Thinking about exponential growth and “Freedom Day”

The UK government is holding fast to its plan to drop all pandemic restrictions as of 19 July, even in the face of rapidly increasing infection rates, hospitalisation rates, and Covid deaths — all up by 25-40% over last week. And numerous medical experts are being quoted in the press supporting the decision. What’s going on?

To begin with, Johnson has boxed himself in. He promised “Freedom Day” to coincide with the summer solstice, and then was forced to climb down, just as he was from his initial “infect everyone, God will recognise his own” plan last March, on realising that his policies would yield an unsustainable level of disruption. The prime minister has, by now, no reputation for consistency or decisiveness left to protect, but even so he probably feels at the very least that a further delay would undermine his self-image as the nation’s Fun Dad. At the same time, the the new opening has been hollowed out, transformed from the original “Go back to living your lives like in pre-pandemic days” message to “Resume taxable leisure activities, with the onus on individuals and private businesses to enforce infection-minimisation procedures.” Thus we have, just today, the Transport Secretary announcing that he expected rail and bus companies to insist on masking, even while the government was removing the legal requirement.

But what are they hoping to accomplish, other than a slight reduction in the budget deficit? The only formal justification offered is that of Health Secretary Sajid Javid, who said on Monday

that infection rates were likely to get worse before they got better – potentially hitting 100,000 a day – but said the vaccination programme had severely weakened the link between infections, hospitalisations and deaths.
Javid acknowledged the risks of reopening further, but said his message to those calling for delay was: “if not now, then when?”.

“Weakened the link” is an odd way of putting a situation where cases, hospitalisations, and Covid deaths are all growing exponentially at the same rate. What has changed is the gearing, the chain and all of its links is as strong as ever. In light of that exponential growth, what should we make Javid’s awkward channeling of Hillel the Elder?

I’ll talk about “masking” as synecdoche for all measures to reduce the likelihood of a person being infected or transmitting Covid. We need to consider separately the questions of when masking makes sense from an individual perspective, and from a public perspective. The individual perspective is straightforward:

On the societal level it’s more complicated, but I do find the argument of England’s Chief Medical Officer Chris Whitty… baffling:

“The slower we take it, the fewer people will have Covid, the smaller the peak will be, and the smaller the number of people who go into hospital and die,” he said.
By moving slowly, he said modelling suggested the pressure on the NHS would not be “unsustainable”.
Prof Whitty said there was less agreement on the “ideal date” to lift restrictions as there is “no such thing as an ideal date” .
However, he said a further delay would mean opening up when schools return in autumn, or in winter, when the virus has an advantage and hospitals are under more pressure.

We may argue about how much effect government regulations have on the rate of the virus spreading, but I have never before heard anyone argue that the rate of change of government regulation is relevant. Of course, too rapid gyrations in public policy may confuse or anger the public. But how the rapidity of changing the rules relates to the size of the peak seems exceptionally obscure. To the extent that you are able to have any effect with the regulations, that effect should be seen directly in R0, and so in the weekly growth or contraction of Covid cases. If masking can push down the growth rate its effect is essentially equivalent at any time in terms of the final infection rate, but masking early gives fewer total cases.

To see this, consider a very simple model: With masking cases grow 25%/week, without masking they shrink 20%/week. So if we have 1000 cases/day now, then after some weeks of masking and the same number of weeks without masking, we’ll be back to 1000 cases/day at the end. But the total number of cases will be very different. Suppose there are 10 weeks of each policy, and we have four possibilities: masking first, unmasking first, alternating (MUMU…), alternating (UMUM…). The total number of cases will be:

StrategyTotal cases
masking first57 000
unmasking first513 000
alternating (MU…)127 000
alternating (UM…)154 000

Of course, the growth rate will not remain constant. The longer we delay, the more people are immune. In the last week close to 2 million vaccine doses have been administered in the UK. That means that a 4-week delay means about 4 million extra people who are effectively immune. If we mask first, the higher growth rate will come later, thus the growth rate will be lower, and more of the cases will be mild.

The only thing I can suppose is that someone did an economic cost-benefit analysis, and decided that the value of increased economic activity was greater than the cost of lives lost and destroyed. Better to let the younger people — who have patiently waited their turn to be vaccinated — be infected, and obtain their immunity that way, than to incur the costs of another slight delay while waiting for them to have their shot at the vaccine.

The young were always at the lowest risk in this pandemic. They were asked to make a huge sacrifice to protect the elderly. Now that the older people have been protected, there is no willingness to sacrifice even another month to protect the lives and health of the young.

