What is the UK government trying to do with COVID-19?

It would be a drastic understatement to say that people are confused by the official advice coming with respect to social-distancing measures to prevent the spread of SARS-CoV-2. Some are angry. Some are appalled. And that includes some very smart people who understand the relevant science better than I do, and probably at least as well as the experts who are advising the government. Why are they not closing schools and restaurants, or banning sporting events — until the Football Association decided to ban themselves — while at the same time signalling that they will be taking such measures in the future? I’m inclined to start from the presumption that there’s a coherent and sensible — though possibly ultimately misguided (or well guided but to-be-proved-retrospectively wrong) — strategy, and I find it hard to piece together what they’re talking about with “herd immunity” and “nudge theory”.

Why, in particular, are they talking about holding the extreme social-distancing measures in reserve until later? Intuitively you would think that slowing the progress of the epidemic can be done at any stage, and the sooner you start the more effective it will be.

Here’s my best guess about what’s behind it, which has the advantage of also providing an explanation why the government’s communication has been so ineffective: Unlike most other countries, which are taking the general approach that the goal is to slow the spread of the virus as much as possible (though they may disagree about what is possible), the UK government wants to slow the virus, but not too much.

The simplest model for the evolution of the number of infected individuals (x) is a differential equation

Here A is the fraction immune at which R0 (the number that each infected person infects) reaches 1, so growth enters a slower phase. The solution is

Basically, if you control the level of social interaction, you change k, slowing the growth of the cumulative rate parameter K(t). There’s a path that you can run through, at varying rates, until you reach the target level A. So, assuming the government can steer k as they like, they can stretch out the process as they like, but they can’t change the ultimate destination. The corresponding rate of new infections — the key thing that they need to hold down, to prevent collapse of the NHS — is kx(A–x). (It’s more complicated because of the time delay between infection, symptoms, and recovery, raising the question of whether such a strategy based on determining the timing of epidemic spread is feasible in practice. A more careful analysis would use the three-variable SIR model.)

Suppose now you think that you can reduce k by a certain amount for a certain amount of time. You want to concentrate your effort in the time period where x is around A/2. But you don’t want to push k too far down, because that slows the whole process down, and uses up the influence. The basic idea is, there’s nothing we can do to change the endpoint (x=A); all you can do is steer the rate so that

1. The maximum rate of new infections (or rather, of total cases in need of hospitalisation) is as low as possible;
2. The peak is not happening next winter, when the NHS is in its annual flu-season near-collapse;
3. The fraction A of the population that is ultimately infected — generally taken to be about 60% in most renditions — includes as few as possible of the most at-risk members of the public. That also requires that k not be too small, because keeping the old and the infirm segregated from the young and the healthy can only be done for a limited time. (This isn’t Florida!)

Hence the messaging problem: It’s hard to say, we want to reduce the rate of spread of the infection, but not too much, without it sounding like “We want some people to die.”

There’s no politic way to say, we’re intentionally letting some people get sick, because only their immunity will stop the infection. Imagine the strategy were: Rather than close the schools, we will send the children off to a fun camp where they will be encouraged to practice bad hygiene for a few weeks until they’ve all had CoViD-19. A crude version of school-based vaccination. If it were presented that way, parents would pull their children out in horror.

It’s hard enough getting across the message that people need to take efforts to remain healthy to protect others. You can appeal to their sense of solidarity. Telling people they need to get sick so that other people can remain healthy is another order of bewildering, and people are going to rebel against being instrumentalised.

Of course, if this virus doesn’t produce long-term immunity — and there’s no reason necessarily to expect that it will — then this strategy will fail. As will every other strategy.