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February  2001, 1(1): 103-124. doi: 10.3934/dcdsb.2001.1.103

Analysis of IVGTT glucose-insulin interaction models with time delay

Jiaxu Li 1, , Yang Kuang 2, and  Bingtuan Li 3,

1. 

Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804, United States
2. 

Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, United States
3. 

Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, United States

Received  October 2000 Revised  January 2001 Published  January 2001

In the last three decades, several models on the interaction of glucose and insulin have appeared in the literature, the mostly used one is generally known as the "minimal model" which was first published in 1979 and modified in 1986. Recently, this minimal model has been questioned by De Gaetano and Arino [4] from both physiological and modeling aspects. Instead, they proposed a new and mathematically more reasonable model, called "dynamic model". Their model makes use of certain simple and specific functions and introduces time delay in a particular way. The outcome is that the model always admits a globally asymptotically stable steady state. The objective of this paper is to find out if and how this outcome depends on the specific choice of functions and the way delay is incorporated. To this end, we generalize the dynamical model to allow more general functions and an alternative way of incorporating time delay. Our findings show that in theory, such models can possess unstable positive steady states. However, for all conceivable realistic data, such unstable steady states do not exist. Hence, our work indicates that the dynamic model does provide qualitatively robust dynamics for the purpose of clinic application. We also perform simulations based on data from a clinic study and point out some plausible but important implications.
Keywords: insulin, qualitative analysis., Glucose, minimum model, delay differential equations.
Mathematics Subject Classification: 34K20, 92C50, 92D2.
Citation: Jiaxu Li, Yang Kuang, Bingtuan Li. Analysis of IVGTT glucose-insulin interaction models with time delay. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 103-124. doi: 10.3934/dcdsb.2001.1.103
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