Auto-antonyms: Drawing a line

The BBC is being scourged for having been — 25 years ago — once insufficiently open and honest with its rightful liege lords (in this case, the then Princess of Wales). Apparently they tried to trick her into giving an interview, though the BBC says it has a handwritten note from her saying she didn’t mind, and she would have done the interview anyway.

In the course of discussing this Diana’s brother is quoted saying “he ‘draws a line’ between the [BBC] interview and her death two years later.” It’s an interesting phrase, of a rare sort that I call auto-antonyms. The same words might mean two diametrically opposite things. I think he means to say that the two events are causally connected — as though by a straight line. But he could also mean that even though one event followed the other, he considers them to be completely separate — to emphasise which he would draw a (dividing) line between them.

Oxbridge Delenda Est

I’ve been thinking for a long time that for all their merits as individual institutions, and all the advantages they offer to their faculty (like myself), students, and alumni (like myself), the hierarchical structure of tertiary education that defines their role, from which they benefit, and which they nurture, is fundamentally destructive.

I wrote an essay on this theme, and it has now appeared in the political magazine Current Affairs.

Can’t look away

Many years ago I read to my daughter a children’s book in which a little girl learning to ride a bicycle keeps running into objects like trees and lampposts. A bicycle instructor explains to her that when you become too fixated on an obstacle it exerts a strong psychological pull, so that the very exigency of evading it leads to a crash.

I used to wonder whether this was a real phenomenon. I don’t anymore…

Actually, I’ve long thought the second Iraq War was an example of the same phenomenon. There was no possibility that there wouldn’t be a war, because once they’d started to consider it Bush and Blair couldn’t bare not to see how it would turn out.

Adrift on the Covid Sea

Political leaders in many countries — but particularly in the US and UK — are in thrall above all to the myth of progress. Catastrophes may happen, but then they get better. And to superficial characters like Johnson and Trump, the improvements seem automatic. It’s like a law of nature.

So, we find ourselves having temporarily stemmed the flood of Covid infections, with governments laying out fantastic plans for “reopening”. Even though nothing significant has changed. The only thing that could make this work — absent a vaccine — would be an efficient contact tracing system or a highly effective treatment for the disease. None of which we have. But we still have a timeline for opening up pubs and cinemas (though less important facilities like schools are still closed, at least for many year groups).

It’s like we had been adrift for days in a lifeboat on the open ocean, carefully conserving our supplies. And there’s still no rescue in sight, but Captain Johnson announces that since we’re all hungry from limiting our food rations, and the situation has now stabilised, we will now be transitioning toward full rations.

COVID-19 and the Dunkirk spirit

My recent post suggesting that the government may have some reasonable thinking behind their go-slow-but-not-too-slow strategy had two underlying errors:

  1. I assumed they knew what the NHS capacity is, and were trying not to linger too long in the period where there is plenty of spare capacity. In fact, resources already appear to be overstretched, particularly protective equipment (PPE), even though the epidemic has barely started, and there are just a few thousand cases in total so far.
  2. I neglected to reckon with — what was otherwise obvious to me — Johnson’s Churchill complex. Johnson doesn’t have all that much in common with Churchill, but one thing the two do share is a mania for all manner of harebrained wheezes rather than careful dependable planning. Keynes famously said “Worldly wisdom teaches that it is better to fail with the crowd than to succeed unconventionally”. Johnson is one of those rare individuals who would rather fail unconventionally — or, at least, is willing to hazard a strong risk of failure for the compensation of being seen as brilliantly unconventional.

Now the government says they miscalculated, after a paper from Imperial College’s Covid-19 Response Team found that the previous strategy would exceed available ICU capacity by a factor of 8! Did they misplace a decimal point? So suddenly the schools, gyms, and everything else that was announcing plans to cope with staying open through the epidemic is shutting down.

I find it genuinely shocking that the UK does not have a strategic reserve of PPE and ventilators, particularly the latter, as the shortage of ventilators was widely discussed in the press in 2009, in the context of the H1N1 pandemic.

Anyway, I’m glad I’m not in the US…

Californians panic-buying guns

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(Ax). (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.

Who’s behind Brexit?

One of the newspaper covers promoting pro-Brexit celebration:

It’s hard to miss that the jubilant lady draped in the Union Jack has a US flag right behind her. A message to those who still suppose Brexit will bring “independence”. (In case they didn’t get the message when the Prime Minister stood up in parliament and pretended to take Jared Kushner’s Middle East “peace plan” seriously.)

And in case any Jews or Muslims might have thought they would be part of this “glorious new Britain”, they have a fucking CRUSADER in their masthead!