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.

Update on Delta in Germany

The Robert Koch Institute produces estimates of variants of concern on Wednesdays. My projection from two weeks ago turns out to have been somewhat too optimistic. At that point I remarked that there seemed to be about 120 delta cases per day, and that that number hadn’t been increasing: The dominance of delta was coming from the reduction of other cases.

This no longer seems to be true. According to the latest RKI report, the past week has seen only a slight reduction in total cases, compared to two weeks ago, to about 604/day. And the proportion of delta continues to double weekly, so that we’re now at 59%, meaning almost 360 Delta cases/day. The number of Delta cases has thus tripled in two weeks, while the number of other cases has shrunk by a similar factor. The result is a current estimated R0 close to 1, but a very worrying prognosis. We can expect that in another two weeks, if nothing changes, we’ll have 90% Delta, around 1100 cases in total, and R0 around 1.6.

Of course, vaccination is already changing the situation. How much? By the same crude estimate I used last time — counting single vaccination as 50% immune, and looking back 3 weeks (to account for the 4 days back to the middle of the reporting period, 10 days from vaccination to immunity, and another 7 days average for infections to turn into cases), the above numbers apply to a 40% immune population. Based on vaccinations to date the population in 2 weeks will be 46% immune, reducing the R0 for Delta to around 1.5. In order to push it below 1.0 we would need to immunise 1/3 of the remaining population, so we need at least 64% fully immunised. At the current (slowed) rate of vaccination, if it doesn’t decelerate further, that will take until around the middle of September, by which point we’ll be back up around 10,000 cases/day.

Delta may not mean change

Germany is in a confusing place with its pandemic developments. Covid cases have been falling as rapidly here as they have been rising in the UK: More than 50% reduction in the past week, dropping the new cases to 6.6 per 100,000 averaged over the week, under 1000 cases per day for the first time since the middle of last summer. At the same time, the Delta variant is rapidly taking over. Last week the Robert Koch Institute reported 8%, this week it’s 15%. Virologist Christian Drosten, speaking on the NDW Coronavirus podcast this week (before the new Delta numbers were available) spoke of the 80% level in England that, he said, marked the watershed between falling and rising case numbers.

I think this is the wrong back-of-the-envelope calculation, because it depends on the overall expansion rate of the virus, and the difference between Delta and Alpha, which is likely particularly large in the UK because of the large number of people who have received just one AstraZeneca dose, which seems to be particularly ineffective against Delta. There’s another simple calculation that we can do specifically for the German situation: In the past week there have been about 810 cases per day, of which 15.1% Delta, so, about 122 Delta cases per day. The previous week there were about 1557 cases per day, of which 7.9% Delta, so also about 123 Delta cases. That suggests that under current conditions (including weather, population immunity, and social distancing) Delta is not expanding. This may mean that current conditions are adequate to keep Delta in check, while Alpha and other variants are falling by more than 50% per week.

This suggests a very optimistic picture: that total case numbers will continue to fall. Within a few weeks Delta will be completely dominant, but the number of cases may not be much more than around 100 per day. And that ignores the increasing immunity: The infections reported this week occurred in the previous week, and the immunity is based on the vaccinations two weeks before that. With about 1% of the population being vaccinated every day, we should have — relative to the approximately 70% non-immune population* 20 days ago — already have about 15% reduced transmission by the first week in July. And at current vaccination rates we can expect, by the end of July that will be 30% reduced, providing some headroom for further relaxation of restrictions without an explosion of Delta cases.

That does raise the question, though, of why the general Covid transmission rate in Germany seems to be lower than in the UK. I don’t see any obvious difference in the level of social-distancing restrictions. Is it just the difference between single-dose AZ versus Biontech? If so, we should see a rapid turnaround in the UK soon.

* I’m very roughly counting each dose as 50% immunity.

