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The Final Oral Examination
for the Degree of Master of Science
(Department of Mathematics and Statistics)

Songwan Joun
BSc (Kyungpook National University, 2017)

“Multi-SNP prediction model for lung function decline”

Friday, April 16, 2021
11:00 A.M.
Conducted Virtually

Supervisory Committee:
Dr. Xuekui Zhang, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Xing Li, Department of Mathematics and Statistics, University of Saskatchewan (Member)

Chair of Oral Examination:
Dr. Ke Xu, Economics, University of Victoria

Notice of the Final Oral Examination for the Degree of Master of Science of

KAI YZENBRANDT
BSc (University of Victoria, 2018)

“Minimax D-optimal Designs for Regression Models with Heteroscedastic Errors”

Department of Mathematics and Statistics

Wednesday, April 14, 2021 1:30 P.M. Conducted Virtually

Supervisory Committee:
Dr. Julie Zhou, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Laura Cowen, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Xiaodai Dong, Department of Electrical and Computer Engineering, UVic

Chair of Oral Examination:
Dr. Timothy Iles, Department of Pacific and Asian Studies, UVic

Abstract
Minimax D-optimal designs for regression models with heteroscedastic errors are studied and constructed. These designs are robust against possible misspecification of the error variance in the model. We propose a flexible assumption for the error variance and use a minimax approach to define robust designs. As usual it is hard to find robust designs analytically, since the associated design problem is not a convex optimization problem. However, the minimax D- optimal design problem has an objective function as a difference of two convex functions. An effective algorithm is developed to compute minimax D-optimal designs under the least squares estimator and generalized least squares estimator. The algorithm can be applied to construct minimax D-optimal designs for any linear or nonlinear regression model with heteroscedastic errors. In addition, several theoretical results are obtained for the minimax D-optimal designs.

Notice of the Final Oral Examination for the Degree of Master of Science of

XINQI XIE
BSc (University of Victoria, 2018)

“Within-host Dynamics of HIV/AIDS”

Department of Mathematics and Statistics
Tuesday, April 13, 2021 10:00 A.M.
Conducted Virtually

Supervisory Committee:
Dr. Pauline van den Driessche, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Junling Ma, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner:
Prof. Libin Rong, Department of Mathematics, University of Florida

Chair of Oral Examination:
Dr. Nathan Lachowsky, School of Public Health and Social Policy, UVic

This thesis first investigates within-host HIV models for the acute stage. These models incorporate the immune responses and helper T cells produced from the activation of naive CD4 T cells. Because both naive CD4 T cells and helper T cells are susceptible classes, backward bifurcation and bistability may occur. We start with a simple model that ignores the CD8 T cell dynamics, then extend it to include this dynamics. We also extend our model to consider the latent infection of naive CD4 T cells. Backward bifurcation occurs in all these models. We numerically investigate the stability of viral equilibria, and show the bistability caused by backward bifurcation. Increasing the in flow of CTLs prevents the backward bifurcation. With a large homeostatic source of healthy naive CD4 T cells, the disease is easier to establish when the basic reproduction number is less than one. Reducing the reproduction number below one is not sufficient to control the infection of HIV. Secondly, this thesis investigates the development of AIDS caused by viral diversity, as proposed by Wodarz et al. using a model that does not include the details of immune responses. We extend their model to include density dependence, and show that the viral load increases with viral diversity. To study if this result still holds with more realistic HIV dynamics, we incorporate viral diversity into our first model. We conclude theoretically that the total viral load is positively correlated with the number of viral strains, and viral diversity can drive the development of AIDS. We also find that the total CD4 T cell count does not always decrease with viral diversity. Thus further investigation is needed to fully understand the development of AIDS.

See announcement and abstract (PDF file)

See announcement and abstract (PDF file)

See announcement and abstract (PDF file).

Notice of the Final Oral Examination for the Degree of Master of Science of

KURTIS MANKE
BSc (Thompson Rivers University, 2017)

“Generating function approach for the effective degree SIR Model”

Department of Mathematics and Statistics

Thursday, December 10, 2020 2:00 P.M. Conducted Remotely

Supervisory Committee:
Dr. Junling Ma, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Slim Ibrahim, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner:
Dr. Hao Wang, Department of Mathematical and Statistical Sciences, University of Alberta

Chair of Oral Examination:
Dr. Timothy Iles, Department of Pacific and Asian Studies, UVic

Abstract
The effective degree model has been applied to both SIR and SIS type diseases (those which confer permanent immunity and those which do not, respectively) with great success. The original model considers a large system of ODEs to keep track of the number of infected and susceptible neighbours of an individual. In this thesis, we use a generating function approach on the SIR effective degree model to transform the system of ODEs into a single PDE. This has the advantage of allowing the consideration of infinite networks. We derive existence and uniqueness of solutions to the PDE. Furthermore, we show that the linear stability of the PDE is governed by the same disease threshold derived by the ODE model, and we also show the nonlinear instability of the PDE agrees with the same disease threshold.

Notice of the Final Oral Examination for the Degree of Master of Science of

ROBERT SNEIDERMAN

BSc (McGill University, 2018)

“Data Analysis of Salmonid Environmental DNA Measurements Obtained via Controlled Experiments and from several Pacific Streams”

Department of Mathematics and Statistics

Friday, November 20, 2020 1:00 P.M.

Conducted Remotely

Supervisory Committee:
Dr. Mary Lesperance, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Morgan Hocking, School of Environmental Studies, UVic (Co-Supervisor)
Dr. Laura Cowen, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Caren Helbing, Department of Biochemistry and Microbiology, UVic

Chair of Oral Examination:
Dr. Ruohong Jiao, School of Earth and Ocean Sciences, UVic

Abstract

Standard sampling and monitoring of fish populations are invasive and time-consuming techniques. The ongoing development of statistical techniques to analyze eDNA introduces a possible solution to these challenges. We analyzed and created statistical models for qPCR data obtained from two controlled experiments conducted on samples of Coho salmon.

The first experiment analyzed was a density experiment whereby varying numbers of Coho (1, 2, 4, 8, 16, 32 and 65 fish) were placed in separate tanks and eDNA measurements were taken. The second experiment dealt with dilution, whereby three Coho were placed into tanks, removed and eDNA was then sampled at dilution volumes of 20kL, 40kL, 80kL, 160kL and 1000kL.

Finally, we analyzed a set of field data from several streams in the Pacific North West for the presence of Coho salmon. In the field models, we considered the impact of environmental covariates as well as eDNA concentrations.

Our analysis suggests that eDNA concentration can be used as a reliable proxy to estimate Coho biomass.