A critical review of mathematical models and data used in diabetology

The literature dealing with mathematical modelling for diabetes is abundant. During the last decades, a variety of models have been devoted to different aspects of diabetes, including glucose and insulin dynamics, management and complications prevention, cost and cost-effectiveness of strategies and epidemiology of diabetes in general. Several reviews are published regularly on mathematical models used for specific aspects of diabetes. In the present paper we propose a global overview of mathematical models dealing with many aspects of diabetes and using various tools. The review includes, side by side, models which are simple and/or comprehensive; deterministic and/or stochastic; continuous and/or discrete; using ordinary differential equations, partial differential equations, optimal control theory, integral equations, matrix analysis and computer algorithms

An Ordinal Multivariate Analysis of Accident Counts as Functions of Traffic Approach Volumes at Intersections

This research is concerned with the statistical analysis of accident counts at non-signalized intersections. The objective is to develop a method for determining general (non-linear) relationships between approach volumes and accident counts. The method must accommodate the testing of whether intersections of differing physical designs have higher or lower rates of accidents than predicted by traffic levels. It is assumed that only aggregate data are available: (1) counts of total accidents by type (e.g., injury versus property damage) without details concerning the locational position(s) of the vehicle(s) involved; and (2) aggregate traffic intensity on each intersection entry without details concerning turning volumes. The method involves the application of non-linear multivariate methods to variables treated as ordinal scales. A case study application involving four-leg and three-leg ("T") non-signalized major arterial intersections in the Netherlands is described. The effect of bicycle traffic on accident rates is included in the case study analysis. The results indicate that there are three groups of each of the two types of intersections based on traffic flow patterns. For each group, a different functional form was found to relate accident rates and specific variables measuring traffic volumes. There were no significant differences among the physical design categories of the intersections in each group that were not accounted for by differences in traffic intensities

A new general glucose homeostatic model using a proportional-integral-derivative controller

The glucose-insulin system is a challenging process to model due to the feedback mechanisms present, hence the implementation of a model-based approach to the system is an on-going and challenging research area. A new approach is proposed here which provides an effective way of characterising glycaemic regulation. The resulting model is built on the premise that there are three phases of insulin secretion, similar to those seen in a proportional-integral-derivative (PID) type controller used in engineering control problems. The model relates these three phases to a biological understanding of the system, as well as the logical premise that the homeostatic mechanisms will maintain very tight control of the system. It includes states for insulin, glucose, insulin action and a state to simulate an integral function of glucose. Structural identifiability analysis was performed on the model to determine whether a unique set of parameter values could be identified from the available observations, which should permit meaningful conclusions to be drawn from parameter estimation. Although two parameters - glucose production rate and the proportional control coefficient - were found to be unidentifiable, the former is not a concern as this is known to be impossible to measure without a tracer experiment, and the latter can be easily estimated from other means. Subsequent parameter estimation using Intravenous Glucose Tolerance Test (IVGTT) and hyperglycaemic clamp data was performed and subsequent model simulations have shown good agreement with respect to these real data