# American Institute of Mathematical Sciences

November  2015, 20(9): 2765-2791. doi: 10.3934/dcdsb.2015.20.2765

## Age-structured and delay differential-difference model of hematopoietic stem cell dynamics

 1 Inria, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 novembre 1918, F-69200 Villeurbanne Cedex, France, France 2 Department of Mathematics, University Aboubekr Belkaid, Tlemcen, Algeria

Received  September 2014 Revised  June 2015 Published  September 2015

In this paper, we investigate a mathematical model of hematopoietic stem cell dynamics. We take two cell populations into account, quiescent and proliferating one, and we note the difference between dividing cells that enter directly to the quiescent phase and dividing cells that return to the proliferating phase to divide again. The resulting mathematical model is a system of two age-structured partial differential equations. By integrating this system over age and using the characteristics method, we reduce it to a delay differential-difference system, and we investigate the existence and stability of the steady states. We give sufficient conditions for boundedness and unboundedness properties for the solutions of this system. By constructing a Lyapunov function, the trivial steady state, describing cell's dying out, is proven to be globally asymptotically stable when it is the only equilibrium. The stability analysis of the unique positive steady state, the most biologically meaningful one, and the existence of a Hopf bifurcation allow the determination of a stability area, which is related to a delay-dependent characteristic equation. Numerical simulations illustrate our results on the asymptotic behavior of the steady states and show very rich dynamics of this model. This study may be helpful in understanding the uncontrolled proliferation of blood cells in some hematological disorders.
Citation: Mostafa Adimy, Abdennasser Chekroun, Tarik-Mohamed Touaoula. Age-structured and delay differential-difference model of hematopoietic stem cell dynamics. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 2765-2791. doi: 10.3934/dcdsb.2015.20.2765
##### References:
 [1] M. Adimy, O. Angulo, C. Marquet and L. Sebaa, A mathematical model of multistage hematopoietic cell lineages, Discrete and Continuous Dynamical Systems - Series B, 19 (2014), 1-26. doi: 10.3934/dcdsb.2014.19.1. [2] M. Adimy and F. Crauste, Global stability of a partial differential equation with distributed delay due to cellular replication, Nonlinear Analysis: Theory, Methods & Applications, 54 (2003), 1469-1491. doi: 10.1016/S0362-546X(03)00197-4. [3] M. Adimy and F. Crauste, Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay, Discrete and Continuous Dynamical Systems - Series B, 8 (2007), 19-38. doi: 10.3934/dcdsb.2007.8.19. [4] M. Adimy, F. Crauste, H. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM Journal on Applied Mathematics, 70 (2010), 1611-1633. doi: 10.1137/080742713. [5] M. Adimy, F. Crauste and S. Ruan, A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia, SIAM Journal on Applied Mathematics, 65 (2005), 1328-1352. doi: 10.1137/040604698. [6] J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two-delay models for erythropoiesis, Math Biosci, 128 (1995), 317-346, URL http://www.sciencedirect.com/science/article/pii/002555649400078E. [7] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086. [8] S. Bernard, J. Bélair and M. C. Mackey, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling, Journal of Theoretical Biology, 223 (2003), 283-298, URL http://www.ams.org/mathscinet-getitem?mr=2079467. doi: 10.1016/S0022-5193(03)00090-0. [9] S. Bernard, J. Bélair and M. C. Mackey, Bifurcations in a white-blood-cell production model, C. R. Biol., 327 (2004), 201-210, URL http://linkinghub.elsevier.com/retrieve/pii/S1631069104000381. doi: 10.1016/j.crvi.2003.05.005. [10] F. J. Burns and I. F. Tannock, On the existence of a G$_0$-phase in the cell cycle, Cell Proliferation, 3 (1970), 321-334. doi: 10.1111/j.1365-2184.1970.tb00340.x. [11] C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis - I. Periodic chronic myelogenous leukemia, Journal of Theoretical Biology, 237 (2005), 117-132. doi: 10.1016/j.jtbi.2005.03.033. [12] F. Ficara, M. J. Murphy, M. Lin and M. L. Cleary, Pbx1 regulates self-renewal of long-term hematopoietic stem cells by maintaining their quiescence, Cell Stem Cell, 2 (2008), 484-496, URL http://www.sciencedirect.com/science/article/pii/S1934590908001161. doi: 10.1016/j.stem.2008.03.004. [13] K. Gu and Y. Liu, Lyapunov-Krasovskii functional for uniform stability of coupled differential-functional equations, Automatica, 45 (2009), 798-804. doi: 10.1016/j.automatica.2008.10.024. [14] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer New York, 1993. doi: 10.1007/978-1-4612-4342-7. [15] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, San Diego, 1993. [16] J. Lei and M. C. Mackey, Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia, Journal of Theoretical Biology, 270 (2011), 143-153. doi: 10.1016/j.jtbi.2010.11.024. [17] M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis., Blood, 51 (1978), 941-956. [18] L. Pujo-Menjouet, S. Bernard and M. C. Mackey, Long period oscillations in a $G_0$ model of hematopoietic stem cells, SIAM J. Appl. Dyn. Syst., 4 (2005), 312-332. doi: 10.1137/030600473. [19] L. Pujo-Menjouet and M. C. Mackey, Contribution to the study of periodic chronic myelogenous leukemia, C. R. Biol., 327 (2004), 235-244, URL http://linkinghub.elsevier.com/retrieve/pii/S1631069104000253. doi: 10.1016/j.crvi.2003.05.004. [20] H. Takizawa, R. R. Regoes, C. S. Boddupalli, S. Bonhoeffer and M. G. Manz, Dynamic variation in cycling of hematopoietic stem cells in steady state and inflammation, J. Exp. Med., 208 (2011), 273-284, URL http://jem.rupress.org/content/208/2/273.full. [21] P. Vegh, J. Winckler and F. Melchers, Long-term "in vitro'' proliferating mouse hematopoietic progenitor cell lines, Immunology Letters, 130 (2010), 32-35, URL http://www.sciencedirect.com/science/article/pii/S0165247810000544. doi: 10.1016/j.imlet.2010.02.001. [22] A. Wilson, E. Laurenti, G. Oser, R. C. van der Wath, W. Blanco-Bose, M. Jaworski, S. Offner, C. F. Dunant, L. Eshkind, E. Bockamp, P. Lió, H. R. MacDonald and A. Trumpp, Hematopoietic stem cells reversibly switch from dormancy to self-renewal during homeostasis and repair, Cell, 135 (2008), 1118-1129, URL http://www.sciencedirect.com/science/article/pii/S009286740801386X.

