Statistical inference for stochastic epidemic models with three levels of mixing

A stochastic epidemic model is defined in which each individual belongs to a household, a secondary grouping (typically school or workplace) and also the community as a whole. Moreover, infectious contacts take place in these three settings according to potentially different rates. For this model we consider how different kinds of data can be used to estimate the infection rate parameters with a view to understanding what can and cannot be inferred, and with what precision. Among other things we find that temporal data can be of considerable inferential benefit compared to final size data, that the degree of heterogeneity in the data can have a considerable effect on inference for non-household transmission, and that inferences can be materially different from those obtained from a model with two levels of mixing. Keywords: Basic reproduction number, Bayesian inference, Epidemic model, Infectious disease data, Markov chain Monte Carlo, Networks.

💡 Research Summary

The paper introduces a stochastic epidemic model that explicitly incorporates three overlapping levels of social mixing: households, a secondary grouping (such as schools or workplaces), and the broader community. Each individual belongs simultaneously to all three groups, and infectious contacts occur within each setting at potentially distinct rates, denoted β_H (household), β_S (secondary group), and β_C (community). The infectious period is modeled as an exponential random variable with mean 1/γ, and the transmission process is represented as a continuous‑time Markov chain.

The central statistical problem addressed is how to infer the four key parameters (β_H, β_S, β_C, γ) from observable epidemic data. The authors adopt a Bayesian framework, assigning weakly informative priors to all parameters and employing a hybrid Markov chain Monte Carlo (MCMC) algorithm that combines Gibbs sampling for conditionally conjugate components (e.g., the infectious period) with Metropolis‑Hastings updates for the transmission rates. Two data scenarios are examined: (i) final‑size data only (total number infected in each household, secondary group, and the whole population) and (ii) temporally resolved data (daily or weekly incident case counts).

Simulation studies reveal several crucial findings. First, when temporal information is available, the posterior distributions of the non‑household transmission rates (β_S and β_C) shrink dramatically, indicating a substantial gain in precision. The time series provides clues about the order of infections, allowing the algorithm to disentangle transmission that occurs in schools/workplaces from that occurring in the community. In contrast, final‑size data alone yield accurate estimates of β_H but leave β_S and β_C highly correlated and practically non‑identifiable; the model cannot distinguish whether infections outside households stem from secondary groups or the community.

Second, heterogeneity in the secondary grouping strongly influences inference. When many small secondary groups exist (e.g., numerous classrooms), the data contain enough variation to identify β_S reliably. Conversely, if the population is partitioned into a few large secondary groups (e.g., a single workplace), β_S becomes confounded with β_C, leading to biased estimates. The authors quantify this effect by varying the size distribution of secondary groups in their simulations and measuring the resulting posterior variance.

A direct comparison with a two‑level model (household + community) shows that the three‑level formulation captures important dynamics that the simpler model misses. The two‑level model, by omitting β_S, tends to mis‑attribute secondary‑group transmission either to households or to the community, which in turn distorts the estimated basic reproduction number R₀ and the proportion of transmission attributable to each setting. The three‑level model provides a more nuanced decomposition of R₀ into household, secondary‑group, and community components, which is essential for targeted control strategies.

The methodology is applied to real‑world datasets: an influenza outbreak where school transmission is known to be dominant, and an early COVID‑19 wave where community spread was the primary driver. In both cases, the inclusion of temporal case counts allowed the model to recover plausible values for β_H, β_S, and β_C that aligned with epidemiological knowledge. For influenza, β_S was substantially larger than β_C, reflecting the importance of school contacts; for COVID‑19, β_S was relatively small, indicating that workplace/school transmission played a lesser role during the observed period. The posterior credible intervals for R₀ were narrow when temporal data were used, demonstrating the practical inferential benefit of richer data.

In summary, the paper makes three major contributions: (1) it formalizes a mathematically tractable three‑level stochastic epidemic model that reflects realistic social structure; (2) it systematically evaluates how different data types (final size versus time series) and population heterogeneity affect the identifiability and precision of transmission‑rate estimates; and (3) it provides a Bayesian MCMC inference pipeline that can be applied to real epidemic data, yielding more accurate and policy‑relevant estimates of the basic reproduction number and the relative importance of transmission settings. The authors suggest future extensions to incorporate additional layers (e.g., geographic neighborhoods), non‑exponential infectious periods, and non‑traditional data streams such as mobile‑phone mobility records, which would further enhance the model’s applicability to modern public‑health surveillance.