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.
I
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.
II
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.
III
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.