show all references

##### References:
 [1] M. Adimy, O. Angulo, C. Marquet and L. Sebaa, A mathematical model of multistage hematopoietic cell lineages, Discrete and Continuous Dynamical Systems - Series B, 19 (2014), 1-26. doi: 10.3934/dcdsb.2014.19.1. [2] M. Adimy and F. Crauste, Global stability of a partial differential equation with distributed delay due to cellular replication, Nonlinear Analysis: Theory, Methods & Applications, 54 (2003), 1469-1491. doi: 10.1016/S0362-546X(03)00197-4. [3] M. Adimy and F. Crauste, Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay, Discrete and Continuous Dynamical Systems - Series B, 8 (2007), 19-38. doi: 10.3934/dcdsb.2007.8.19. [4] M. Adimy, F. Crauste, H. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM Journal on Applied Mathematics, 70 (2010), 1611-1633. doi: 10.1137/080742713. [5] M. Adimy, F. Crauste and S. Ruan, A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia, SIAM Journal on Applied Mathematics, 65 (2005), 1328-1352. doi: 10.1137/040604698. [6] J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two-delay models for erythropoiesis, Math Biosci, 128 (1995), 317-346, URL http://www.sciencedirect.com/science/article/pii/002555649400078E. [7] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086. [8] S. Bernard, J. Bélair and M. C. Mackey, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling, Journal of Theoretical Biology, 223 (2003), 283-298, URL http://www.ams.org/mathscinet-getitem?mr=2079467. doi: 10.1016/S0022-5193(03)00090-0. [9] S. Bernard, J. Bélair and M. C. Mackey, Bifurcations in a white-blood-cell production model, C. R. Biol., 327 (2004), 201-210, URL http://linkinghub.elsevier.com/retrieve/pii/S1631069104000381. doi: 10.1016/j.crvi.2003.05.005. [10] F. J. Burns and I. F. Tannock, On the existence of a G$_0$-phase in the cell cycle, Cell Proliferation, 3 (1970), 321-334. doi: 10.1111/j.1365-2184.1970.tb00340.x. [11] C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis - I. Periodic chronic myelogenous leukemia, Journal of Theoretical Biology, 237 (2005), 117-132. doi: 10.1016/j.jtbi.2005.03.033. [12] F. Ficara, M. J. Murphy, M. Lin and M. L. Cleary, Pbx1 regulates self-renewal of long-term hematopoietic stem cells by maintaining their quiescence, Cell Stem Cell, 2 (2008), 484-496, URL http://www.sciencedirect.com/science/article/pii/S1934590908001161. doi: 10.1016/j.stem.2008.03.004. [13] K. Gu and Y. Liu, Lyapunov-Krasovskii functional for uniform stability of coupled differential-functional equations, Automatica, 45 (2009), 798-804. doi: 10.1016/j.automatica.2008.10.024. [14] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer New York, 1993. doi: 10.1007/978-1-4612-4342-7. [15] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, San Diego, 1993. [16] J. Lei and M. C. Mackey, Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia, Journal of Theoretical Biology, 270 (2011), 143-153. doi: 10.1016/j.jtbi.2010.11.024. [17] M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis., Blood, 51 (1978), 941-956. [18] L. Pujo-Menjouet, S. Bernard and M. C. Mackey, Long period oscillations in a $G_0$ model of hematopoietic stem cells, SIAM J. Appl. Dyn. Syst., 4 (2005), 312-332. doi: 10.1137/030600473. [19] L. Pujo-Menjouet and M. C. Mackey, Contribution to the study of periodic chronic myelogenous leukemia, C. R. Biol., 327 (2004), 235-244, URL http://linkinghub.elsevier.com/retrieve/pii/S1631069104000253. doi: 10.1016/j.crvi.2003.05.004. [20] H. Takizawa, R. R. Regoes, C. S. Boddupalli, S. Bonhoeffer and M. G. Manz, Dynamic