Single Delay Model Research Articles

Consistency of compact and extended models of glucose-insulin homeostasis: The role of variable pancreatic reserve.

Published compact and extended models of the glucose-insulin physiologic control system are compared, in order to understand why a specific functional form of the compact model proved to be necessary for a satisfactory representation of acute perturbation experiments such as the Intra Venous Glucose Tolerance Test (IVGTT). A spectrum of IVGTT’s of virtual subjects ranging from normal to IFG to IGT to frank T2DM were simulated using an extended model incorporating the population-of-controllers paradigm originally hypothesized by Grodsky, and proven to be able to capture a wide array of experimental results from heterogeneous perturbation procedures. The simulated IVGTT’s were then fitted with the Single-Delay Model (SDM), a compact model with only six free parameters, previously shown to be very effective in delivering precise estimates of insulin sensitivity and secretion during an IVGTT. Comparison of the generating, extended-model parameter values with the obtained compact model estimates shows that the functional form of the nonlinear insulin-secretion term, empirically found to be necessary for the compact model to satisfactorily fit clinical observations, captures the pancreatic reserve level of the simulated virtual patients. This result supports the validity of the compact model as a meaningful analysis tool for the clinical assessment of insulin sensitivity.

Goodwin accelerator model revisited with fixed time delays

Dynamics of Goodwin’s accelerator business cycle model is reconsidered. The model is characterized by a nonlinear accelerator and an investment time delay. The role of the nonlinearity for the birth of persistent oscillations is fully discussed in the existing literature. On the other hand, not much of the role of the delay has yet been revealed. The purpose of this paper is to show that the delay really matters. In the original framework of Goodwin [6], it is first demonstrated that there is a threshold value of the delay: limit cycles arise for smaller values than the threshold and so do sawtooth oscillations for larger values. In the extended framework in which a consumption or saving delay, in addition to the investment delay, is introduced, three main results are demonstrated under assumption of the identical length of investment and consumption delays. The dynamics with consumption delay is basically the same as that of the single delay model. Second, in the case of saving delay, the steady state can coexist with the stable and unstable limit cycles in the stable case. Third, in the unstable case, there is an interval of delay in which the limit cycle or the sawtooth oscillation emerges depending on the choice of the constant initial function.

Equilibrium Points and Their Stability Properties of a Multiple Delays Model

In this paper, we present a mathematical model with delay which describes the dynamics of three fish populations. The equilibrium points and their stability properties of the non-delay model are studied using Eigenvalue analysis. For the system with single delay model and multiple delays model, the existence of the interior equilibrium point and its stability are studied using the characteristic equation for the linearized system.

The Asymptotic Behavior in a Nonlinear Cobweb Model with Time Delays

We study the effects of production delays on the local as well as global dynamics of nonlinear cobweb models in a continuous-time framework. After reviewing a single delay model, we proceed to two models with two delays. When the two delays are used to form an expected price or feedback for price adjustment, we have a winding stability switching curve and in consequence obtain repetition of stability losses and gains via Hopf bifurcation. When the two delays are involved in two interrelated markets, we find that the stability switching occurs on straight lines and complicated dynamics can arise in unstable markets.

Voltage and Temperature-Aware SSTA Using Neural Network Delay Model

With the emergence of voltage scaling as one of the most powerful power reduction techniques, it has been important to support voltage scalable statistical static timing analysis (SSTA) in deep submicrometer process nodes. In this paper, we propose a single delay model of logic gate using neural network which comprehensively captures process, voltage, and temperature variation along with input slew and output load. The number of simulation programs with integrated circuit emphasis (SPICE) required to create this model over a large voltage and temperature range is found to be modest and 4× less than that required for a conventional table-based approach with comparable accuracy. We show how the model can be used to derive sensitivities required for linear SSTA for an arbitrary voltage and temperature. Our experimentation on ISCAS 85 benchmarks across a voltage range of 0.9-1.1 V shows that the average error in mean delay is less than 1.08% and average error in standard deviation is less than 2.85%. The errors in predicting the 99% and 1% probability point are 1.31% and 1%, respectively, with respect to SPICE. The two potential applications of voltage-aware SSTA have been presented, i.e., one for improving the accuracy of timing analysis by considering instance-specific voltage drops in power grids and the other for determining optimum supply voltage for target yield for dynamic voltage scaling applications.

Global stability of a delay model of glucose–insulin interaction

Recently, there were several differential delay models of the glucose–insulin system analyzed in the literature. One of the mostly used models is generally known as the m i n i m a l m o d e l which was first published in 1979 and modified in 1986. This minimal model has been challenged by De Gaetano and Arino in 2000 from both physiological and modeling aspects. Instead, they proposed a new and mathematically more reasonable model, called d y n a m i c m o d e l . Following dynamic model, Li, Kuang and Li proposed a more popular, general and realistic model. Panunzi, Palumbo and De Gaetano believed that it requires a reasonably simple model, with as few parameters to be estimated as practicable, and with a qualitative behavior consistent with physiology, and they proposed a discrete single delay model (SDM). These models were proved to have the reasonable qualitative behavior, such as uniform persistence and global stability of equilibrium. In this paper we investigate a more general model, which includes SDM and one of the models in Li et al. (2001) [7] as its special cases. The model is shown to admit globally stable equilibrium under certain conditions of the parameters. For the two special models (SDM and one in [7]), it is shown that these conditions are slightly better. It is also shown that the model admits oscillating behavior due to the existence of Hopf-bifurcation.

