Stability crossing boundaries of delay systems modeling immune dynamics in leukemia

Discrete and Continuous Dynamical Systems - Series B

Volumn 13, Issue 1, 2010, Pages 129-156

  Niculescu, Silviu Iulian ^a   Kim, Peter S ^b   Keqin, G U ^c   Lee, Peter P ^d   Levy, Doron ^e  

^a CNRS   (France)

^b UNIVERSITY OF UTAH   (United States)

^c SOUTHERN ILLINOIS UNIVERSITY EDWARDSVILLE   (United States)

^d STANFORD UNIVERSITY SCHOOL OF MEDICINE   (United States)

^e University of Maryland   (United States)

Author keywords

Asymptotic stability; Crossing curves; Delay; Leukemia models; Quasipolynomial; Reversal; Switch

Indexed keywords

EID: 76649144226     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2010.13.129     Document Type: Article

References (34)

1. M. Adimy, F. Crauste and S. Ruan, A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia, SIAM J. Appl. Math., 65 (2005), 1328-1352.
2. M. Adimy, F. Crauste and S. Ruan, Periodic oscillations in leukopoiesis models with two delays, J. Theor. Biol., 242 (2006), 288-299.
3. S. Barnett, "Polynomials and Linear Control Systems," Marcel Dekker, Inc., New York, NY, USA, 1983.
4. 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.
5. S. Bernard, J. Bélair and M. C. Mackey, Oscillations in cyclical neutropenia: new evidence based on mathematical modeling, J. Theor. Biol., 223 (2003), 283-298.
6. S. Bernard, J. Bélair and M. C. Mackey, Bifurcations in a white-blood cell production model, C. R. Biol., 227 (2004), 201-210.
7. F. G Boese, Stability with respect to the delay: On the paper of K. L. Cooke and p. van den Driessche, J. Math. Anal. Appl., 228 (1998), 293-321.
8. J. W. Bruce and P. J. Giblin, "Curves and Singularities," Cambridge University Press, Cambridge, UK, 1984.
9. S. A. Campbell, Stability and bifurcation in the harmonic oscillator with multiple, delayed feedback loops, Dynamics of Continuous, Discrete, and Impulsive Systems, 5 (1999), 225-235.
10. C. I.-U. Chen, H. T. Maecker and P. P. Lee, Development and dynamics of robust T cell responses to CML under imatinib treatment, Blood, 111 (2008), 5342-5349.
11. D. L. Chao, S. Forrest, M. P. Davenport and A. S. Perelson, Stochastic stage-structured modeling of the adaptive immune system, Proc. IEEE Comput. Soc. Bioinform. Conf., 2 (2003), 124-131.
12. C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis-I. Periodic chronic myelogenous leukemia, J. Theor. Biol., 237 (2005), 117-132.
13. K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90.
14. R. Datko, A procedure for determination of the exponential stability of certain differentialdifference equations, Quart. Appl. Math., 36 (1978), 279-292.
15. R. DeConde, P. S. Kim, D. Levy and P. P. Lee, Post-transplantation dynamics of the immune response to chronic myelogenous leukemia, J. Theor. Biol., 236 (2005), 39-59.
16. O. Diekmann, S. A. van Gils, S. M. Verduyn-Lunel and H.-O. Walther, "Delay equations, Functional-, Complex and Nonlinear Analysis," volume 110 of Appl. Math. Sciences, Springer-Verlag, New York, NY, 1995.
17. K. Engelborghs, T. Luzyanina and G. Samaey, DDE-BIFTOOL v. 2.00: A Matlab package for bifurcation analysis of delay differential equations, , 2001.
18. K. Gu, V. L. Kharitonov and J. Chen, "Stability and Robust Stability of Time-Delay Systems," Birkhäuser, Boston, MA, 2003.
19. K. Gu, S.-I. Niculescu and J. Chen, On stability of crossing curves for general systems with two delays, J. Math. Anal. Appl., 311 (2005), 231-253.
20. J. K. Hale and W. Huang, Global geometry of the stable regions for two delay differential equations, J. Math. Anal. Appl., 178 (1993), 344-362.
21. J. K. Hale, E. F. Infante and F. S.-P. Tsen, Stability in linear delay equations, J. Math. Anal. Appl., 105 (1985), 533-555.
22. J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," volume 99 of Appl. Math. Sciences. Springer-Verlag, New York, NY, 1993.
23. P. S. Kim, R. DeConde, D. Levy and P. P. Lee, Post-transplantation dynamics of the immune response to chronic myelogenous leukemia, presented at Int. Conf. on Diff. Eqs. and Applications in Math. Biology, 2004.
24. P. S. Kim, P. P. Lee and D. Levy, Mini-transplants for chronic myelogenous leukemia: A modeling perspective, in "Biology and Control Theory: Current Challenges,"(eds. Queinnec et al.), Lecture Notes in Control and Information Sciences, LNCIS 357, Springer, Berlin, 3-20, 2007.
25. Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, Boston, MA, 1993.
26. M. S. Lee and C. S. Hsu, On the τ -decomposition method of stability analysis for retarded dynamical systems, SIAM J. Control, 7 (1969), 242-259.
27. N. MacDonald, "Biological Delay Systems: Linear Stability Theory," Cambridge University Press, Cambridge, UK, 1989.
28. K. Murali-Krishna, J. D. Altman, M. Suresh, D. J. D. Sourdive, D. J. D. Zajac, J. D. Miller, J. Slansky and R. Ahmed, Counting antigen-specific CD8+ T cells: A re-evaluation of bystander activation during viral infection, Immunity, 8 (1998), 177-187.
29. J. D. Murray, "Mathematical Biology: I. An Introduction," Springer-Verlag, New York, NY, 3rd edition, 2002.
30. J. D. Murray, "Mathematical Biology: II. Spatial Models and Biomedical Applications," Springer-Verlag, New York, NY, 3rd edition, 2003.
31. S.-I. Niculescu, "Delay Effects on Stability. A Robust Control Approach," volume 269 of Lecture Notes in Control and Information Sciences, Springer-Verlag, Heidelberg, Germany, 2001.
32. N. Olgac, A. F. Ergenc and R. Sipahi, Delay scheduling: a new concept for stabilization in multiple delay systems, J. Vib. Control, 11 (2005), 1159-1172.
33. G. Stépán, "Retarded Dynamical Systems: Stability and Characteristic Functions," volume 210 of "Pitman Research Notes in Mathematics," Longman Scientific & Technical, Harlow, New York, 1989.
34. O. Toker and H. Özbay, Complexity issues in robust stability of linear delay-differential systems, Math. Contr. Signals Syst., 9 (1996), 386-400.