American Institute of Mathematical Sciences

Advanced
January  2010, 13(1): 129-156. doi: 10.3934/dcdsb.2010.13.129

Stability crossing boundaries of delay systems modeling immune dynamics in leukemia

Silviu-Iulian Niculescu 1, , Peter S. Kim 2, , Keqin Gu 3, , Peter P. Lee 4, and  Doron Levy 5,

1. 

Laboratoire des Signaux et Systèmes (UMR CNRS 8506), Centre National de la Recherche Scientifique-Supélec, Gif-sur-Yvette, France
2. 

Department of Mathematics, University of Utah, Salt Lake City, Utah, 84112-0900, United States
3. 

Department of Mechanical and Industrial Engineering, Southern Illinois University at Edwardsville, Edwardsville, Illinois, 62026-1805, United States
4. 

Division of Hematology, School of Medicine, Stanford University, Stanford, California 94305, United States
5. 

Department of Mathematics and Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742

Received  October 2008 Revised  May 2009 Published  October 2009

This paper focuses on the characterization of delay effects on the asymptotic stability of some continuous-time delay systems encountered in modeling the post-transplantation dynamics of the immune response to chronic myelogenous leukemia. Such models include multiple delays in some large range, from one minute to several days. The main objective of the paper is to study the stability of the crossing boundaries of the corresponding linearized models in the delay-parameter space by taking into account the interactions between small and large delays. Weak, and strong cell interactions are discussed, and analytic characterizations are proposed. An illustrative example together with related discussions completes the presentation.
Keywords: leukemia models., Delay, asymptotic stability, switch, crossing curves, reversal, quasipolynomial.
Mathematics Subject Classification: Primary: 34K20; Secondary: 92C50, 93D09, 93D9.
Citation: Silviu-Iulian Niculescu, Peter S. Kim, Keqin Gu, Peter P. Lee, Doron Levy. Stability crossing boundaries of delay systems modeling immune dynamics in leukemia. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 129-156. doi: 10.3934/dcdsb.2010.13.129
[1]

Heikki Haario, Leonid Kalachev, Marko Laine. Reduction and identification of dynamic models. Simple example: Generic receptor model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 417-435. doi: 10.3934/dcdsb.2013.18.417
[2]

Somkid Intep, Desmond J. Higham. Zero, one and two-switch models of gene regulation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 495-513. doi: 10.3934/dcdsb.2010.14.495
[3]

R. A. Everett, Y. Zhao, K. B. Flores, Yang Kuang. Data and implication based comparison of two chronic myeloid leukemia models. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1501-1518. doi: 10.3934/mbe.2013.10.1501
[4]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727
[5]

Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020032
[6]

Tómas Chacón-Rebollo, Macarena Gómez-Mármol, Samuele Rubino. On the existence and asymptotic stability of solutions for unsteady mixing-layer models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 421-436. doi: 10.3934/dcds.2014.34.421
[7]

Wei Feng, Michael T. Cowen, Xin Lu. Coexistence and asymptotic stability in stage-structured predator-prey models. Mathematical Biosciences & Engineering, 2014, 11 (4) : 823-839. doi: 10.3934/mbe.2014.11.823
[8]

Mostafa Adimy, Fabien Crauste. Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 19-38. doi: 10.3934/dcdsb.2007.8.19
[9]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
[10]

Tetsuya Ishiwata. On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 865-873. doi: 10.3934/dcdss.2011.4.865
[11]

Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109
[12]

Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493
[13]

Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219
[14]

Anatoly Neishtadt. On stability loss delay for dynamical bifurcations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 897-909. doi: 10.3934/dcdss.2009.2.897
[15]

Marc Chamberland, Anna Cima, Armengol Gasull, Francesc Mañosas. Characterizing asymptotic stability with Dulac functions. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 59-76. doi: 10.3934/dcds.2007.17.59
[16]

Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki. Lyapunov functionals for multistrain models with infinite delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 507-536. doi: 10.3934/dcdsb.2017025
[17]

Qingming Gou, Wendi Wang. Global stability of two epidemic models. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 333-345. doi: 10.3934/dcdsb.2007.8.333
[18]

Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Dirichlet problems with a crossing reaction. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2749-2766. doi: 10.3934/cpaa.2014.13.2749
[19]

Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 239-269. doi: 10.3934/dcdss.2014.7.239
[20]

Alina Chertock, Changhui Tan, Bokai Yan. An asymptotic preserving scheme for kinetic models with singular limit. Kinetic & Related Models, 2018, 11 (4) : 735-756. doi: 10.3934/krm.2018030

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences