The compositions of in-host microbial communities (microbiota) play a significant role in host health, and a better understanding of the microbiota's role in a host's transition from health to disease or vice versa could lead to novel medical treatments. One of the first steps toward this understanding is modeling interaction dynamics of the microbiota, which can be exceedingly challenging given the complexity of the dynamics and difficulties in collecting sufficient data. Methods such as principal differential analysis, dynamic flux estimation, and others have been developed to overcome these challenges for ordinary differential equation models. Despite their advantages, these methods are still vastly underutilized in mathematical biology, and one potential reason for this is their sophisticated implementation. While this work focuses on applying principal differential analysis to microbiota data, we also provide comprehensive details regarding the derivation and numerics of this method. For further validation of the method, we demonstrate the feasibility of principal differential analysis using simulation studies and then apply the method to intestinal and vaginal microbiota data. In working with these data, we capture experimentally confirmed dynamics while also revealing potential new insights into those dynamics. We also explore how we find the forward solution of the model differential equation in the context of principal differential analysis, which amounts to a least-squares finite element method. We provide alternative ideas for how to use the least-squares finite element method to find the forward solution and share the insights we gain from highlighting this piece of the larger parameter estimation problem. / Ph. D. / In this age of “big data,” scientists increasingly rely on mathematical models for analyzing the data and drawing insights from them. One particular area where this is especially true is in medicine where researchers have found that the naturally occurring bacteria within individuals play a significant role in their health and well-being. Understanding the bacteria’s role requires that we understand their interactions with each other and with their hosts so that we can predict how changes in bacterial populations will affect individuals’ health. Given the number and complexity of these interactions, creating good models for them is a difficult task that traditional methods often fail to complete. The goal of this work is to promote the awareness of alternatives to the traditional modeling methods and to present a particular alternative in a way that is accessible to readers to encourage its and other methods’ use. With this goal and the medical application as the focus of our work, we discuss some of the traditional methods for constructing such models, discuss some of the more modern alternatives, and describe in detail the method we use. We explain the derivation of the method and apply it to both an example problem and two separate experimental studies to demonstrate its usefulness, and in the case of the experimental studies, we gain interesting insights into the bacterial interactions captured by the data. We then focus on some of the method’s numerical details to further highlight its strengths and to identify how we can improve its application.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/78674 |
| Date | 04 August 2017 |
| Creators | Krueger, Justin Michael |
| Contributors | Mathematics, Chung, Matthias, Pop, Mihai, Gugercin, Serkan, Chung, Julianne |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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