“Like lepers”

From yesterday’s Guardian:

Contentious advice will be removed from the government website, a spokesperson confirmed on Tuesday evening. It had advised against all but essential travel to and from eight areas of England where the Covid variant identified in India has been spreading. Instead, people will be advised to “minimise travel”.

This gets to something that I observed last spring when I (like many people) decided to immerse myself in the literature of epidemics — The Last Man, La Peste, Journal of the Plague Year, Pale Horse Pale Rider: In an epidemic, people can’t help but think of contagion as a moral failing, and so, thinking of themselves as blameless, underestimate the possibility that they could unwittingly infect someone else. This makes it socially uncomfortable to insist that others practice necessary hygiene, because it sounds like you’re accusing them of secret turpitude. Strangers should wear masks, but “we’re friends”, or even — much too casually — “you’re in my bubble”.

Thus two Bolton residents, whose hotel reservations on the Isle of Wight had been cancelled purely from abhorrence of their municipal origins, complained to the press: “Bolton people are being treated like lepers”.

On a literal level we have here people who are feared to be at high risk of carrying an explosively contagious infection that produces an acute disease with no very good treatment, that is often rapidly fatal; complaining that they are being unjustly shunned as though they might be carrying a different, mildly contagious infection, that produces a chronic disease that can be completely cured with medication.

But, of course, what they really mean is, we’re being treated like morally culpable potential disease carriers. This is a status that has traditionally been conferred on carriers of leprosy, something we do not question, but that only highlights our moral — hence also virological — purity.

Does a shot of vitamin B protect against Covid?

Thirty years ago there were some who envisioned a new united Germany combining the best of east and west: A vibrant market economy with an expanded commitment to economic justice, a confident democratic federal government balanced by a greater variety of states, and a commitment to individual liberty reinforced by the recent experience of dictatorship. A bridge between the solid democracy in America and the rising democracy in Russia.

Instead, Covid has revealed the modern Federal Republic as the combined worst of both systems: A timid central government in thrall to private business interests, unable to take decisive action to protect public health because of a lack of confidence that their authority would be seen as sufficiently legitimate. A resurgent right wing inspired by American and Russian ethnonationalists to express their individual liberty by rejecting even elementary public health measures. And now, a public vaccination campaign dominated by the financial and status needs of private physicians, and prioritising those people with high levels of what the East Germans called Vitamin B — “B” for “Beziehungen“, connections, the way scarce resources were distributed under socialism.

When the vaccination program started it was concentrated in large vaccination centres (Impfzentren). These were highly efficient, providing rapid throughput and simple logistics, and the official priorities of the Permanent Vaccination Commission (Ständige Impfkomission, or STIKO) — covering people over 60 years old, those with serious medical conditions putting them at elevated risk of Covid complications (including pregnant women and their companions), police, teachers, and government officials — could be securely monitored.

There were two major problems with this system: First, the physicians in private practice, for whom vaccinations were traditionally a great money spinner, felt that they were losing money and influence; Second, there was no unobtrusive path to providing priority immunisation to people who were important, influential, or just rich, threatening to lead to the sort of blatant corruption that just embarrasses everyone. This led the government* to bring the GPs into the vaccination program, paying them upwards of €50 per vaccine. The GPs, unsurprisingly, rushed to vaccinate their friends and favourite patients — particularly those patients with private insurance, who they are generally keen to hold on to, as the private insurance covers all manner of treatments that the public insurance won’t pay for, and the payment levels are generally significantly higher.

How should one respond to this? The Ethics commission is very concerned… that people who haven’t been clever enough to work the system might be jealous of the superior people who have. Here is a comment from a recent podcast interview with commission chair Alena Buyx:

AB: We shouldn’t confuse the people who have gone the extra mile and somehow managed to get it organised, or had a stroke of luck, with those who have cheated.

Spiegel: … It could be that someone who isn’t so resourceful… for various reasons, it could be social background, it could be language, it could be some lack of access to information –and I can understand that they might feel he or she feels like you’re taking something away from them.

