Here is my response to the paper from Oxford (Lourenco et al.) arguing that novel coronavirus infection may already be widespread in the UK and Italy. The result is based on fitting a disease spreading model, called an SIR model, to the cumulative number of deaths. SIR models usually consist of ordinary differential equations (ODEs) for the fraction of people in a given population who are susceptible to the infectious agent (S), the number infected (I), and the number recovered (R). There is one other state in the model, which is the fraction who die from the disease (D). The SIR model considers transitions between these states. In the case of ODEs, the states are treated as continuous quantities, which is not a bad approximation for a large population, and each equation in the model describes the rate of change of a state (hence differential equation). There are parameters in the model governing the rate of different interactions in each equation, for example there is a parameter for the rate of increase in S whenever an S interacts with an I, and then there is a rate of loss of an I, which transitions into either R or D. The Oxford group model D somewhat differently. Instead of a transition from I into D they consider that a fraction of (1-S) will die with some delay between time of infection and death.
They estimate the model parameters by fitting the model to the cumulative number of deaths. They did this instead of fitting directly to I because that is unreliable as many people who have Covid-19 have not been tested. They also only fit to the first 15 days from the first recorded death since they want to model what happens before social distancing was implemented. They find that the model is consistent with a scenario where the probability that an infected person gets severe enough to be flagged is low and thus the disease is much more wide spread than expected. I redid the analysis without assuming that the parameters need to have particular values (called priors in Bayesian inference and machine learning) and showed that a wide range of parameters will fit the data. This is because the model is under-constrained by death data alone so even unrealistic parameters can work. To be fair, the authors only proposed that this is a possibility and thus the population should be tested for anti-bodies to the coronavirus (SARS-CoV-2) to see if indeed there may already be herd immunity in place. However, the press has run with the result and that is why I think it is important to examine the result more closely.