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.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s