Qualitative behavior of a family of delay-differential models of the Glucose-Insulin system

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March 2007, 7(2): 399-424. doi: 10.3934/dcdsb.2007.7.399

Qualitative behavior of a family of delay-differential models of the Glucose-Insulin system

Pasquale Palumbo ^1, , Simona Panunzi ^2, and Andrea De Gaetano ^2,

1.	CNR-IASI, BioMatLab, UCSC - Largo A., Gemelli 8, 00168 Roma, Italy
2.	Istituto di Analisi dei Sistemi ed Informatica "A. Ruberti", Consiglio Nazionale delle Ricerche (IASI-CNR), BioMatLab - UCSC - Largo A. Gemelli 8, 00168 Roma, Italy, Italy

Received February 2006 Revised November 2006 Published December 2006

Abstract
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A family of delay-differential models of the glucose-insulin system is introduced, whose members represent adequately the Intra-Venous Glucose Tolerance Test and allied experimental procedures of diabetological interest. All the models in the family admit positive bounded unique solutions for any positive initial condition and are persistent. The models agree with the physics underlying the experiments, and they all present a unique positive equilibrium point.
Local stability is investigated in a pair of interesting member models: one, a discrete-delays differential system; the other, a distributed-delay system reducing to an ordinary differential system evolving on a suitably defined extended state space. In both cases conditions are given on the physical parameters in order to ensure the local asymptotic stability of the equilibrium point. These conditions are always satisfied, given the actual parameter estimates obtained experimentally. A study of the global stability properties is performed, but while from simulations it could be conjectured that the models considered are globally asymptotically stable, sufficient stability criteria, formally derived, are not actually satisfied for physiological parameters values. Given the practical importance of the models studied, further analytical work may be of interest to conclusively characterize their behavior.

Keywords: Glucose-Insulin Homeostasis, Stability analysis., Integro-Differential Systems, Delay-Differential Systems.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Pasquale Palumbo, Simona Panunzi, Andrea De Gaetano. Qualitative behavior of a family of delay-differential models of the Glucose-Insulin system. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 399-424. doi: 10.3934/dcdsb.2007.7.399

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