After many thousands of deaths and tens of thousands of cases it has been a relief to see daily new cases of COVID and weekly death counts drop to double or even single figures.
But as we try and find a balance between lockdown and our normal lives, we are faced with a new problem: if you attempt to perform analysis on counts of less than 10 then you are likely committing a statistical sin. That’s not to say small numbers tell you nothing, but that we must accept that the uncertainty associated with a count of 10 will be much greater than for a count of 1000. This has significant implications for government decision making and for public perception.
But there’s a more subtle problem that creeps in when you’re dealing with small numbers: viral infection is a random process. Before exploring that, it’s worth understanding why our recent experience of the pandemic might cause us to overlook its random nature because the growing numbers of infections in the earlier stages exhibited predictable, exponential growth.
The process of viral infection is characterised by what is called the R number, which tells you how many people on average an infected person will go on to infect. With no immunity or measures to prevent infection in place, it seems COVID-19 has an R of about 3.
Now, if you infect 3 other people and they go on to infect another 3 each then that results in 9 infections. If you continue this you see numbers increase exponentially, literally, as powers of 3: 1, 3, 9, 27, 81 and so on. An unchecked viral outbreak with an R of 3 in which a person is infectious for a week can go from just one infected person to exceed a million in three months (3 raised to the power of 13 is 1.6 million).
No R in decision-making
Lockdown measures have brought R under 1 and this means we have seen a decline over the summer. Now, by relaxing lockdown, we reclaim some freedoms while hoping we can keep R below 1 (not always possible perhaps?). The approach of governments around the world is to relax the lockdown in stages while being vigilant for any sign for an increase in cases. Trying to estimate R itself is very difficult and so it should be regarded as a number useful for understanding the mechanics of viral outbreaks but not one that offers a useful metric for decision making.
So, is it reasonable to be seeking to drive infection rates down to a very low level and then maximise people’s freedom so that R stays around 1 so infection levels neither increase nor decrease? A straightforward statistical analysis shows that on average, yes, this seems plausible. You don’t even need a fancy computer model to confirm it; you can prove it with school level mathematics. In fact, you can probably see it intuitively based on the fact that an R of 1 simply means that one person infects just one other, on average. A little maths illustrates the point: 3 to the power 13 is 1.6 million whereas 1 to the power 13 is still 1!
Now for a bit of role-playing. Imagine you are a government minister and you know lockdown had brought R under 1 but now you fret that easing the lockdown might have caused R to be nudged over 1. You are presented with this graph of infection levels from five towns which all started with an infection level of 10. What is the R number for each one and which should be considered for a local lockdown?
And here’s the surprise. All five have an R of 1 and have been generated from the same model (specifically, what is known in epidemiology as a stochastic SIS model). That is, the differences between the five outcomes are solely due to chance. When you run simple computer models like this one you will rarely see the number of infections stay at a constant level when R is 1.
So, we appear to have a paradox. Surely either such a simple model is wrong or else the mathematical proof and our intuition are both wrong?
The answer is that both are correct. The words that confound our intuition above are “on average”. Although the average does indeed remain constant for an R of 1, the spread around that average does not. In fact, it can be proved mathematically that it grows with time.
A random walk or following the science?
But that’s not all. The proof sheds light on why we are forced to confront the randomness at smaller numbers. The spread as a fraction of the number of infections is larger for a small number of infections. For example, if you expect an average of 10 infections then you can expect a spread of 3 either way, that is 30%; at 10,000 the spread would be 100 either way, which is 1%. This also explains why the random effects aren’t apparent with exponential increase because it quickly produces large numbers and so the spread about the average will be relatively small.
If this still seems surprising to you, it may help to realise that this is an example of what is often called a random walk process. Imagine you are at the north pole in a snowstorm and have no way of navigating and you stumble around at random. Since you are as likely to stumble in one direction as any other, your “average” location will be at the north pole where you started. In reality, you will likely never set foot near it again and each footstep increases the maximum distance you might have moved from the pole.
The important implication of this is that looking at counts of numbers alone when infections are at low levels can easily mislead us. Any pattern can be extracted from random noise and not unnaturally our fears may pick out certain patterns. To distinguish such statistical ghosts from the start of a real resurgence of the virus requires knowing when cases are connected and this is the fundamental reason why confidence in a system for testing, tracing and isolating is vital in reopening society.
This is only one of many uncertainties that must be understood in deciding how we balance restoring our freedoms against accepting risks as we emerge from lockdown. It is not something that mathematics nor statistics nor science can decide for us. Government ministers who say they are just following the science are deceiving us and possibly themselves. It is their unenviable responsibility to weigh up advice, acknowledge the uncertainty and decide what we will do. If that turns out badly because of the uncertainty when their decision was made we should be understanding, but if it was because they discounted advice for political reasons it is our job as the electorate to hold them to account.
Scottish numbers August 13