AB: These are things that one couldn’t have imagined earlier. We have vaccine envy and also vaccine guilty conscience. But all I can say is: Good People, every vaccination is a good vaccination… Those who have been lucky, or who have profited from this “flexibility” — if they haven’t cheated anyone — they should enjoy their good fortune.

From there to full social Darwinism is just a small step, and that step was taken by one Christoph S. in the comments section of the national newspaper Die Welt:

In my social circle — definitely well off — is just about everyone vaccinated, and always the whole family, including the university-age children. None of them in the vaccination centre, always in the GP practice or through doctors they know personally. In other words, since the GPs have been doing vaccinations the prioritisation has fallen away de facto, at least for the “higher” levels of the population. This is not pretty, but as long as they’re managing to vaccinate up to a million people a day in Germany, I find it acceptable. One shouldn’t make a fuss about the people who try to cut ahead at the vaccination centres; Germany has much bigger problems than someone getting vaccinated a few days early. And, by the way, this is how it’s always been, that those who make the most noise prevail, and presumably that’s why Homo sapiens has managed to survive.***

* Just to be clear, this is not the official justification. This is a purely speculative exercise on my part. It’s hard to think of any other justification, though. It’s not as though the GPs were otherwise unoccupied, with huge amounts of spare capacity for taking on vaccination duty.

Continue reading “Does a shot of vitamin B protect against Covid?”

How to vaccinate all the Germans in two easy steps

One might despair at how hopelessly behind Europe in general, and Germany in particular, is with its vaccination campaign. According to the data below from the Robert Koch Institute, they recovered last week from the collapse of the week before due to the brief rejection of the AstraZeneca vaccine, and resumed their very modest acceleration, but that seems to have stopped, and they’re now back to the rate of the previous week of about 1.5 million vaccines per week, a rate that would get them through the entire adult population in around… 2 years.

RKI Vaccine statistics 1/4/2021

But not to worry! says Der Spiegel. They quote an expert — Sebastien Dullien, scientific director of the Institute for Macroeconomics and Economic Research (Institut für Makroökonomie und Konjunkturforschung (IMK) der Hans-Böckler-Stiftung), for which I’ll have to take their word that he’s somehow an expert on vaccines and public health, because his job (and his Wikipedia page) make it seem that he’s an expert on finance and economics — who claims that the vaccination of the entire German adult population will be complete before the middle of the summer. “Es ist realistisch, alle impfbereiten erwachsenen Deutschen bis Ende Juli durchgeimpft zu haben.” [It is realistic, that we can have all willing adult Germans vaccinated by the end of July.) Sounds good! He goes on to say “Dafür müssen nur zwei Bedingungen erfüllt werden.” [This depends on just two conditions being fulfilled.] Okay, two conditions. I hope the conditions are fulfilled… What are they?

Der Impfstoff muss kommen, und er muss verimpft werden.
[We have to get the vaccine, and then we have to vaccinate people with it.]

It’s this kind of reduction of complex problems into manageable sub-problems that only the truly great minds can deliver. This goes on my list of “How-to-do-it” solutions to complex problems. (Previous entries here, here, and here.)

Actually, this is amazingly close to the Monty Python original, where the kiddie show How to Do It explained “how to rid the world of all known diseases”. Their method was more elaborate, though, involving five steps:

First of all, become a doctor, and discover a marvelous cure for something. And then, when the medical profession starts to take notice of you, you can jolly well tell them what to do and make sure they get everything right, so there will never be any diseases ever again.

Absence of caution: The European vaccine suspension fiasco

Multiple European countries have now suspended use of the Oxford/AstraZeneca vaccine, because of scattered reports of rare clotting disorders following vaccination. In all the talk of “precautionary” approaches the urgency of the situation seems to be suddenly ignored. Every vaccine triggers serious side effects in some small number of individuals, occasionally fatal, and we recognise that in special systems for compensating the victims. It seems worth considering, when looking at the possibility of several-in-a-million complications, how many lives may be lost because of delayed vaccinations.

