### Abstract

This paper is devoted to the analysis of a mathematical model of blood cell production in the bone marrow (hematopoiesis). The model is a system of two age-structured partial differential equations. Integrating these equations over the age, we obtain a system of two nonlinear differential equations with distributed time delay corresponding to the cell cycle duration. This system describes the evolution of the total cell populations. By constructing a Lyapunov functional, it is shown that the trivial equilibrium is globally asymptotically stable if it is the only equilibrium. It is also shown that the nontrivial equilibrium, the most biologically meaningful one, can become unstable via a Hopf bifurcation. Numerical simulations are carried out to illustrate the analytical results. The study may be helpful in understanding the connection between the relatively short cell cycle durations and the relatively long periods of peripheral cell oscillations in some periodic hematological diseases.

Original language | English (US) |
---|---|

Pages (from-to) | 1328-1352 |

Number of pages | 25 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 65 |

Issue number | 4 |

DOIs | |

State | Published - 2005 |

### Fingerprint

### Keywords

- Asymptotic stability
- Blood cells
- Differential equations
- Distributed delay
- Hematopoiesis
- Hopf bifurcation
- Lyapunov functional

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*65*(4), 1328-1352. https://doi.org/10.1137/040604698

**A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia.** / Adimy, Mostafa; Crauste, Fabien; Ruan, Shigui.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 65, no. 4, pp. 1328-1352. https://doi.org/10.1137/040604698

}

TY - JOUR

T1 - A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia

AU - Adimy, Mostafa

AU - Crauste, Fabien

AU - Ruan, Shigui

PY - 2005

Y1 - 2005

N2 - This paper is devoted to the analysis of a mathematical model of blood cell production in the bone marrow (hematopoiesis). The model is a system of two age-structured partial differential equations. Integrating these equations over the age, we obtain a system of two nonlinear differential equations with distributed time delay corresponding to the cell cycle duration. This system describes the evolution of the total cell populations. By constructing a Lyapunov functional, it is shown that the trivial equilibrium is globally asymptotically stable if it is the only equilibrium. It is also shown that the nontrivial equilibrium, the most biologically meaningful one, can become unstable via a Hopf bifurcation. Numerical simulations are carried out to illustrate the analytical results. The study may be helpful in understanding the connection between the relatively short cell cycle durations and the relatively long periods of peripheral cell oscillations in some periodic hematological diseases.

AB - This paper is devoted to the analysis of a mathematical model of blood cell production in the bone marrow (hematopoiesis). The model is a system of two age-structured partial differential equations. Integrating these equations over the age, we obtain a system of two nonlinear differential equations with distributed time delay corresponding to the cell cycle duration. This system describes the evolution of the total cell populations. By constructing a Lyapunov functional, it is shown that the trivial equilibrium is globally asymptotically stable if it is the only equilibrium. It is also shown that the nontrivial equilibrium, the most biologically meaningful one, can become unstable via a Hopf bifurcation. Numerical simulations are carried out to illustrate the analytical results. The study may be helpful in understanding the connection between the relatively short cell cycle durations and the relatively long periods of peripheral cell oscillations in some periodic hematological diseases.

KW - Asymptotic stability

KW - Blood cells

KW - Differential equations

KW - Distributed delay

KW - Hematopoiesis

KW - Hopf bifurcation

KW - Lyapunov functional

UR - http://www.scopus.com/inward/record.url?scp=24344475745&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=24344475745&partnerID=8YFLogxK

U2 - 10.1137/040604698

DO - 10.1137/040604698

M3 - Article

VL - 65

SP - 1328

EP - 1352

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 4

ER -