In this article, we estimate the parameters of a simple random network and a stochastic epidemic on that network using data consisting of recovery times of infected hosts. The SEIR epidemic model we fit has exponentially distributed transmission times with gamma distributed latent (exposed) and infective periods on a network where every tie exists with the same probability, independent of other ties. We employ a Bayesian framework and MCMC integration to make estimates of the joint posterior distribution of the model parameters. We discuss the accuracy of the estimates of different parameters under various prior assumptions and, in particular, show that it is possible in many scientifically interesting cases to accurately recover the network parameter p. We demonstrate some of the important aspects of our approach by studying a measles outbreak in Hagelloch, Germany in 1861 consisting of 188 affected individuals. We provide an R package to carry out these analyses, which is available publicly on the Comprehensive R Archive Network (CRAN).

Summary In this article, we demonstrate a statistical method for fitting the parameters of a sophisticated network and epidemic model to disease data. The pattern of contacts between hosts is described by a class of dyadic independence Exponential-family Random Graph Models (ERGMs) while the transmission process that runs over the network is modeled as a stochastic Susceptible-Exposed-Infectious-Removed (SEIR) epidemic. We fit these models to very detailed data from the 1861 measles outbreak in Hagelloch, Germany. The network models include parameters for all recorded host covariates including age, sex, household and classroom membership and household location while the SEIR epidemic model has exponentially distributed transmission times with gamma distributed latent and infective periods. This approach allows us to make meaningful statements about the structure of the population—separate from the transmission process—as well as to provide estimates of various biological quantities of interest, such as the effective reproductive number, R. Using reversible jump Markov chain Monte Carlo, we produce samples from the joint posterior distribution of all the parameters of this model—the network, transmission tree, network parameters and SEIR parameters—and perform Bayesian model selection to find the best-fitting network model. We compare our results with those of previous analyses and show that the ERGM network model better fits the data than a Bernoulli network model previously used. We also provide a software package, written in R, that performs this type of analysis.

Markov chain Monte Carlo (MCMC) algorithms offer a very general approach for sampling from arbitrary distributions. However, designing and tuning MCMC algorithms for each new distribution, can be challenging and time consuming. It is particularly difficult to create an efficient sampler when there is strong dependence among the variables in a multivariate distribution. We describe a two-pronged approach for constructing efficient, automated MCMC algorithms: (1) we propose the “factor slice sampler”, a generalization of the univariate slice sampler where we treat the selection of a coordinate basis (factors) as an additional tuning parameter, and (2) we develop an approach for automatically selecting tuning parameters in order to construct an efficient factor slice sampler. In addition to automating the factor slice sampler, our tuning approach also applies to the standard univariate slice samplers. We demonstrate the efficiency and general applicability of our automated MCMC algorithm with a number of illustrative examples.

We present the R package epinet, which provides tools for analyzing the spread of epidemics through populations. We assume that the relationships among individuals in a population are modeled by a contact network described by an exponential-family random graph model and that the disease being studied spreads across the edges of this network from infectious to susceptible individuals. We use a susceptible-exposed-infectiousremoved compartmental model to describe the progress of the disease within each host. We describe the functionality of the package, which consists of routines that perform simulation, plotting, and inference. The main inference routine utilizes a Bayesian approach and a Markov chain Monte Carlo algorithm. We demonstrate the use of the package through two examples, one involving simulated data and one using data from an actual measles outbreak.