Modelling and Bayesian analysis of the Abakaliki smallpox data

Abstract

The celebrated Abakaliki smallpox data have appeared numerous times in the epidemic modelling literature, but in almost all cases only a specific subset of the data is considered. The only previous analysis of the full data set relied on approximation methods to derive a likelihood and did not assess model adequacy. The data themselves continue to be of interest due to concerns about the possible re-emergence of smallpox as a bioterrorism weapon. We present the first full Bayesian statistical analysis using data-augmentation Markov chain Monte Carlo methods which avoid the need for likelihood approximations and which yield a wider range of results than previous analyses. We also carry out model assessment using simulation-based methods. Our findings suggest that the outbreak was largely driven by the interaction structure of the population, and that the introduction of control measures was not the sole reason for the end of the epidemic. We also obtain quantitative estimates of key quantities including reproduction numbers.

Importance sampling for partially observed temporal epidemic models

“…When all event times are known, the likelihood is trivial to write down and conditional distributions for the parameters can often be derived, allowing for efficient Gibbs sampling. The greatest strength of the data-augmented approach is its flexibility; non-Markov models are handled as easily as Markov ones along with potentially large amounts of heterogeneity in the population and spreading process (Jewell et al 2009;Lau et al 2015;Stockdale et al 2017). The downside is that there is strong dependence between the missing data and the parameters that means mixing can become very slow and convergence can become an issue as the amount of missing data increases (McKinley et al 2014;Pooley et al 2015;Walker et al 2017).…”

Bayesian Nonparametrics for Stochastic Epidemic Models

“…These data have been considered by numerous authors, almost always to illustrate new statistical methodology, and are usually taken to consist of the symptom-appearance times of 30 individuals among a homogeneously-mixing susceptible population of 120 individuals. More extensive analyses of the full data set, which includes information on population structure, vaccination and other aspects, can be found in Eichner and Dietz (2003) and Stockdale et al (2017). The results are shown in Figure 3, the key finding of which is that there is no evidence to suggest any material variation in the infection rate during the outbreak.…”

Latent likelihood ratio tests for assessing spatial kernels in epidemic models

“…Such data which can be modelled by an SEIR model are encountered in many real-world situations, for example, concerning diseases with long latent periods or because of delay in reporting of cases, such as Foot-and-Mouth Disease (Keeling 2001;Chis Ster et al 2009;Ferguson 2001;Ferguson et al 2001;Morris et al 2001;Ster and Ferguson 2007;Streftaris and Gibson 2004a;Tildesley et al 2008) and Citrus Canker (Neri et al 2014;Gottwald et al 2002a, b). In many such datasets, the situation does arise where the times of infectiousness and removal are observed, but not the time of exposure, for example, in diseases where infectivity occurs only after the onset of symptoms, for example smallpox and avian influenza (Rorres et al 2011;Stockdale et al 2017;Boys and Giles 2007), or in insect or plant infestations (Brown et al 2013;Lau et al 2014), where the invading species has to reach a certain phase of its life cycle before producing eggs/seeds/spores. We therefore specify z to incorporate the times and location of the unobserved transitions from S to E (termed exposure events) and use MCMC to sample from π(θ, z|y).…”