Epidemics often occur rapidly, with new cases being observed daily. Due to the
frequently severe social and economic consequences of an outbreak, this is an area of research that benefits greatly from online inference. This motivates research into the construction of fast, adaptive methods for performing real-time statistical analysis of epidemic data.

The aim of this thesis is to develop sequential Monte Carlo (SMC) methods for infectious disease outbreaks. These methods utilize the observed removal times of individuals, obtained throughout the outbreak. The SMC algorithm adaptively generates samples from the evolving posterior distribution, allowing for the real-time estimation of the parameters underpinning the outbreak. This is achieved by transforming the samples when new data arrives, so that they represent samples from the posterior distribution which incorporates all of the data.

To assess the performance of the SMC algorithm we additionally develop a novel
Markov chain Monte Carlo (MCMC) algorithm, utilising adaptive proposal schemes to improve its mixing. We test the SMC and MCMC algorithms on various simulated outbreaks, finding that the two methods produce comparable results in terms of parameter estimation and disease dynamics. However, due to the parallel nature of the SMC algorithm it is computationally much faster.

The SMC and MCMC algorithms are applied to the 2001 UK Foot-and-Mouth outbreak: notable for its rapid spread and requirement of control measures to contain the outbreak. This presents an ideal candidate for real-time analysis. We find good agreement between the two methods, with the SMC algorithm again much quicker than the MCMC algorithm. Additionally, the performed inference matches well with previous work conducted on this data set.

Overall, we find that the SMC algorithm developed is suitable for the real-time
analysis of an epidemic and is highly competitive with the current gold-standard of MCMC methods, whilst being computationally much quicker.