Contributions to the mathematical theory of epidemic dynamics

Abstract

I derive a novel model of disease transmission in dynamic host populations ("model MM") from mechanistic assumptions. This model addresses several concepts simultaneously: (1) infection of susceptible individuals occurs through 2 mechanisms, contact with "point" sources of infectious material resulting in a constant risk, and contact with infectious individuals resulting in additional risk that varies with the size of the infectious population; (2) that transmission may cross species boundaries; (3) that population size can effect risk; and, (4) that individual covariates affect risk. I demonstrate this model's theoretical application by forecasting possible outcomes of a bovine tuberculosis (Mycobacterium bovis) epidemic in a white-tailed deer (Odocoileits virgmiamis) population using data from a recent epidemic in Michigan. I conclude that the best use of epidemic forecasting exercises is to identify critical gaps in knowledge for future research. Next, I propose 50 additional candidate models of the epidemic process. I present the structure of these models by first defining 4 general classes of epidemic models: multinomial, structured multinomial, nested logistic regression, and biological-mechanistic. I then challenge these models using 2 classic data sets of binomial chain epidemic data from the literature (measles, "common cold") and evaluate the bias-variance tradeoff of each model/data set combination using Akaike's Information Criterion (AIC). Model selection results are not consistent between data sets and no universal model emerges. This conclusion raises questions about the rational of invoking popular epidemic models in attempts to explain or predict specific host/parasite dynamics. I then use Monte Carlo methods to evaluate the ability of AIC and it's Bayesian compliment, BIC, to detect model MM as the generating process of data of the same sample sizes and similar rates of infection as found in these 2 data sets. BIC outperformed AIC when the generating process was simple and in the candidate set of models. I conclude that sample sizes were insufficient to strongly rule out model MM as the underlying the generating process. Finally, I summarize the literature on epidemic dynamics. I conclude that many fundamental questions remain open to debate. In response, I make several recommendations to direct future research.