
-
Previous Article
Existence of generalized homoclinic solutions for a modified Swift-Hohenberg equation
- DCDS-S Home
- This Issue
-
Next Article
An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints
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 |
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} $).
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
R. S. Go,
Idiopathic cyclic thrombocytopenia, Blood Reviews, 19 (2005), 53-59.
doi: 10.1016/j.blre.2004.05.001. |
[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. |
[13] |
E. N. Gryazina,
The D-decomposition theory, Automation and Remote Control, 65 (2004), 1872-1884.
doi: 10.1023/B:AURC.0000049874.93222.2c. |
[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. |
[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. |
[17] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998.
doi: 10.1007/b98848. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[27] |
M.-F. Tsan,
Kinetics and distribution of platelets in man, American Journal of Hematology, 17 (1984), 97-104.
doi: 10.1002/ajh.2830170114. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
R. S. Go,
Idiopathic cyclic thrombocytopenia, Blood Reviews, 19 (2005), 53-59.
doi: 10.1016/j.blre.2004.05.001. |
[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. |
[13] |
E. N. Gryazina,
The D-decomposition theory, Automation and Remote Control, 65 (2004), 1872-1884.
doi: 10.1023/B:AURC.0000049874.93222.2c. |
[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. |
[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. |
[17] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998.
doi: 10.1007/b98848. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[27] |
M.-F. Tsan,
Kinetics and distribution of platelets in man, American Journal of Hematology, 17 (1984), 97-104.
doi: 10.1002/ajh.2830170114. |











[1] |
Ariadna Farrés, Àngel Jorba. On the high order approximation of the centre manifold for ODEs. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 977-1000. doi: 10.3934/dcdsb.2010.14.977 |
[2] |
Ismail Abdulrashid, Abdallah A. M. Alsammani, Xiaoying Han. Stability analysis of a chemotherapy model with delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 989-1005. doi: 10.3934/dcdsb.2019002 |
[3] |
Bin Fang, Xue-Zhi Li, Maia Martcheva, Li-Ming Cai. Global stability for a heroin model with two distributed delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 715-733. doi: 10.3934/dcdsb.2014.19.715 |
[4] |
Miljana Jovanović, Vuk Vujović. Stability of stochastic heroin model with two distributed delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020016 |
[5] |
Marek Bodnar, Monika Joanna Piotrowska, Urszula Foryś, Ewa Nizińska. Model of tumour angiogenesis -- analysis of stability with respect to delays. Mathematical Biosciences & Engineering, 2013, 10 (1) : 19-35. doi: 10.3934/mbe.2013.10.19 |
[6] |
Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 |
[7] |
Joseph M. Mahaffy, Timothy C. Busken. Regions of stability for a linear differential equation with two rationally dependent delays. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4955-4986. doi: 10.3934/dcds.2015.35.4955 |
[8] |
Frederic Mazenc, Gonzalo Robledo, Michael Malisoff. Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1851-1872. doi: 10.3934/dcdsb.2018098 |
[9] |
Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157 |
[10] |
Freddy Dumortier, Robert Roussarie. Bifurcation of relaxation oscillations in dimension two. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 631-674. doi: 10.3934/dcds.2007.19.631 |
[11] |
Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 311-330. doi: 10.3934/cpaa.2017015 |
[12] |
Junyuan Yang, Yuming Chen, Jiming Liu. Stability analysis of a two-strain epidemic model on complex networks with latency. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2851-2866. doi: 10.3934/dcdsb.2016076 |
[13] |
Alexander Nabutovsky and Regina Rotman. Lengths of geodesics between two points on a Riemannian manifold. Electronic Research Announcements, 2007, 13: 13-20. |
[14] |
Tyrus Berry, Timothy Sauer. Consistent manifold representation for topological data analysis. Foundations of Data Science, 2019, 1 (1) : 1-38. doi: 10.3934/fods.2019001 |
[15] |
Guy Katriel. Stability of synchronized oscillations in networks of phase-oscillators. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 353-364. doi: 10.3934/dcdsb.2005.5.353 |
[16] |
Keonhee Lee, Ngoc-Thach Nguyen, Yinong Yang. Topological stability and spectral decomposition for homeomorphisms on noncompact spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2487-2503. doi: 10.3934/dcds.2018103 |
[17] |
Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933 |
[18] |
Alexandr A. Zevin, Mark A. Pinsky. Qualitative analysis of periodic oscillations of an earth satellite with magnetic attitude stabilization. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 293-297. doi: 10.3934/dcds.2000.6.293 |
[19] |
Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223 |
[20] |
Dingheng Pi. Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 881-905. doi: 10.3934/dcdsb.2018211 |
2018 Impact Factor: 0.545
Tools
Article outline
Figures and Tables
[Back to Top]