I start with the case fatality rate (CFR) from this metaanalysis, and multiply them by the current overall weekly case rate, which is 1.78 cases/thousand population in the EU (according to data from the ECDC). This ignores the differences between countries, and differences between age groups in infection rate, and certainly underestimates the infection rate for obvious reasons of selective testing.

Age group0-3435-4445-5455-6465-7475-8485+
CFR (per thousand)0.040.682.37.52585283
Expected fatalities per week per million population0.071.24.11345151504
Number of days delay to match VFR120070206.41.80.60.2

Let’s assume now that all of the blood clotting problems that have occurred in the EEA — 30 in total, according to this report — among the 5 million receiving the AZ vaccine were actually caused by the vaccine, and suppose (incorrectly) that all of those people had died.* That would produce a vaccine fatality rate (VFR) of 6 per million. We can double that to account for possible additional unreported cases, or other kinds of complications that have not yet been recognised. We can then calculate how many days of delay would cause as many extra deaths as the vaccine itself might cause.

The result is fairly clear: even the most extreme concerns raised about the AZ vaccine could not justify even a one-week delay in vaccination, at least among the population 55 years old and over. (I am also ignoring here the compounding effect of onward transmission prevented by vaccination, which makes the delay even more costly.) As is so often the case, “abundance of caution” turns out to be the opposite of cautious.

* I’m using only European data here, to account for the contention that there may be a specific problem with European production of the vaccine. The UK has used much more of the AZ vaccine, with even fewer problems.

The first principle of statistical inference

When I first started teaching basic statistics, I thought about how to explain the importance of statistical hypothesis testing. I focused on a textbook example (specifically, Freedman, Pisani, Purves Statistics, 3rd ed., sec 28.2) of a data set that seems to show more women being right-handed than men. I pointed out that we could think of many possible explanations: Girls are pressured more to conform, women are more rational — hence left-brain-centred. But before we invest too much time and credibility in abstruse theories to explain the phenomenon, we should first make sure that the phenomenon is real, that it’s not just the kind of fluctuation that could happen by accident. (It turns out that the phenomenon is real. I don’t know if either of my explanations is valid, or if anyone has a more plausible theory.)

I thought if this when I heard about the strange Oxford-AstraZeneca vaccine serendipity that was announced this week. The third vaccine success announced in as many weeks, the researchers announced that they had found about a 70% efficacy, which is good, but not nearly as impressive as the 95% efficacy of the mRNA vaccines announced earlier in the month. But the strange thing was, they found that a subset of the test subjects who received only a half dose at the first injection, and a full dose later, showed a 90% efficacy. Experts have been all over the news media trying to explain how some weird idiosyncrasies of the human immune system and the chimpanzee adenovirus vector could make a smaller dose more effective. Here’s a summary from Science:

Researchers especially want to know why the half-dose prime would lead to a better outcome. The leading hypothesis is that people develop immune responses against adenoviruses, and the higher first dose could have spurred such a strong attack that it compromised the adenovirus’ ability to deliver the spike gene to the body with the booster shot. “I would bet on that being a contributor but not the whole story,” says Adrian Hill, director of Oxford’s Jenner Institute, which designed the vaccine…

Some evidence also suggests that slowly escalating the dose of a vaccine more closely mimics a natural viral infection, leading to a more robust immune response. “It’s not really mechanistically pinned down exactly how it works,” Hill says.

Because the different dosing schemes likely led to different immune responses, Hill says researchers have a chance to suss out the mechanism by comparing vaccinated participants’ antibody and T cell levels. The 62% efficacy, he says, “is a blessing in disguise.”

Others have pointed out that the populations receiving the full dose and the half dose were substantially different: The half dose was given by accident to a couple of thousand subjects at the start of the British arm of the study. These were exclusively younger, healthier individuals, something that could also explain the higher efficacy, in a less benedictory fashion.

