The serial interval (SI), defined as the symptom time between an infector and an infectee, is widely used to better understand transmission patterns of an infectious disease. Estimating the SI for tuberculosis (TB) is complicated by the slow progression from asymptomatic infection to active, symptomatic disease, and the fact that there is only a 5-10% lifetime risk of developing active TB disease. Furthermore, the time of symptom onset for infectors and infectees is rarely observed accurately. In this dissertation, we first conduct a systematic literature review to demonstrate the limited methods currently available to estimate the serial interval for TB as well as the few estimates that have been published. Secondly, under the assumption of an ideal scenario where all SIs are observed with precision, we evaluate the effect of prior information on estimating the SI in a Bayesian framework. Thirdly, we apply cure models, proposed by Boag in 1949, to estimate the SI for TB in a Bayesian framework. We show that the cure models perform better in the presence of credible prior information on the proportion of the study population that develop active TB disease, and should be chosen over traditional survival models which assume that all of the study population will eventually have the event of interest—active TB disease. Next, we modify the method by Reich et al. in 2009 by using a Riemann sum to approximate the likelihood function that involves a double integral. In doing so, we are able to reduce the computing time of the approximation method by around 50%. We are also able to relax the assumption of uniformity on the censoring intervals. We show that when using weights that are consistent with the underlying skewness of the intervals, the proposed approaches consistently produce more accurate estimates than the existing approaches. We provide SI estimates for TB using empirical datasets from Brazil and USA/Canada.

When an outbreak of an infectious disease occurs, public health officials need to understand the dynamics of disease transmission in order to launch an effective response. Two quantities that are often used to describe transmission are the basic reproductive number and the distribution of the serial interval. The basic reproductive number, R0, is the average number of secondary cases a primary case will infect, assuming a completely susceptible population. The serial interval (SI) provides a measure of temporality, and is defined as the time between symptom onset between a primary case and its secondary case. Investigators typically collect outbreak data in the form of an epidemic curve that displays the number of cases by each day (or other time scale) of the outbreak. Occasionally the epidemic curve data is more expansive and includes demographic or other information. A contact trace sample may also be collected, which is based on a sample of the cases that have their contact patterns traced to determine the timing and sequence of transmission. In addition, numerous large scale social mixing surveys have been administered in recent years to collect information about contact patterns and infection rates among different age groups. These are readily available and are sometimes used to account for population heterogeneity. In this dissertation, we modify the methods presented in White and Pagano (2008) to account for additional data beyond the epidemic curve to estimate R0 and SI. We present two approaches that incorporate these data through the use of a Bayesian framework. First, we consider informing the prior distribution of the SI with contact trace data and examine implications of combining data that are in conflict. The second approach extends the first approach to account for heterogeneity in the estimation of R0. We derive a modification to the White and Pagano likelihood function and utilize social mixing surveys to inform the prior distributions of R0. Both approaches are assessed through a simulation study and are compared to alternative approaches, and are applied to real outbreak data from the 2003 SARS outbreak in Hong Kong and Singapore, and the influenza A(H1N1)2009pdm outbreak in South Africa.