Publication Date

Summer 7-29-2025

Abstract

Ordinary Differential Equations (ODEs) are central to the mathematical modeling of various real-world phenomena, from mechanical systems governed by Newton’s laws to epidemic dynamics described by SIR-type ODEs. Since many ODEs do not admit closed-form analytic solutions, we approximate them numerically (e.g., with Euler’s, Runge–Kutta, or other such methods). This raises the key question: How accurate are these numerical solutions? In particular, reliably estimating the error in some quantity of interest (QoI) at time T without having an exact solution is of great scientific interest.

The first main contribution of this thesis is the development and analysis of adjoint-based error representations that estimate the error in a chosen QoI at final time T. Specifically, we derive four key theorems addressing the scalar linear, vector linear, scalar nonlinear, and vector nonlinear ODE cases. Each theorem shows how to express the true error in the QoI through an integral involving only numerically computed solutions of both the state ODE and a corresponding adjoint ODE. We validate these theoretical results with several examples, including linear ODEs (e.g., a scalar first-order linear equation), nonlinear ODEs (e.g., the logistic equation), and systems of ODEs (e.g., the harmonic oscillator and two-body orbital problems). Our findings confirm that the adjoint-based estimates reliably capture the final-time error, thus offering a rigorous framework for quantifying errors in ODE-based models across scientific and engineering applications.

The second main contribution is the numerical implementation of these ideas. We show how to solve the state ODE forward in time and the adjoint ODE backward in time, then apply Gauss–Legendre quadrature to approximate the final-time error integral. This process involves a nested Gauss–Legendre scheme on [0,T], subdividing the main time intervals and using local quadrature nodes to approximate the integral accurately.

The third main contribution of this thesis is the illustration and validity of these theorems on the SIR‐type epidemic models (classical SIR, SIR with demographics, multi‐stage SIR) in epidemiology. These numerical experiments help us estimate the final-time T error in the infected class. We show how the adjoint‐based error estimates perform in classical SIR, SIR with demographics, and multi‐stage SIR contexts, confirming the (error in the QoI for non-linear vector ODEs), hence showing the validity in realistic epidemiological systems. Therefore, more importantly, showing us the potential of our theorems in real-world application contexts. By better understanding these models, public health authorities can make more informed decisions about how best to contain outbreaks, reduce the number of infected individuals, and, in the process, ensure healthier communities.

Degree Name

Mathematics

Level of Degree

Masters

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Jehanzeb Chaudhary

Second Committee Member

Helen Wearing

Third Committee Member

Owen Lewis

Language

English

Keywords

adjoint‐based a posteriori error estimation, multistage SIR epidemic model, Ordinary Differential Equations (ODE), Gauss-Legendre quadrature, numerical stability and mesh refinement, quantity of interest (QoI)

Document Type

Thesis

Comments

This is a resubmission of my thesis as a revision, following the successful completion of all edits requested by the thesis coordinator, Rikk Murphy. All requested changes have been fully implemented.

Thank you

Daniel Alcala

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