Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Stability crossing boundaries of delay systems modeling immune dynamics in leukemia

Pages: 129 - 156, Volume 13, Issue 1, January 2010      doi:10.3934/dcdsb.2010.13.129

       Abstract        Full Text (631.5K)       Related Articles

Silviu-Iulian Niculescu - Laboratoire des Signaux et Systèmes (UMR CNRS 8506), Centre National de la Recherche Scientifique-Supélec, Gif-sur-Yvette, France (email)
Peter S. Kim - Department of Mathematics, University of Utah, Salt Lake City, Utah, 84112-0900, United States (email)
Keqin Gu - Department of Mechanical and Industrial Engineering, Southern Illinois University at Edwardsville, Edwardsville, Illinois, 62026-1805, United States (email)
Peter P. Lee - Division of Hematology, School of Medicine, Stanford University, Stanford, California 94305, United States (email)
Doron Levy - Department of Mathematics and Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, United States (email)

Abstract: 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:  Delay, asymptotic stability, switch, reversal, crossing curves, quasipolynomial, leukemia models.
Mathematics Subject Classification:  Primary: 34K20; Secondary: 92C50, 93D09, 93D99.

Received: October 2008;      Revised: May 2009;      Available Online: October 2009.