scholar

[3]

J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two delay models for erythropoiesis, Math. Biosciences, 128 (1995), 317-346.  doi: 10.1016/0025-5564(94)00078-E.  Google Scholar

[4]

A. Besse, Modélisation Mathématique de La Leucémie Myéloïde Chronique, Ph.D thesis, Université Claude Bernard Lyon 1, 2017. Google Scholar

[5]

L. Boullu, M. Adimy, F. Crauste and L. Pujo-Menjouet, Oscillations and asymptotic convergence for a delay differential equation modeling platelet production, Discrete and Continuous Dynamical Systems Series B, 24 (2019), 2417-2442.  doi: 10.3934/dcdsb.2018259.  Google Scholar

[6]

L. Boullu, L. Pujo-Menjouet and J. H. Wu, A model for megakaryopoiesis with state-dependent delay, SIAM J. Appl. Math., 79 (2019), 1218-1243.  doi: 10.1137/18M1201020.  Google Scholar

[7]

T. C. Busken and J. M. Mahaffy, Regions of stability for a linear differential equation with two rationally dependent delays, Discrete and Continuous Dynamical Systems, 35 (2015), 4955-4986.  doi: 10.3934/dcds.2015.35.4955.  Google Scholar

[8]

S. A. Campbell, Calculating centre manifolds for delay differential equations using MapleTM, Delay Differential Equations, Springer, New York, (2009), 221-244. doi: 10.1007/978-0-387-85595-0_8.  Google Scholar

[9]

F. J. de Sauvage, K. Carver-Moore, S. M. Luoh, A. Ryan, M. Dowd, D. L. Eaton and M. W. Moore, Physiological regulation of early and late stages of megakaryocytopoiesis by thrombopoietin, The Journal of Experimental Medicine, 183 (1996), 651-656.  doi: 10.1084/jem.183.2.651.  Google Scholar

[10]

H. A. El-Morshedy, G. Röst and A. Ruiz-Herrera, Global dynamics of delay recruitment models with maximized lifespan, Zeitschrift für angewandte Mathematik und Physik, 67 (2016), Art. 56, 15 pp. doi: 10.1007/s00033-016-0644-0.  Google Scholar

[11]

R. S. Go, Idiopathic cyclic thrombocytopenia, Blood Reviews, 19 (2005), 53-59.  doi: 10.1016/j.blre.2004.05.001.  Google Scholar

[12]

R. Grozovsky, A. J. Begonja, K. F. Liu, G. Visner, J. H. Hartwig, H. Falet and K. M. Hoffmeister, The Ashwell-Morell receptor regulates hepatic thrombopoietin production via JAK2-STAT3 signaling, Nature Medicine, 21 (2015), 47-54.  doi: 10.1038/nm.3770.  Google Scholar

[13]

E. N. Gryazina, The D-decomposition theory, Automation and Remote Control, 65 (2004), 1872-1884.  doi: 10.1023/B:AURC.0000049874.93222.2c.  Google Scholar

[14]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[15]

K. Kaushansky, Megakaryopoiesis and thrombopoiesis, in Williams Hematology, McGraw-Hill, 9th edition, (2016), 1815-1828. Google Scholar

[16]

D. J. Kuter, The biology of thrombopoietin and thrombopoietin receptor agonists, International Journal of Hematology, 98 (2013), 10-23.  doi: 10.1007/s12185-013-1382-0.  Google Scholar

[17]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998. doi: 10.1007/b98848.  Google Scholar

[18]

G. P. Langlois, M. Craig, A. R. Humphries, M. C. Mackey, J. M. Mahaffy, J. Bélair, T. Moulin, S. R. Sinclair and L. L. Wang, Normal and pathological dynamics of platelets in humans, Journal of Mathematical Biology, 75 (2017), 1411-1462.  doi: 10.1007/s00285-017-1125-6.  Google Scholar

[19]

J. Li, D. E. van der Wal, G. H. Zhu, M. Xu, I. Yougbare, L. Ma, B. Vadasz, N. Carrim, R. Grozovsky, M. Ruan, L. Y. Zhu, Q. S. Zeng, L. L. Tao, Z.-M. Zhai, J. Peng, M. Hou, V. Leytin, J. Freedman, K. M. Hoffmeister and H. Y. Ni, Desialylation is a mechanism of Fc-independent platelet clearance and a therapeutic target in immune thrombocytopenia, Nature Communications, 6 (2015), 7737. doi: 10.1038/ncomms8737.  Google Scholar

[20]

J. M. Mahaffy, J. Bélair and M. C. Mackey, Hematopoietic model with moving boundary condition and state dependent delay: Applications in erythropoiesis, Journal of Theoretical Biology, 190 (1998), 135-146.  doi: 10.1006/jtbi.1997.0537.  Google Scholar

[21]

J. M. Mahaffy, K. M. Joiner and P. J. Zak, A geometric analysis of stability regions for a linear differential equation with two delays, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 779-796.  doi: 10.1142/S0218127495000570.  Google Scholar

[22]

S. E. McKenzie, S. M. Taylor, P. Malladi, H. Yuhan, D. L. Cassel, P. Chien, E. Schwartz, A. D. Schreiber, S. Surrey and M. P. Reilly, The role of the human Fc receptor FcgRIIA in the immune clearance of platelets: A transgenic mouse model, The Journal of Immunology, 162 (1999), 4311-4318.   Google Scholar

[23]

L. Pitcher, K. Taylor, J. Nichol, D. Selsi, R. Rodwell, J. Marty, D. Taylor, S. Wright, D. Moore, C. Kelly and A. Rentoul, Thrombopoietin measurement in thrombocytosis: Dysregulation and lack of feedback inhibition in essential thrombocythaemia, British Journal of Haematology, 99 (1997), 929-932.  doi: 10.1046/j.1365-2141.1997.4633267.x.  Google Scholar

[24]

H. Y. Shu, L. Wang and J. H. Wu, Global dynamics of Nicholson's blowflies equation revisited: Onset and termination of nonlinear oscillations, J. Differential Equations, 255 (2013), 2565-2586.  doi: 10.1016/j.jde.2013.06.020.  Google Scholar

[25]

H. Y. Shu, L. Wang and J. H. Wu, Bounded global Hopf branches for stage-structured differential equations with unimodal feedback, Nonlinearity, 30 (2017), 943-964.  doi: 10.1088/1361-6544/aa5497.  Google Scholar

[26]

Y. L. Song and J. Jiang, Steady-state, Hopf and steady-state-hopf bifurcations in delay differential equations with applications to a damped harmonic oscillator with delay feedback, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250286, 31 pp. doi: 10.1142/S0218127412502860.  Google Scholar

[27]

M.-F. Tsan, Kinetics and distribution of platelets in man, American Journal of Hematology, 17 (1984), 97-104.  doi: 10.1002/ajh.2830170114.  Google Scholar

Figure 2.1.  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 $
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Figure 2.2.  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
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Figure 2.3.  Region of stability for the null solution of (2.1) with $ A = 0.55 $
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Figure 2.4.  Region of stability for the null solution of (2.1) with A = 1
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Figure 2.5.  Region of stability for the null solution of (2.1) with A = 2
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Figure 4.1.  Parametric portraits for the phase portraits near the double Hopf bifurcation (from [1], Figure 3.3)
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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
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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
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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
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Figure 5.1.  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
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Figure 5.2.  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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