# Thoughts on the Oxford COVID-19 Model

William Waites <wwaites @ inf.ed.ac.uk>
Laboratory for Foundations of Computer Science
School of Informatics, University of Edinburgh

Lourenço et al have published (2020/03/24) a draft paper, Fundamental principles of epidemic spread highlight the immediate need for large-scale serological surveys to assess the stage of the SARS-CoV-2 epidemic, in which they fit a straightforward SIRf model to COVID-19 mortality data in the UK and Italy. They find a range of parameters, in particular, $$\rho$$, the fraction of the population at risk of severe disease, that fit the empirical data well.

The correct value of $$\rho$$ has important implications. If it is very small, about 0.1%, their model implies that a large fraction of the population has already had the disease, probably with mild or no symptoms, and are now immune. Soon the epidemic will be over and we will be able to relax. We would like this to be the case. If $$\rho$$ is much larger, 10%, then a much smaller fraction of the population has already been infected, suggesting that a prolonged period of stringent control measures and likely fatalities.

In order to ascertain the value of $$\rho$$ from the available data, the authors infer the date at which the first case was reported. For small values, this date will be relatively earlier, and for larger values, relatively later. It turns out that the smaller value most closely matches the first reported case in the UK. If this is true, it is very good news.

However. This procedure for estimating $$\rho$$ assumes that the first reported case was the progenitor of all (or most) subsequent cases. It is quite possible that the first case, and any localised outbreak surrounding it, did not take hold. This is a continuous model expressed as a system of Ordinary Differential Equations (ODEs). The model itself is not published, but previous work cited in respect to this used ODEs and then a Markov Chain Monte Carlo (MCMC) method to fit the model parameters. ODEs, however, have the limitation that they only work for large populations. A single case is not a large population. Stochastic simulation methods should properly be used in such a situation to account for local effects that happen at low copy numbers (such as an infection lineage dying out). This point is related to Shen et al.'s criticism of Ferguson et al.'s model of non-pharmaceutical interventions. Even then, the result of many stochastic simulations will simply be a distribution of trajectories of the system (population) that cannot tell us with any certainty what happened in this particular real-world instance.

The authors stress that the real solution to this is empirical data. They are absolutely correct that we need serological tests to determine the true fraction of the population that has been infected with the disease and has since recovered. It is natural to want to be optimistic, we can hope that the true value of $$\rho$$ is indeed very small. But we need measurement and testing to find this out; we cannot conclude this from the available data and this model.

2020/03/24