Date of Award

2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

Laurent Hébert-Dufresne

Second Advisor

Christopher M. Danforth

Abstract

Mathematical modeling of disease dynamics provides powerful tools to understand, predict, and evaluate emerging diseases. These insights aid public health officials, along with other modelers. With a plethora of models to choose from, it is important to consider a model that encapsulates the stochastic nature of disease dynamics. Stochasticity not only conveys chances of stochastic extinction, but provides probabilistic outcomes, essential for capturing the stochastic nature of the real world. In this thesis, three stochastic models are presented, each addressing uncertainties in mechanisms and interpretation of these models, to aid other modelers and decision makers.Starting with the source of infection spread, we address the uncertainties in human papillomavirus (HPV) within-host dynamics. The motivation behind this mechanistic cellular dynamics model is that HPV infection progression information, viral load and extinction are not well documented for model inputs. Through a master equation approach, we establish extinction probability, persistence, and viral load metrics from moment information to inform population-level model parameters. Furthermore, the structure of the skin layer, possibly indicating an older individual, impacts differing viral load information and disease propagation. The subsequent topic of this thesis models interventions on stochastic branching processes on disease spread contact patterns. From an extension of a temporal prob- ability generating function approach, we evaluate different interventions, resulting in cumulative count probability distributions. When comparing intervention strategies, probability distribution output makes comparison difficult. Nevertheless, we establish several metrics to compare these temporal and probabilistic forecasts, providing clear definition on what a decision maker may want to mitigate. Lastly, the final chapter addresses uncertainty in the giant component analysis application of probability generating functions (PGFs): a sensitivity analysis of the polynomial roots. The condition number of these roots can be evaluated when small perturbations are applied to the coefficients. Two probability distributions are presented as case studies to assess which systems may be more prone to giant component variation, or in the context of disease modeling, final outbreak size variation. This not only evaluates the sensitivity of PGF applications for the first time, but establishes a way to examine the sensitivity of a branching process for other applications. This thesis explores the uncertainties for disease progression outcomes, case count distribution comparison, and branching process final outbreak size sensitivity. Each chapter contributes method expansion or evaluation, along with commentary on declaring clear assumptions for each method.

Language

en

Number of Pages

136 p.

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