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Lexicographic Sensitivity Functions for Nonsmooth Models in Mathematical Biology The University of Maine DigitalCommons@UMaine Electronic Theses and Dissertations Fogler Library Spring 5-8-2021 Lexicographic Sensitivity Functions for Nonsmooth Models in Mathematical Biology Matthew D. Ackley University of Maine, [email protected] Follow this and additional works at: https://digitalcommons.library.umaine.edu/etd Part of the Biology Commons, Dynamic Systems Commons, Immunology and Infectious Disease Commons, Numerical Analysis and Computation Commons, and the Ordinary Differential Equations and Applied Dynamics Commons Recommended Citation Ackley, Matthew D., "Lexicographic Sensitivity Functions for Nonsmooth Models in Mathematical Biology" (2021). Electronic Theses and Dissertations. 3355. https://digitalcommons.library.umaine.edu/etd/3355 This Open-Access Thesis is brought to you for free and open access by DigitalCommons@UMaine. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of DigitalCommons@UMaine. For more information, please contact [email protected]. LEXICOGRAPHIC SENSITIVITY FUNCTIONS FOR NONSMOOTH MODELS IN MATHEMATICAL BIOLOGY By Matthew Ackley B.A. University of Maine, 2019 A THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Arts (in Mathematics) The Graduate School The University of Maine May 2021 Advisory Committee: Peter Stechlinski, Assistant Professor of Mathematics, Advisor David Hiebeler, Professor of Mathematics Brandon Lieberthal, Lecturer of Mathematics LEXICOGRAPHIC SENSITIVITY FUNCTIONS FOR NONSMOOTH MODELS IN MATHEMATICAL BIOLOGY By Matthew Ackley Thesis Advisor: Professor Peter Stechlinski An Abstract of the Thesis Presented in Partial Fulfillment of the Requirements for the Degree of Master of Arts (in Mathematics) May 2021 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 (C1). 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. ACKNOWLEDGEMENTS First and foremost I would like to thank my advisor, Dr. Peter Stechlinski, for his abundance of help and guidance in the creation of this work. Through multiple article publications, presentations, and many hours of meetings, Peter has been an excellent mentor — second to none — and I will truly miss our work together. I would also like to thank Dr. David Hiebeler and Dr. Brandon Lieberthal for their work as members of my committee and for providing insightful advice and thoughts as this thesis was assembled. Finally, I would like to thank my family in Rockport, along with my significant other, Sara, for their incredible and incalculable amount of support over the last several years. Figures and tables appearing in Chapter 4 have been reprinted from the source Matthew Ackley and Peter Stechlinski. Lexicographic derivatives of nonsmooth glucose-insulin kinetics under normal and artificial pancreatic responses. Applied Mathematics and Computation, 395:125876, 2021. Copyright ©2021 Elsevier. Similarly, figures and tables appearing in Chapter 5 have been reprinted from the source Matthew Ackley and Peter Stechlinski. Determining key parameters in riots using lexicographic directional differentiation. SIAM Journal on Applied Mathematics, In Production. Copyright ©2021 Society for Insutrial and Applied Mathematics. Reprinted with permission. All rights reserved. The permission of both journals is appreciated. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS ................................................................... ii LIST OF TABLES ............................................................................ v LIST OF FIGURES ........................................................................... vi LIST OF THEOREMS AND DEFINITIONS ............................................... ix Chapter 1. INTRODUCTION ........................................................................ 1 2. PRELIMINARIES AND BACKGROUND.............................................. 6 2.1 Notation .............................................................................. 6 2.2 Classical sensitivity theory of ordinary differential equations .................... 7 2.3 Clarke’s generalized derivative ...................................................... 12 3. LEXICOGRAPHIC SENSITIVITY FUNCTIONS FOR NONSMOOTH ORDINARY DIFFERENTIAL EQUATIONS ........................................... 24 3.1 Lexicographically smooth functions and the lexicographic derivative ............ 24 3.2 Computation of lexicographic derivatives .......................................... 27 3.3 Generalized sensitivity analysis of nonsmooth ordinary differential equations ............................................................................. 38 4. QUANTIFYING PARAMETRIC SENSITIVITIES OF A NONSMOOTH MODEL OF GLUCOSE-INSULIN INTERPLAY ....................................... 45 iii 4.1 Model background ................................................................... 45 4.2 Model formulation for healthy subjects ............................................ 48 4.3 Variations for modeling subjects with type 1 diabetes ............................ 58 4.4 Discussion ............................................................................ 68 5. QUANTIFYING PARAMETRIC SENSITIVITIES OF A NONSMOOTH MODEL OF RIOT SPREAD ............................................................. 70 5.1 Model background ................................................................... 70 5.2 Model formulation ................................................................... 73 5.3 Lexicographic sensitivity analysis................................................... 77 5.4 Discussion ............................................................................ 90 6. CONCLUSION ............................................................................ 93 REFERENCES ................................................................................ 99 BIOGRAPHY OF THE AUTHOR .......................................................... 100 iv LIST OF TABLES ∗ 4.1 Parameters in (4.1) are labeled and described above. Note that pi represents a reference parameter [1]. .......................................... 49 4.2 The parameters introduced to the minimal model when formulated ∗ to represent a diabetic patient. Note that pi represents a reference parameter [1]. ................................................................... 59 4.3 The different considerations of the insulin infusion term u [1]. .............. 60 4.4 Sensitivity metrics for glucose and insulin concentrations with respect to model input parameters: the absolute maximum of relative sensitivity over time horizon, the average relative sensitivity over time horizon, and the absolute variance of relative sensitivity over time horizon [1]. ................................................................ 66 5.1 Site-independent parameters used in (5.1), with reference values from [12] [2]. .......................................................................... 75 v LIST OF FIGURES 2.1 Solutions of the system (2.5) with differing choices of reference parameter p∗. In black, p∗ = (0; −1); in blue, p∗ = (1; 1); and in red, p∗ = (1; 2). ...................................................................... 10 2.2 Varying behavior in solutions to (2.5) for different choices of p at different fixed times. At top-left, t = 0 and solutions take the form of the x = p2 plane. At top-right, t = 1 and solutions begin to diverge. At bottom, by t = 3, solutions have diverged greatly (note the difference in scale between the figures)...................................
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