Date on Master's Thesis/Doctoral Dissertation

8-2015

Document Type

Doctoral Dissertation

Degree Name

Ph. D.

Department

Mathematics

Degree Program

Applied and Industrial Mathematics, PhD

Committee Chair

Li, Jiaxu

Committee Co-Chair (if applicable)

Li, Bingtuan

Committee Member

Li, Bingtuan

Committee Member

Hu, Changbing

Committee Member

Gill, Ryan

Committee Member

Kalbfleisch, Ted

Subject

Glucose; Insulin; Biology--Mathematical models

Abstract

Three dynamic models are proposed to study the mechanism of glucose-insulin regulatory system and the possible causes of diabetes mellitus. The progression of diabetes comes along with the apoptosis of pancreatic beta-cells. A dynamical system model is formulated based on physiology and studied by geometric singular perturbation theory. The analytical studies reveal rich analytical features, such as persistence of solutions, Hopf bifurcation and backward bifurcation, while numerical studies successfully fit available longitudinal T2DM data of Pima Indian tribe. These studies together not only validate our model, but also point out key intrinsic factors leading to the development of T2DM. We found that the intermittent rests of beta-cells in insulin secretion are essential for the cells to survive through the observation of the existence of a limit cycle. A delay differential equation model for IVGTT is also studied thoroughly to determine the range of time delay and the globally asymptotic stability by Liapunov function. The third kinetic model aims to investigate the scaling effect of local insulin in islet on proliferation and apoptosis of beta-cells. It is revealed that the local concentration of monomeric insulin within the islet is in the biologist defined picomolar ‘sweet spot’ range of insulin doses, which activate the insulin receptors and have the most potent effects on beta-cells in vitro.

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