Date of Award

Spring 5-8-2021

Level of Access Assigned by Author

Open-Access Thesis

Degree Name

Master of Arts (MA)

Department

Mathematics

Advisor

Peter Stechlinski

Second Committee Member

David Hiebeler

Third Committee Member

Brandon Lieberthal

Abstract

Systems of ordinary differential equations (ODEs) may be used to model a wide variety of real-world phenomena in biology and engineering. Classical sensitivity theory is well-established and concerns itself with quantifying the responsiveness of such models to changes in parameter values. By performing a sensitivity analysis, a variety of insights can be gained into a model (and hence, the real-world system that it represents); in particular, the information gained can uncover a system's most important aspects, for use in design, control or optimization of the system. However, while the results of such analysis are desirable, the approach that classical theory offers is limited to the case of ODE systems whose right-hand side functions are at least once continuously differentiable. This requirement is restrictive in many real-world systems in which sudden changes in behavior are observed, since a sharp change of this type often translates to a point of nondifferentiability in the model itself. To contend with this issue, recently-developed theory employing a specific class of tools called lexicographic derivatives has been shown to extend classical sensitivity results into a broad subclass of locally Lipschitz continuous ODE systems whose right-hand side functions are referred to as lexicographically smooth. In this thesis, we begin by exploring relevant background theory before presenting lexicographic sensitivity functions as a useful extension of classical sensitivity functions; after establishing the theory, we apply it to two models in mathematical biology. The first of these concerns a model of glucose-insulin kinetics within the body, in which nondifferentiability arises from a biochemical threshold being crossed within the body; the second models the spread of rioting activity, in which similar nonsmooth behavior is introduced out of a desire to capture a "tipping point" behavior where susceptible individuals suddenly begin to join a riot at a quicker rate after a threshold riot size is crossed. Simulations and lexicographic sensitivity functions are given for each model, and the implications of our results are discussed.

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