Geometric Ideas in the Stability Analysis of Delay Models in Biosciences

This chapter focuses on the stability analysis of various continuous-time delay systems encountered in biosciences. Our main interest is to characterize the effects induced by the presence of delays on the systems’ stability. More precisely, we present some intuitive and easy to follow geometric ideas for characterizing the behavior of the characteristic roots (or eigenvalues) of the corresponding linearized models with respect to the imaginary axis for the single and multiple discrete (constant) delays or for some particular classes of distributed delays (gamma-distribution with a gap). Several examples (human respiration, cell-to-cell spread models in well-mixed configurations or immune dynamics in chronic leukemia) complete the presentation.

[1]  K. P. Hadeler,et al.  Delay equations in biology , 1979 .

[2]  J. Hale,et al.  Stability in Linear Delay Equations. , 1985 .

[3]  Doktor Rerum Naturalium,et al.  A Global Model for the Cardiovascular and Respiratory System , 1998 .

[4]  N. Macdonald.  Biological Delay Systems: Linear Stability Theory , 1989 .

[5]  S. Lunel,et al.  Delay Equations. Functional-, Complex-, and Nonlinear Analysis , 1995 .

[6]  B Vielle,et al.  Delay equation analysis of human respiratory stability. , 1998, Mathematical biosciences.

[7]  Doron Levy,et al.  Mini-Transplants for Chronic Myelogenous Leukemia: A Modeling Perspective , 2007 .

[8]  Wim Michiels,et al.  Asymptotic stability of some distributed delay systems: an algebraic approach , 2006 .

[9]  N. Macdonald,et al.  An interference effect of independent delays , 1987 .

[10]  S. Niculescu.  Delay Effects on Stability: A Robust Control Approach , 2001 .

[11]  Peilin Fu,et al.  High-Order Analysis Of Critical Stability Properties of Linear Time-Delay Systems , 2007, 2007 American Control Conference.

[12]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[13]  Jie Chen,et al.  On stability crossing curves for general systems with two delays , 2004 .

[14]  Irinel-Constantin Morarescu,et al.  Remarks on the stability crossing curves of linear systems with distributed delay , 2005 .

[15]  R. Datko.  A procedure for determination of the exponential stability of certain differential-difference equations , 1978 .

[16]  Kirk S. Walton,et al.  Direct method for TDS stability analysis , 1987 .

[17]  R I Shrager,et al.  HIV-1 infection kinetics in tissue cultures. , 1996, Mathematical biosciences.

[18]  Doron Levy,et al.  Post-transplantation dynamics of the immune response to chronic myelogenous leukemia. , 2005, Journal of theoretical biology.

[19]  H. Peitgen,et al.  Functional Differential Equations and Approximation of Fixed Points , 1979 .

[20]  Fathalla A. Rihan,et al.  Numerical modelling in biosciences using delay differential equations , 2000 .

[21]  Irinel-Constantin Morarescu,et al.  Stability Crossing Curves of Shifted Gamma-Distributed Delay Systems , 2007, SIAM J. Appl. Dyn. Syst..

[22]  Alan S. Perelson,et al.  Stochastic stage-structured modeling of the adaptive immune system , 2003, Computational Systems Bioinformatics. CSB2003. Proceedings of the 2003 IEEE Bioinformatics Conference. CSB2003.

[23]  K. Cooke,et al.  On zeroes of some transcendental equations , 1986 .

[24]  Vladimir L. Kharitonov,et al.  Stability of Time-Delay Systems , 2003, Control Engineering.

[25]  Michiel Hazewinkel,et al.  Mathematics of Biology , 1985, Springer Netherlands.

[26]  N. Macdonald,et al.  Two delays may not destabilize although either delay can , 1986 .

[27]  A. Perelson,et al.  Influence of delayed viral production on viral dynamics in HIV-1 infected patients. , 1998, Mathematical biosciences.

[28]  J. Zagury,et al.  Cytopathic effect of human immunodeficiency virus in T4 cells is linked to the last stage of virus infection. , 1988, Proceedings of the National Academy of Sciences of the United States of America.

[29]  John L. Casti,et al.  Introduction to the theory and application of differential equations with deviating arguments , 1973 .

[30]  Onur Toker,et al.  Mathematics of Control , Signals , and Systems Complexity Issues in Robust Stability of Linear Delay-Differential Systems * , 2005 .

[31]  S. T. Buckland,et al.  An Introduction to the Bootstrap. , 1994 .

[32]  F. G. Boese.  Stability with Respect to the Delay: On a Paper of K. L. Cooke and P. van den Driessche , 1998 .

[33]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[34]  J J Batzel,et al.  Stability of the human respiratory control system I. Analysis of a two-dimensional delay state-space model , 2000, Journal of mathematical biology.

[35]  S. Niculescu,et al.  Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach , 2007 .

[36]  Yang Kuang,et al.  Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependent Parameters , 2002, SIAM J. Math. Anal..

[37]  G. Webb,et al.  A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay , 2003, Journal of mathematical biology.