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Topic

Deterministic and Stochastic Modelling of Infectious Diseases in the Early Stages

Department of Mathematics and Statistics

Date & location

• Friday, April 11, 2025
• 10:00 A.M.
• Clearihue Building, Room B019

Examining Committee

Supervisory Committee

• Dr. Junling Ma, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
• Dr. Pauline van den Driessche, Department of Mathematics and Statistics, UVic (Co-Supervisor)
• Dr. Dean Karlen, Department of Physics and Astronomy, UVic (Outside Member)

External Examiner

• Dr. Michael Li, Department of Mathematical and Statistical Sciences, University of Alberta

Chair of Oral Examination

• Dr. Mihai Sima, Department of Electrical and Computer Engineering, UVic

Abstract

During the early stages of an epidemic, case counts typically grow exponentially, influenced by disease transmissibility, contact patterns, and implemented control measures. Understanding this exponential growth and disentangling the effects of various interventions are critical for public health decision-making. This dissertation investigates the dynamics of the early stages of an epidemic under control measures, addressing two key topics: evaluating the effectiveness of contact tracing and estimating the exponential growth rate of cases.

Contact tracing is a key public health measure to reduce disease transmission. However, due to limited public health capacity, it is mostly effective during the early stage when the case counts are low. In Chapter 2, I develop a novel modelling framework to track contacts in a randomly mixed population. This approach borrows the idea of edge dynamics from network models to track contacts included in a compartmental SIR model for an epidemic spreading. Using COVID-19 as a case study, I evaluate the effectiveness of contact tracing during the early stage when multiple control measures were implemented in Chapter 3. I conduct a simulation study to determine the necessary dataset for parameter estimation. I find that new case counts, cases identified through contact tracing (or voluntary testing), and symptomatic onset counts are necessary for parameter identification. Finally, I apply our models to the early stages of the COVID-19 pandemic in Ontario, Canada.

Chapters 4 and 5 focus on reliably estimating the exponential growth rate during the early stages of an outbreak, a key measure of the speed of disease spread. To establish a suitable likelihood function for accurate growth rate estimation, I derive the probability generating function for new cases using a linear stochastic SEIR model and obtain formulas for its mean and variance in Chapter 4. Numerical simulations show that the binomial or negative binomial distribution closely approximates the distribution of new cases. To determine the most appropriate method for estimating the growth rate, I compare the performance of the negative binomial regression model and the hidden Markov model (HMM) in Chapter 5. My results show that the 95% credible intervals produced by the HMM have a higher probability of covering the true growth rate.