Some numbers which mean something

Yesterday I wrote that the important thing when looking at COVID-19 deaths is the death or infection rate: how many people are dying or have been infected per million, or equivalently what a given person's chance of having died or having become infected is. And I said that the right way to compare various approaches to dealing with the epidemic was to look at these rates – at the numbers normalised by population rather not the raw numbers of people dying or being infected.

This is, eventually, right, but it's not right now: I was wrong.

The reason it's not right is that, in the early stages of an epidemic there are, in any population, essentially an unlimited number of people to get infected. So each person who gets infected infects a certain number of other people (this is the famous R that people talk about) and they in turn infect the same number of other people, and so on.

The result of this is that the number of people who are infected is an exponential: it looks like n = n₀ e^{(t - t₀)/τ}, where

• n is the number of people infected, or who have died at some time t¹;
• n₀ is the number of people infected or dead at t = t₀;
• τ, tau, is a time-constant which depends on things like R, how long it takes people to become infectious and so on.

The total population doesn't appear in this expression, at all. So in any isolated population, during the exponential phase, the number of people infected or dead depends, other things being equal, only on the time after the first infection and this parameter τ.

Of course things are much more complicated in real life, since there are no isolated populations² and so on.

But in the early stages of an epidemic it is the case that the raw number of deaths tells you something useful. Only later, when either something has been done to suppress the epidemic or it is running out of people to infect, does the death rate per head start to matter. And I don't think we are at the point where it runs out of people yet: there were 216,526 confirmed cases in the UK on the 10th of May 2020, so even if that is undercounting by a factor of 10, there are only about 2 million cases out of a population of 66 million: there are still plenty of people to infect.

So, again: I was wrong.