Delay differential equation models in diabetes modeling: a review Athena Makroglou + and Iordanis Karaoustas Department of Mathematics University of Portsmouth 1st Floor Lion Gate Bldg, Portsmouth PO1 3HF, UK Jiaxu Li ∗ Department of Mathematics University of Louisville Louisville KY 40292, USA Yang Kuang † Department of Mathematics and Statistics Arizona State University Tempe AZ 85287-1804, USA. Abstract Delay differential equation models can generate rich dynamics using minimum num- ber of parameters. This characteristic enables such models to play important roles in a growing number of areas of diabetes studies. Such areas include the insulin/glucose regulatory system, intravenous glucose tolerance test (IVGTT), and insulin therapies. In this paper, an extensive review of such models is presented together with some com- putational results and brief summaries of theoretical results for the cases of models for ultradian oscillations of insulin and models for diagnostic tests. 1 Introduction Diabetes mellitus is a disease of the glucose-insulin regulatory system (see for example Fig. 1.1 in [13] for a picture of the plasma glucose-insulin interaction loops). It is classified into two main categories. Type 1 diabetes which is juvenile onset and insulin-dependent and Type 2 diabetes which is adult onset and insulin-independent. Complications of the disease include retinopathy, nephropathy, peripheral neuropathy, blindness The disease is affecting hundreds millions of people worldwide (type 2 diabetes mellitus had an estimated incidence of 151 million in the year 2000, which has motivated many researchers to study the mathematical, computational and medical problems associated with it. Articles about the prevalence and the problems of diabetes appear frequently in various media outlets. + athena.makroglou@port.ac.uk * jiaxu.li@louisville.edu. Work is partially supported by NIH/NIDCR Grant R01-DE019243 and NIH/NIEHS Grant P30ES014443. † kuang@asu.edu. Work is partially supported by DMS-0436341 and DMS/NIGMS-0342388. 1