Mathematical models of health focusing on diabetes: Delay differential equations and data mining

Easton, Jonathan (2015) Mathematical models of health focusing on diabetes: Delay differential equations and data mining. Doctoral thesis, Northumbria University.

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Abstract

Mathematical models have been applied to biology and health to gain a better understanding of physiological systems and disease, as well as to improve levels of treatment and care for certain conditions. This thesis will focus on two different methodologies to investigate models of health, namely delay differential equations andBayesian based data mining.

The first approach uses delay differential equations to model the glucose-insulin regulation system. Many models exist in this area, typically including four exponential functions, and take a number of different forms. The model used here is a system of two delay differential equations with two time delays. The one delay form of this model has previously been widely studied, but less is known about the two delay system from an analytical view point.

This work improves upon the existing models by incorporating Hill functions instead of exponential functions. The new model presented is studied for its appropriateness and robustness to changing parameters such as glucose infusion rate and insulin degradation. A local and global stability of the two-delay system is presented both in general terms and explicitly using Lyapunov functionals and linear matrix inequalities.

The second method employs data mining techniques including a robust and transparent naïve Bayes classifier for classification and prediction of aspects of health. A study into prediction of post-stroke mortality is made on a data set of stroke patients. Interesting results are obtained for the classification of naturally arising mortality periods and an investigation into the role of age as a risk factor for post-stroke mortality. A wide range of risk factors are then investigated for significance which are used to build new predictive models.

These two approaches have the joint aim of improving the understanding of aspects of health through mathematical modelling techniques. A new model of the glucose-insulin regulatory system is developed and for the first time an analysis of the global stability of the two-delay model by use of a Lyapunov functional is provided. The second approach sees typical and robust data mining techniques used to analyse medical data. New models for stroke mortality and prediction of diabetes and obesity are created, which review risk factors and also illustrate the benefit of data mining techniques for analysing medical data.

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