PhD Oral Exam - Terry Easlick, Mathematics

When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.

Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.

Abstract

This thesis is comprised of three parts which collectively serve as a study of stochastic epidemiological models, in particular, the susceptible, infected, recovered/removed (SIR) model.

We propose a unified stochastic SIR model driven by Lévy noise. The structural model allows for time-dependency, nonlinearity, discontinuity, demography and environmental disturbances. We present concise results on the existence and uniqueness of positive global solutions and investigate the extinction and persistence of the novel model. Examples and simulations are provided to illustrate the main results.

This part is twofold; we investigate the parameter estimation and forecasting of two forms of a stochastic SIR model driven by small Lévy noises, and we provide theoretical results on parameter estimation of time-dependent drift for Lévy noise-driven stochastic differential equations. A novel algorithm is introduced for approximating the least-squares estimators, which lack attainable closed-forms; moreover, the presented results ensure the consistency of these approximated Estimators.

We apply the previous results to study the COVID-19 pandemic using data from New York City, New York. This application yields parameter estimation and predictive analysis, including the unknown period for a periodic transmission function and importation/exportation of infection.