Advantages of the single delay model for the assessment of insulin sensitivity from the intravenous glucose tolerance test

BackgroundThe Minimal Model, (MM), used to assess insulin sensitivity (IS) from Intra-Venous Glucose-Tolerance Test (IVGTT) data, suffers from frequent lack of identifiability (parameter estimates with Coefficients of Variation (CV) less than 52%). The recently proposed Single Delay Model (SDM) is evaluated as a practical alternative.MethodsThe SDM was applied to 74 IVGTTs from lean (19), overweight (22), obese (22) and morbidly obese (11) subjects. Estimates from the SDM (KxgI) were compared with the corresponding MM (SI), 1/HOMA-IR index and Euglycemic-Hyperinsulinemic Clamp (M-EHC over 7 subjects) estimates.ResultsKxgI was identifiable in 73 out of 74 subjects (CV = 69% in the 74th subject) and ranged from 1.25 × 10-5 to 4.36 × 10-4min-1pM-1; SI CV was >52% in 36 subjects (up to 2.36 × 109%) and presented 18 extreme values (≤ 1.5 × 10-12 or ≥ 3.99).KxgI correlated well with 1/HOMA-IR (r = 0.56, P < 0.001), whereas the correlations KxgI-SI and 1/HOMA-IR-SI were high (r = 0.864 and 0.52 respectively) and significant (P < 0.001 in both cases) only in the non-extreme SI sub-sample (56 subjects). Correlations KxgI vs. M-EHC and SI vs. M-EHC were positive (r = 0.92, P = 0.004 and r = 0.83, P = 0.02 respectively). KxgI decreased for higher BMI's (P < 0.001), SI significantly so only over the non-extreme-SI sub-sample. The Acute Insulin Response Index was also computed and the expected inverse (hyperbolic) relationship with the KxgI observed.ConclusionsPrecise estimation of insulin sensitivity over a wide range of BMI, stability of all other model parameters, closer adherence to accepted physiology make the SDM a useful alternative tool for the evaluation of insulin sensitivity from the IVGTT.

Stability Analysis of a Max-Min Fair Rate Control Protocol (RCP) in a Small Buffer Regime

In this note we analyse various stability properties of the max-min fair Rate Control Protocol (RCP) operating with small buffers. We first tackle the issue of stability for networks with arbitrary topologies. We prove that the max-min fair RCP fluid model is globally stable in the absence of propagation delays, and also derive a set of conditions for local stability when arbitrary heterogeneous propagation delays are present. The network delay stability result assumes that, at equilibrium, there is only one bottleneck link along each route. Lastly, in the simpler setting of a single link, single delay model, we investigate the impact of the loss of local stability via a Hopf bifurcation.

A mathematical model of a crocodilian population using delay-differential equations

The crocodilia have multiple interesting characteristics that affect their population dynamics. They are among several reptile species which exhibit temperature-dependent sex determination (TSD) in which the temperature of egg incubation determines the sex of the hatchlings. Their life parameters, specifically birth and death rates, exhibit strong age-dependence. We develop delay-differential equation (DDE) models describing the evolution of a crocodilian population. In using the delay formulation, we are able to account for both the TSD and the age-dependence of the life parameters while maintaining some analytical tractability. In our single-delay model we also find an equilibrium point and prove its local asymptotic stability. We numerically solve the different models and investigate the effects of multiple delays on the age structure of the population as well as the sex ratio of the population. For all models we obtain very strong agreement with the age structure of crocodilian population data as reported in Smith and Webb (Aust. Wild. Res. 12, 541-554, 1985). We also obtain reasonable values for the sex ratio of the simulated population.

A discrete Single Delay Model for the Intra-Venous Glucose Tolerance Test

BackgroundDue to the increasing importance of identifying insulin resistance, a need exists to have a reliable mathematical model representing the glucose/insulin control system. Such a model should be simple enough to allow precise estimation of insulin sensitivity on a single patient, yet exhibit stable dynamics and reproduce accepted physiological behavior.ResultsA new, discrete Single Delay Model (SDM) of the glucose/insulin system is proposed, applicable to Intra-Venous Glucose Tolerance Tests (IVGTTs) as well as to multiple injection and infusion schemes, which is fitted to both glucose and insulin observations simultaneously. The SDM is stable around baseline equilibrium values and has positive bounded solutions at all times. Applying a similar definition as for the Minimal Model (MM) SI index, insulin sensitivity is directly represented by the free parameter KxgI of the SDM.In order to assess the reliability of Insulin Sensitivity determinations, both SDM and MM have been fitted to 40 IVGTTs from healthy volunteers. Precision of all parameter estimates is better with the SDM: 40 out of 40 subjects showed identifiable (CV < 52%) KxgI from the SDM, 20 out of 40 having identifiable SI from the MM. KxgI correlates well with the inverse of the HOMA-IR index, while SI correlates only when excluding five subjects with extreme SI values. With the exception of these five subjects, the SDM and MM derived indices correlate very well (r = 0.93).ConclusionThe SDM is theoretically sound and practically robust, and can routinely be considered for the determination of insulin sensitivity from the IVGTT. Free software for estimating the SDM parameters is available.