# Response to Oxford paper on covid-19

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.

## 3 thoughts on “Response to Oxford paper on covid-19”

1. Ishi Crew says:

This is a nice and fairly simple mathematical (though it would take me alot more time to actually go through it—i’d probably try to write it a la ‘systems dynamics’ –i.e. write down a picture or graph to accompany the equations. Some people tend to think in pcitures.

There appears to be a debate between people at Oxford U and Imperial College , along with your paper (and there are others) , about how bad this ‘pandemic’ is. (I’m hoping my fever will be gone in 2 weeks—they say that may be the time it takes, though some say 2 months).

I hear Iran is not quarantining (i guess people go to work and mosques) but some say they are not revealing information about the health situation there.

I’m mostly interested in the idea that the model is ‘under-constrained’. I sort of collect these kinds of ideas. (I’d even say a post from a few years back on this blog about 1+2+3+4+…. = -1/12 is an example of another unconstrained problem; the ‘wine/water paradox’ which goes back to Keynes and others i put in the same class–they have lots of solutions.)

The paper ‘Deglobalization in a hyper-connected world’ in Nature Palgrave Communications Feb 25 2020 while not on COVID has another case in which using very different parameters in a model can lead to similar conclusions. (My favorite example may be seeing some set of sequences that look like random walks or random cuin flips—-they could be created by a deterministic process with known parameters or rules).

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2. […] but has not been identified as such. This was the goal of the Lourenco et al. paper I wrote about previously that pulled me into this pit of despair. I argued that fitting to deaths alone, which is what they […]

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