| In this dissertation, we explore multiple aspects of the inverse problem when applied to infectious disease models. First, we examine estimation of the parameters of Susceptible-Infective-Recovered (SIR) models in the context of least squares (LS). We review the use of asymptotic statistical theory and sensitivity analysis to obtain measures of uncertainty for estimates of the model parameters and basic reproductive number (R 0)---an epidemiologically significant parameter grouping. Uncertainty estimates and sensitivity analysis are used to investigate how the frequency at which data is sampled affects the estimation process and how the accuracy and uncertainty of estimates improves as data is collected over the course of an outbreak. We assess the informativeness of individual data points in a given time series with a view to better understanding when more frequent sampling (if possible) would prove to be most beneficial to the estimation process. We include a more general discussion of parameter identifiablility in both the epidemic and seasonal endemic SIR settings. We propose an algorithm to select parameter subset combinations that can be estimated using an LS inverse problem formulation with a given data set. The algorithm selects the parameter combinations that correspond to sensitivity matrices with full rank and it involves uncertainty quantification by using the inverse of the Fisher Information Matrix. We conclude with an application of the Akaike information criterion to select a model from a series of epidemic models fitted to an outbreak of influenza in a boy's boarding school in England. We find that an uncommonly used epidemic model, a Susceptible-Infective-Confined-Recovered (SICR), model is the best fitted model and produces an estimate of R0 of 4.25. |
