Michael Whitehouse, Lorenzo Rimella, and Nick Whiteley’s contribution to the Discussion of ‘Some statistical aspects of the Covid-19 response’ by Wood et al.

We commend the authors of [10] for their thorough contribution to the retrospective analysis of the statistical response to the Covid-19 crisis.We focus on the issues raised in section 4 around the use of models with increasing extents of heterogeneity.The authors correctly recognise that omitting such considerations can lead to incorrect inference, e.g.poor estimation of final size.It is important, however, to acknowledge the challenges associated with learning model parameters from available data.As opposed to aggregated incidence/prevalence counts in classical epidemic models and metapopulation models [5,2], individual-based models (IBMs), such as the influential CovidSim model [3], explicitly consider individuals' disease states.This provides a refined representation of the population and in principle allows practitioners to work with covariates defined at an individual level (e.g.health records, geographical locations, contact networks).A simple example is a regression modelling of the rates at which the k-th individual infects the n-th: logwhere c n , c k are observed covariates associated with the n, k-th individuals and , b S , b I are model parameters respectively controlling background infection and how covariates contribute to attracting (susceptibility) and transmitting (infectivity) infection.This sliding scale of heterogeneity can be linked to the population distribution of the individual specific susceptibility parameter of the model defined in subsection 4.1 of [10].Since transmission processes are typically only partially observed, exact statistical inference, e.g.aimed at learning regression parameters , b S , b I or informing properties of , requires marginalisation over the latent disease state of the entire population.This incurs a prohibitively expensive cost which is exacerbated by increasing individual heterogeneity, motivating the development of approximate inference methods [4,9,1,7,6,8].Furthermore, the identifiability of models incorporating individual heterogeneity, in general, depends on the observation model, for instance whether individual test results or aggregated incidence/ prevalence counts are reported.In particular, [9,7] show that, in the context of both metapopulation models and IBMs, parameters defining processes with coinciding large population behaviour cannot be distinguished.We are curious about the balance between realism and feasibility and how the research community will address this in the future: given the prohibitive cost of fitting increasingly heterogeneous models, how far should modellers go?In the presence of identifiability issues, how should modellers acknowledge 1