Date of Award

Spring 2015

Degree Type


Degree Name

Doctor of Philosophy (PhD)



First Advisor

Zhilan Feng

Committee Chair

Zhilan Feng

Committee Member 1

Gregery Buzzard

Committee Member 2

Aaron Yip

Committee Member 3

Andrew Hill


The use of discrete-time models (or discrete models) in the field of mathematical epidemiology has been limited while continuous-time models (or continuous models) are often times preferred, particularly because disease dynamics do occur continuously in time and more mathematical tools are available for model analysis. How- ever, discrete models are not only more tractable and easier to understand, but also more directly related to data, particularly when the disease stage distributions are arbitrarily distributed (e.g., when the data cannot be fitted by distributions from a parametric family). Under these circumstances continuous models usually lead to complex system of integral equations. Deterministic and stochastic epidemic models have commonly assumed that the disease stages, particularly the infectious period, have constant exit rates (continuous models) or constant exit probabilities (discrete models), which correspond to exponential and geometric distributions, respectively. The very property of these distributions that makes models tractable, the memoryless property, is biologically unrealistic for most infectious diseases. In fact, it has been shown that models with these simplifying assumptions may generate biased and possibly misleading evaluations for disease control strategies. Realistic alternatives considered in the literature are the Gamma and Negative Binomial distributions, a natural generalization due to their relationship with the above mentioned distributions. The "linear chain trick" can be used to reduce a system of integro-differential equations to a system of ordinary differential equations and a similar idea can be applied in stochastic models to allow for the use of Gamma distribution, while still preserving the Markov property of the process. Few models, however, include distributions beyond these alternatives. The focuses of this thesis is the use of arbitrarily distributed disease stages in discrete models, their formulation and analysis, as well as the study of the impact of a given distribution on model predictions. Chapter 1 includes a brief review of relevant topics and the motivation for this work. In Chapter 2 several SEIR-type models with arbitrarily distributed infectious period are introduced and analyzed. This chapter focuses on the use of the next generation matrix approach to derive analytic expressions for R0 and RC. In Chapter 3 we develop and analyze of a model with quarantine and isolation when arbitrarily distributed disease stages are incorporated. The results obtained in the general framework are then applied to models with specific distributions (e.g., Geometric vs. more realistic distributions), which allow us to investigate the influence of disease stage distributions on the dynamics of single epidemic outbreaks. It is demonstrated that the discrepancies between model predictions can sometimes be substantial. In Chapter 4 a stochastic discrete-time model with n patches and (random) infectious period T is developed. The results obtained are then used to investigate how the distribution of T may affect model outcomes. Specific distributions analyzed include Geometric, Negative Binomial, Poisson and Uniform. The model predictions are contrasted both numerically and analytically by comparing the corresponding R0 values as well as the probability of disease extinction. It is shown analytically that for n=2 the R0 values corresponding to different distributions of T can be ordered based on the probability generating function of T. In addition, numerical simulations are carried out to examine the final epidemic size, duration and peak of the epidemic.