Households represent a key unit of interest in infectious disease epidemiology, in both empirical studies and mathematical modelling. The within-household transmission potential of a disease is often summarised by a secondary attack ratio (SAR). Despite its widespread use, the SAR depends on the household size distribution (HHSD) seen during the study period, making it difficult to generalise to new contexts. Extending estimates of transmission potential to new populations instead requires estimates of person-to-person transmission rates which can be convoluted with data on population structure to parametrise mechanistic transmission models. In this study we present a new Bayesian inference method which uses an MCMC algorithm to infer the transmission intensity by imputing the unreported household structure underlying the epidemic. This method can be run on household epidemiological data reported at varying levels of resolution. For synthetic data from a realistic underlying HHSD, we were able to achieve over 95% coverage in our estimates of transmission rate consistently. We were also able to consistently achieve over 95% coverage for data generated with a pathological underlying HHSD, given strong information about the HHSD. Using an existing dataset which recorded micro-scale household epidemiological outcomes during the COVID-19 pandemic, we show that stratifying observed SARs by household size substantially reduces the uncertainty in estimates. Our findings suggest that researchers conducting household epidemiological studies can improve the utility of results for infectious disease modellers by reporting household-stratified estimates. These results aim to encourage the reporting of higher resolution outputs in epidemiological field work as, in the absence of strong priors, transmission parameters were not easily identifiable from low resolution datasets, which are often reported.

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