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Abstract
The Intra Venous Glucose Tolerance Test (IVGTT) is a simple and
established experimental procedure in which a challenge bolus of
glucose is administered intra-venously and plasma glucose and
insulin concentrations are then frequently sampled. The modeling
of the measured concentrations has the goal of providing
information on the state of the subject's glucose/insulin control
system: an open problem is to construct a model representing
simultaneously the entire control system with a physiologically
believable qualitative behavior. A previously published
single-distributed-delay differential model was shown to have
desirable properties (positivity, boundedness, global stability of
solutions) under the hypothesis of a specific, square-wave delay
integral kernel. The present work extends the previous results to
a family of models incorporating a generic non- negative, square
integrable normalized kernel. Every model in this family describes
the rate of glucose concentration variation as due to both
insulin-dependent and insulin-independent net glucose tissue
uptake, as well as to constant liver glucose production. The rate
of variation of plasma insulin concentration depends on insulin
catabolism and on pancreatic insulin secretion. Pancreatic insulin
secretion at time $t$ is assumed to depend on the earlier effects
of glucose concentrations, up to time $t$ (distributed delay). We
consider a non-negative, square integrable normalized weight
function $\omega$ on $R^+ =[0, \infty)$ as the fraction of
maximal pancreatic insulin secretion at a given glucose
concentration. No change in local asymptotic stability is
introduced by the time delay. Considering an appropriate Lyapunov
functional, it is found that the system is globally asymptotically
stable if the average time delay has a parameter- dependent upper
bound. An example of good model fit to experimental data is shown
using a specific delay kernel.
Mathematics Subject Classification: 39Axx.
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