Modelling COVID

The most common method of modelling the spread of COVID is to consider estimates for how rapidly it is spreading (R0) and how deadly it is (mortality rate).

R0 is the number of new people infected by a given individual.  If R0 is 1 then the number of cases and deaths per day is stable, i.e. total deaths continue to rise but at a constant rate.  An R0 above 1 and the disease is accelerating.  I have found it hard to find a transparent source for this and have therefore built my own simple version which I found to work extremely well and be very instructive.

Parameters 

R0 I estimated R0 initially at 2.7 considering the UK case.
This was observed during the period before we applied measures to change the population’s behaviour.  The speed of the spread changes as we become more aware and change our behaviour.  We would expect the speed of spread of an airborne disease to reduce during various forms of lockdown.

For the pedants amongst you, I am aware that in the scientific literature they tend to say Re (effective) has fallen but this is just semantics and I will just say that R0 changes.

If you recall we went through a stage of around 2 weeks of hand-washing and cancelling football matches and some form of social distancing.   I estimate an R0 of 2.0 from the 3rd of March

On March 23rd we started official lockdown.  Here I use an R0 of 0.8, a big reduction resulting from a dramatic change to economic and social behaviours.

Mortality rate I used 1%

I assume that people are infectious for 5 days and death occurs 18 days after infection.  This produces the following results for what we have seen and gives a projection for the next few weeks.

Cumulative and daily deaths in UK – actual v modelcovid1covid2

We can also consider countries which are further advanced to see if the model works going forward.  Here we see Spain and Italy fit the daily deaths model very well, including the recent period when death rates are declining.

Daily deaths in UK, Spain, Italy (adjusted start date)- actual v model
covid3

I then applied the same model to look at the US.  The state level data is messy and dominated by NY and NJ.  To see what is happening outside of those, I totalled all the states excluding New York and New Jersey and compared to the same model as previous.

Daily deaths in USA (excluding New York/New Jersey) – actual v model
covid4
Until recently, the model works extremely well but worryingly the actual US daily death rate has continued to rise rather than fall.

 

How sensitive is the model to the assumptions?

R0

The model is very sensitive to changes in R0.
For example, if R0 actually remained at 2.7 in the 2 weeks pre-lockdown rather than 2.0, then the peak death rate would be dramatically higher.

covid5

This helps appreciate how hard it is to give accurate forecasts as minor changes in assumptions can give hugely different results.  It further shows how scary this virus it as it could easily be far worse than we are currently experiencing. Given the problems in estimation, it helps explain why the NHS were so concerned that we would exceed capacity and how important it was that the measures we took at least slowed the acceleration of the spread.  The sensitivity of the model to R0 makes forecasting very difficult but conversely makes our confidence much higher on what it has been historically.  We know that R0 cannot have been much higher or lower than the model as otherwise the outcomes would be wildly different.

Mortality rate

The other key assumption is of course mortality rate. I was encouraged that the model is robust to changes in mortality rate, given that this parameter is most controversial.  For lower mortality rates, it would require a higher R0 for the initial period  to match the actual data we are seeing i.e. we have more cases but a lower mortality rate to reach the same number of deaths.   This means we can look separately at the evidence on mortality rate, but it does not materially change the projections for the next few months.covid6

What does this model suggest about the virus and our policy response?

The model is robust across the US, UK, Italy and Spain over different time periods without having to change parameters, apart from considering different start dates, which makes me quite confident in the approach.

Before we had any response to COVID, the R0 was around 2.7 which means that cases and deaths roughly double every 3 days.  The initial period of attempting to slow the spread by washing our hands, stop touching our faces, social distancing etc only seems to have reduced R0 to 2.0 which is still a rapid spread.  But this made enough difference that we did not run out of hospital beds, even if we did run out of PPE.

The number that I think is most important is the R0 ,post lockdown.  This can be seen by how quickly the death rates decline after the peak.  It is clear from the UK, Italy and Spain that the R0 is below 1 as the death rates are declining, and the peak deaths were around 3 weeks after the lockdown.  But I am very concerned that the death rates are coming down so slowly which suggests that the R0 may be 0.8 or perhaps even higher.  The US data is even worse as it implies that the average R0 in 49 of the states may still be over 1.

The reason this is so important is that this is the number that tells us how soon we can end lockdown and how far we can relax.  I will expand on this in my next post.

 

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