# 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.

# The relevant Covid-19 fatality rate

Much has been written in the past few days about whether the case fatality rate (CFR) for Covid-19 is actually much lower than the original estimate of about 3 to 4%. Globally, the CFR is highly variable ranging  from half a  percent in Germany to nearly 10% in Italy. The difference could be due to underlying differences in the populations or to the extent of testing. South Korea, which has done very wide scale testing, has a CFR of around 1.5%. However, whether the CFR is high or low is not the important parameter.  The number we must determine is the population fatality rate because even if most of the people who become infected with SARS-CoV-2 have mild or even no symptoms so the CFR is low, if most people are susceptible and the entire world population gets the virus then even a tenth of a percent of 7 billion is still a very large number.

What we don’t know yet is how much of the population is susceptible. Data from the cruise ship Diamond Princess showed that about 20% of the passengers and crew became infected but there were still some social distancing measures in place after the first case was detected so this does not necessarily imply that 80% of the world population is innately immune. A recent paper from Oxford argues that about half of the UK population may already have been infected and is no longer susceptible. However, I redid their analysis and find that widespread infection although possible is not very likely (details to follow — famous last words) but this can and should be verified by testing for anti-bodies in the population. The bottom line is that we need to test, test and test both for the virus and for anti-bodies before we will know how bad this will be.

# How many Covid-19 cases are too many ?

The US death rate is approximately 900 per 100,000 people. Thus, for a medium sized city of a million there are on average 25 deaths per day. Not all of these deaths will be  preceded by hospital care of course but that gives an idea for the scale of the case load of the health care system. The doubling time for the number of cases of Covid-19 is about 5 days. At this moment, the US has over 25 thousand cases with 193 cases in Maryland, where I live, and over 11 thousand in New York. If the growth rate is unabated then in 5 days there will be almost 400 cases in Maryland and over 50 thousand in the US. The case-fatality rate for Covid-19 is still not fully known but let’s suppose it is 1% and let’s say 5% of those infected need hospital care. This means that 5 days from now there will be an extra 20 patients in Maryland and 2500 patients in the US. New York will be have an extra thousand patients. Many of these patients will need ventilators and most hospitals only have a few. It is easy to see that it will not take too long until every ventilator in the state and US will be in use. Also, with the shortage of protective gear,  some of the hospital staff will contract the virus and add to the problem. As conditions in hospitals deteriorate, the virus will spread to non-covid-19 patients. This is where northern Italy is now and the US is about 10 days behind them. This is the scenario that has been presented to the policy makers and they have chosen to take what may seem like extreme social distancing measures now. We may not be able to stop this virus but if we can slow the doubling time, which is related to how many people are infected by a person with the virus, then we can give the health care system a chance to catch up.