But before we start arguing over these very interesting explanations, much less trying to use them to “suss out the mechanisms” the question they should be asking is, is the effect real? The Science article quotes immunologist John Moore asking “Was that a real, statistically robust 90%?” To ask that question is to answer it resoundingly: No.

They haven’t provided much data, but the AstraZeneca press release does give enough clues:

One dosing regimen (n=2,741) showed vaccine efficacy of 90% when AZD1222 was given as a half dose, followed by a full dose at least one month apart, and another dosing regimen (n=8,895) showed 62% efficacy when given as two full doses at least one month apart. The combined analysis from both dosing regimens (n=11,636) resulted in an average efficacy of 70%. All results were statistically significant (p<=0.0001)

Note two tricks they play here. First of all, they give those (n=big number) which makes it seem reassuringly like they have an impressively big study. But these are the numbers of people vaccinated, which is completely irrelevant for judging the uncertainty in the estimate of efficacy. The reason you need such huge numbers of subjects is so that you can get moderately large numbers where it counts: the number of subjects who become infected. Further, while it is surely true that the “results” were highly statistically significant — that is, the efficacy in each individual group was not zero — this tells us nothing about whether we can be confident that the efficacy is actually higher than what has been considered the minimum acceptable level of 50%, or — and this is crucial for the point at issue here — whether the two groups were different from each other.

They report a total of 131 cases. They don’t say how many cases were in each group, but if we assume that there were equal numbers of subjects getting the vaccine and the treatment in all groups then we can back-calculate the rest. We end up with 98 cases in the full-dose group (of which 27 received the vaccine) and 33 cases in the half-dose group, of which 3 received the vaccine. Just 33! Using the Clopper-Pearson exact method, we obtain 90% confidence intervals of (.781,.975) for the efficacy of the half dose and (.641, .798) for the efficacy of the full dose. Clearly some overlap there, and not much to justify drawing substantive conclusions from the difference between the two groups — which may actually be zero, or close to 0.

Early Trumpist medical treatments

And then I see the disinfectant, where it knocks it out in a minute. One minute! And is there a way we can do something like that, by injection inside or almost a cleaning. Because you see it gets in the lungs and it does a tremendous number on the lungs. So it would be interesting to check that.

When Donald Trump used a Covid-19 press briefing to recommend injecting disinfectants to kill viruses within the human body, people reacted as though this were entirely unprecedented. But it wasn’t, entirely. From Frank Snowden’s Epidemics and Society:

Of all nineteenth-century treatments for epidemic cholera, however, perhaps the most painful was the acid enema, which physicians administered in the 1880s in a burst of excessive optimism after Robert Koch’s discovery of V. cholerae. Optimistic doctors reasoned that since they at last knew what the enemy was and where it was lodged in the body, and since they also understood that bacteria are vulnerable to acid, as Lister had demonstrated, all they needed to destroy the invader and restore patients’ health was to suffuse their bowels with carbolic acid. Even though neither Koch nor Lister ever sanctioned such a procedure, some of their Italian followers nevertheless attempted this treatment during the epidemic of 1884–1885. The acid enema was an experimental intervention that, in their view, followed the logic of Koch’s discoveries and Lister’s practice. The results, however, were maximally discouraging…

Apparently it’s a not uncommon response on someone first learning of the germ theory of disease.

Vaccine probabilities

From an article on the vaccine being developed by Robin Shattock’s group at Imperial College:

The success rate of vaccines at this stage of development is 10%, Shattock says, and there are already probably 10 vaccines in clinical trials, “so that means we will definitely have one”

It could be an exercise for a probability course:

  1. Suppose there are exactly 10 vaccines in this stage of development. What is the probability that one will succeed?
  2. Interpret “probably 10 vaccines” to mean that the number of vaccines in clinical trials is Poisson distributed with parameter 10. What is the probability that one will succeed?