© 2005 Royal Statistical Society 1369–7412/05/67731 J. R. Statist. Soc. B (2005) 67, Part 5, pp. 731–745 Bayesian inference for stochastic multitype epidemics in structured populations via random graphs Nikolaos Demiris Medical Research Council Biostatistics Unit, Cambridge, UK and Philip D. O’Neill University of Nottingham, UK [Received January 2005. Revised June 2005] Summary. The paper is concerned with new methodology for statistical inference for final out- come infectious disease data using certain structured population stochastic epidemic models. A major obstacle to inference for such models is that the likelihood is both analytically and numer- ically intractable. The approach that is taken here is to impute missing information in the form of a random graph that describes the potential infectious contacts between individuals. This level of imputation overcomes various constraints of existing methodologies and yields more detailed information about the spread of disease. The methods are illustrated with both real and test data. Keywords: Bayesian inference; Epidemics; Markov chain Monte Carlo methods; Metropolis– Hastings algorithm; Random graphs; Stochastic epidemic models 1. Introduction This paper is concerned with the problem of inferring information about the spread of disease given data on the final outcome of an epidemic in a structured population. Before outlining our approach, we begin by briefly recalling relevant background material. Stochastic epidemic models that incorporate structured populations have become a subject of considerable research activity in recent years. Examples include independent household mod- els (e.g. Longini and Koopman (1982), Becker and Dietz (1995) and Becker and Hall (1996)), models with two levels of mixing (e.g. Ball et al. (1997), Ball and Lyne (2001) and Demiris and O’Neill (2005)), random-network models (e.g. Andersson (1999) and Britton and O’Neill (2002)) and social cluster models (e.g. Schinazi (2002)). The basic motivation for such work is that, in contrast with epidemic models that assume a homogeneously mixing population of individuals, most human populations contain inherent structure because individuals usually spend their time in various groups such as dwelling places, workplaces and child care facilities. Our focus here is on so-called two-level mixing models, which are defined formally below. These models, which were introduced in Ball et al. (1997), describe a population that is par- titioned into groups in which infectious contacts can occur both locally within a group and globally between groups. The basic inference problem is then to estimate the local and global Address for correspondence: Philip D. O’Neill, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. E-mail: pdo@maths.nott.ac.uk