A Bifurcational Analysis of the Onset of Type 1 Diabetes
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Date
2018-01-01
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Department
Mathematics and Statistics
Program
Mathematics, Applied
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Access limited to the UMBC community. Item may possibly be obtained via Interlibrary Loan thorugh a local library, pending author/copyright holder's permission.
This item is likely protected under Title 17 of the U.S. Copyright Law. Unless on a Creative Commons license, for uses protected by Copyright Law, contact the copyright holder or the author.
Abstract
The purpose of this paper is to combine two models of diabetes and analyze the periodic behavior and the bifurcations produced by the newly combined model. The first of these two models by Mahaffy \cite{Mahaffy2007} analyzes the onset of type 1 diabetes (T1D) at the cellular level due to an immune response, where as the second, the Topp model \cite{Topp}, analyzes the coupled dynamics between beta cell mass, insulin, and glucose. Both models include an equation for beta cell mass which is the key equation in combining the two and will enable us to look at how insulin and glucose levels change and relate to the onset of T1D. The resulting model provides a plethora of mathematically interesting properties such as various different bifurcations and bifurcation types as well as chaos. In terms of biology, we show that the combined model produces a situation in which beta cells actually recover after the initial attack on the pancreas. We are able look at the concentration of certain cell types in the blood at different stages during the onset. Our goal is to use the mathematical properties mentioned above to conclude that the combination of the Mahaffy and Topp models, and thus coupling glucose and insulin to immune cells, leads to a case of recovery of beta cells as well as forcing beta cell recovery by controlling the degradation of beta cell peptides. This exhibits the impact that certain parameter changes have on pathways to T1D. From this analysis, we can conclude that there are two types of recovery from T1D before it sets in and becomes permanent. The first is cyclic recovery in which beta cell mass, insulin concentrations, and glucose concentrations oscillate as they return to their healthy steady state values and low levels of effector T-cells remain in the blood stream but not high enough levels to induce full blown T1D. The second is noncyclic recovery in which beta cell mass, insulin concentrations, and glucose concentrations return to healthy steady state values but do not oscillate, which means that no effector T-cells remain in the blood after a certain time period.