The range of time delay and the global stability of the equilibrium for an IVGTT model

Abstract

Diabetes mellitus has become a prevalent disease in the world. Diagnostic protocol for the onset of diabetes mellitus is the initial step in the treatments. The intravenous glucose tolerance test (IVGTT) has been considered as the most accurate method to determine the insulin sensitivity and glucose effectiveness. It is well known that there exists a time delay in insulin secretion stimulated by the elevated glucose concentration level. However, the range of the length of the delay in the existing IVGTT models are not fully discussed and thus in many cases the time delay may be assigned to a value out of its reasonable range. In addition, several attempts had been made to determine when the unique equilibrium point is globally asymptotically stable. However, all these conditions are delay-independent. In this paper, we discuss the range of the time delay and provide easy-to-check delay-dependent conditions for the global asymptotic stability of the equilibrium point for a recent IVGTT model through Liapunov function approach. Estimates of the upper bound of the delay for global stability are given in corollaries. In addition, the numerical simulation in this paper is fully incorporated with functional initial conditions, which is natural and more appropriate in delay differential equation system.

The Effects of Physical Exercise on the Insulin-Dependent Diabetes Mellitus Subjects Using the Modified Minimal Model

“…In this paper, experimental data on NGT subject published in reference [9]. The first column is the time in a minute to sample the blood with a two-minute shift.…”

“…The fourth and fifth columns are the data for the subject 7. The sixth and seventh columns are the data for subject 8 [9]. …”

Is Dynamic Autocrine Insulin Signaling Possible? A Mathematical Model Predicts Picomolar Concentrations of Extracellular Monomeric Insulin within Human Pancreatic Islets

“…The transfer of peripheral insulin monomers should be same as the monomeric insulin concentration that remains in the circulation. We therefore obtain the following model for relative concentrations of local free insulin hexamers, dimers and monomers, and peripheral insulin:with the initial condition The transport from newly secreted insulin to peripheral insulin is assumed to be proportional to the level of monomers, , and the secretion of insulin hexamers is stimulated by glucose, which follows the dynamics of Hill's function which has been employed in other models [27], [37], [38], [39], [40], [41]. The division of by 6 is due to the secretion of insulin from the -cells as insulin hexamers.…”

Finite-time control of plasma glucose in insulin therapies for diabetes

“…In our simulations, we set T = 120 min and the other parameter values are chosen and adjusted from [16,17,20,23,24] and [18] (refer to Table 1). Just like in [16,23] and [4], a unit conversion is made.…”

“…For example, the glucose-insulin regulatory system of normal people was studied in [14][15][16][17][18], and the insulin sensitivity is considered as a whole in [19][20][21][22]. In order to determine the main cause of the sustained oscillation of the endocrine metabolic system, delay differential equation models are proposed in [23][24][25][26]. Doran et al [27] formulated a mathematical model to study the insulin therapy for critically ill patients in ICU and insulin infusion was considered.…”