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.
The functional form of R is a tanh function. There are 4 parameters:
- The initial value.
- The final value.
- The timing of the transition, i.e. when did lock down begin.
- 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.