variation in cycling of hematopoietic stem cells in steady state and inflammation, J. Exp. Med., 208 (2011), 273-284, URL http://jem.rupress.org/content/208/2/273.full. [21] P. Vegh, J. Winckler and F. Melchers, Long-term "in vitro'' proliferating mouse hematopoietic progenitor cell lines, Immunology Letters, 130 (2010), 32-35, URL http://www.sciencedirect.com/science/article/pii/S0165247810000544. doi: 10.1016/j.imlet.2010.02.001. [22] A. Wilson, E. Laurenti, G. Oser, R. C. van der Wath, W. Blanco-Bose, M. Jaworski, S. Offner, C. F. Dunant, L. Eshkind, E. Bockamp, P. Lió, H. R. MacDonald and A. Trumpp, Hematopoietic stem cells reversibly switch from dormancy to self-renewal during homeostasis and repair, Cell, 135 (2008), 1118-1129, URL http://www.sciencedirect.com/science/article/pii/S009286740801386X.
 [1] Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385 [2] Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046 [3] Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure and Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657 [4] Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264 [5] Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577 [6] Shaoli Wang, Jianhong Wu, Libin Rong. A note on the global properties of an age-structured viral dynamic model with multiple target cell populations. Mathematical Biosciences & Engineering, 2017, 14 (3) : 805-820. doi: 10.3934/mbe.2017044 [7] Piotr Gwiazda, Karolina Kropielnicka, Anna Marciniak-Czochra. The Escalator Boxcar Train method for a system of age-structured equations. Networks and Heterogeneous Media, 2016, 11 (1) : 123-143. doi: 10.3934/nhm.2016.11.123 [8] Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295 [9] R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure and Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147 [10] Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859 [11] Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641 [12] Luciano Abadías, Carlos Lizama, Pedro J. Miana, M. Pilar Velasco. On well-posedness of vector-valued fractional differential-difference equations. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2679-2708. doi: 10.3934/dcds.2019112 [13] Andrey B. Muravnik. On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 541-561. doi: 10.3934/dcds.2006.16.541 [14] Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 [15] Lesia V. Baranovska. Pursuit differential-difference games with pure time-lag. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1021-1031. doi: 10.3934/dcdsb.2019004 [16] Sun Yi, Patrick W. Nelson, A. Galip Ulsoy. Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter. Mathematical Biosciences & Engineering, 2007, 4 (2) : 355-368. doi: 10.3934/mbe.2007.4.355 [17] Yuan Yuan, Xianlong Fu. Mathematical analysis of an age-structured HIV model with intracellular delay. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2077-2106. doi: 10.3934/dcdsb.2021123 [18] Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 [19] Jinliang Wang, Jiying Lang, Yuming Chen. Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3721-3747. doi: 10.3934/dcdsb.2017186 [20] Frédérique Billy, Jean Clairambault, Franck Delaunay, Céline Feillet, Natalia Robert. Age-structured cell population model to study the influence of growth factors on cell cycle dynamics. Mathematical Biosciences & Engineering, 2013, 10 (1) : 1-17. doi: 10.3934/mbe.2013.10.1

2021 Impact Factor: 1.497

## Metrics

• PDF downloads (110)
• HTML views (0)
• Cited by (12)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]