Understanding R: Why we need more intelligent leaders

This week I have heard people claim that locking down earlier would not have made much difference. This is wrong. Even locking down a few days earlier would have made an enormous difference, and I can prove it!

In early March, as I wrote on March 12th, we knew that cases were growing at over 30% per day. This was widely reported at the time, most notably by the Financial Times on March 10th. And we knew that the only way to stop this was to stop people interacting, i.e. lock them down, as they had done in China and elsewhere. In mathematical terms, reducing human interaction has the effect of reducing the so called R number – the reproductive factor of the virus.

As I tried to make clear back then, understand R is bit like understanding the interest on a bank account. Except that, whereas for money we would like a positive interest rate, for a virus we want it to be negative. In early March the rate of increase in Covid-19 cases was around 30 – 33% per day. The maths works exactly like compound interest on a bank account. Say the daily increase is 31%. If you have 1 case on day 1, then on day 2 you have 1.31, on day 3 you have 1 x 1.31 x 1.31, and so on.

If the R number is 3.8, then each case will create another 3.8 cases, on average. To relate this to a daily increase in cases we need to know how long it takes for each case to make the next person infectious. This time is known as the serial interval, and for Covid it is about 5 days. To relate the R value to the rate of increase per day, the 31%, we do the compound interest calculation 5 times, i.e.

R = 1.31 x 1.31 x 1.31 x 1.31 x 1.31 = 3.8 .

Or if we know the R number, then to work out the number of cases tomorrow, we simply multiple the number of cases today by the 5th root of the R number. In the Wikipedia article this is called the Simple Model! For anyone that has ever written a computer code or done a compound interest calculation, it is very simple!

Now if you have watched the daily briefings then you will have seen some charts (I call them graphs!) of cases and deaths over time. There are many sites keeping track of this data such as Our World in Data. Below is a graph of deaths per day for the UK. The actual data is shown as red dots. We use a 7 days moving average to smooth out the weekend effect. How well does our Simple Model do in explaining the actual data? The answer is that the Simple model is perfect, if we allow the R number to change over time. Using the R values, shown in the top graph, and then doing our simple ‘compound interest’ calculation exactly reproduces the data as shown by the blue line in the main graph. The agreement is impressive because we have made it so. But this does show that our Simple Model is sufficient to explain the observational data.

Figure caption: The red dots show UK deaths from Covid-19 using a 7-day moving average. Data from ECDC. The top graph shows the R value over time illustrating the effect of lock down. The green lines show the effect of implementing a lock down 5 days earlier.

The interesting conclusion is that the way the R number behaved over time is relatively simple, and does exactly what we might have expected. In particular, before lock down the R value was high. Initially the R number was about 3.8, corresponding to the 31 % increase per as reported at the time by the Financial Times and others. After lock down (indicated by the vertical dashed line). The R value fell gradually and eventually stabilised at a lower level. The observant will notice that in my model the R number was beginning to fall even before the government lock down was implemented. If you remember, there was some fall off of activity before the official lock down – football games were cancelled, many workplaces had already implemented home working, etc. There are also questions about the speed of transition from high to low R. This is quite complex – factors such as distribution and usage of PPE, screens in supermarkets, the separation of Covid and non-Covid patients in hospitals and care homes, were all crucial. It would have been possible to achieve a faster transition if lock down policies had been more effective. Eventually the R number appears to have stabilised at about 0.85. An R below 1 – the situation we want, indicated by the horizontal dashed line in the graph – is like a negative interest rate. The number of cases falls rather than rises. If R increases above 1 then the cases will start to rise again. For those interested, I give a bit more detail on the functional form of R in the Notes below.

The useful aspect of this model is that we can now use it to say what would have happened if lock down had been enacted says 5 days earlier. This is shown by the green curves. The effect is dramatic!

Back on March 12th I said that Covid 19 was a test of leadership. Leaders needed to act fast and decisively. The UK leadership did not act fast enough. The consequence was that the impact of the virus was far higher than it needed to be, and that is tragic.

Notes:

The functional form of R is a tanh function. There are 4 parameters:

  1. The initial value.
  2. The final value.
  3. The timing of the transition, i.e. when did lock down begin.
  4. The speed or effectiveness of the lock down.

The tanh function was chosen for simplicity. We could also use an exponential decay from high to low which gives similar results.

Comments

  1. Andrew -

    That is remarkably simple and clear, and a damning indictment of the government’s slow response when it was plain as a pikestaff that immediate action was required. The thing that got it home to me was the 10 March “act now” Medium article by Tomas Puyeo, which I think I first read around the time of the Budget on the 12th.

    What would the number of deaths have been under your green model?

    1. Charles Adams -

      Thanks. According to this model, locking down 5 days earlier would have reduced the total deaths by a factor of 4.5.

      I agree that the Tomas Puyeo article was excellent. The first week of March, I was thinking we need to act now. The second week, like many others, I was starting to despair and by the third week, I knew it was too late. The die was cast.

    2. Andrew -

      The Budget was 11th March of course! Seems so long ago now.

      (On reflection, I think I did first read the Puyeo piece on the 12th though – perhaps from the link in the comments on your post that day.)

  2. Bill Hughes -

    I think Sage, ex Sage and the alternative Sage scientists are pretty well agreed that tens of thousands of lives could have been saved if the official lockdown was a week earlier ie on March 16 not 23.

  3. Tony_B -

    It is claimed Cummings is brilliant, if so, I wonder what his thoughts were on how long it would take to reach herd immunity? Three further weeks, perhaps? Or is he functionally innumerate; perhaps not as he would be aware of Malthus’s discussion of unrestrained population growth.

    It’s clear he, his team and Johnson never thought through the social and political consequences, or they are psychopaths to consider the virus freely sweeping through the population as a plan. We will never hear their true thoughts on these matters, either way they are unfit for office.

  4. Graham -

    Excellent analysis. The Puyeo article should have alerted the scientists if not the donkeys. I wonder what might have happened if we had shut the borders and imposed quarantining early March?

  5. Michael G -

    An apparently completely irrelevant article, which might possibly be relevant to R. As we get older, our chance of dying in the next 12 months increases each year. But in extreme old age, it stops increasing. If you survive to 100, your chance of dying next year is the same as last year. But statisticians pointed out that you would get the same flattening of the curve if a small fraction of centenarians misremembered their age. Almost certainly this is not the explanation for the age plateau, but is it relevant to R?
    R is an aggregate of a large number of individual R numbers.
    Surely the tail of the COVID curve is determined by the distribution of these R numbers? Charles would be able to answer or model this, but my suspicion is that the tail of the COVID curve will be higher and longer than a simple model would predict, because of the distribution curve of R numbers. (Just as more centenarians live to be 101 than you would expect).
    https://www.the-scientist.com/news-opinion/new-study-questions-whether-death-rate-levels-off-in-old-age-65264?utm_campaign=TS_DAILY%20NEWSLETTER_2018&utm_source=hs_email&utm_medium=email&utm_content=68563686&_hsenc=p2ANqtz-_US8AY4AMQVUh4VSNqtzt6YYtWkNEPP2i7awNOGFbzSjhXkMDxOuULIfBKgGtx58M-H8NUlPZSFqXpEX1-h0bDRZ4d72XjMLoKxXXicAOzqYSdsw8&_hsmi=68563686

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