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?

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.

The unexpected epidemic: A political paradox

An epidemiologist says, “A new pandemic will definitely sweep the world some time this century. But you won’t know until the day it starts when it will be. So you’d better start preparing now.”

The president is downcast. He doesn’t like preparing, but he also doesn’t like when the stock-market falls and people on TV blame him for millions of deaths and blah blah blah. What can he do?

His son-in-law comes to him and says, “I read a book on this. This prediction of an unexpected epidemic can’t happen. Imagine it’s 2099 and there hasn’t been a pandemic yet. Then people would know it has to happen in 2099. So it has to happen earlier. But now, suppose we get to 2098 without a pandemic. We know it can’t happen in 2099, so we would know for sure it must be 2098, which would contradict what the so-called expert told us.” And so, step by step, he shows that the unexpected pandemic can never happen.

You know the rest: The president disbands the National Security Council pandemic preparedness team and writes a celebratory tweet. And then in 2020 a pandemic arrives, and the president announces that “this is something that you can never really think is going to happen.”

(For the original version see Quine’s “On a so-called paradox“. For an account of some of the many times experts warned that a pandemic was coming and would be disastrous, see here.)

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.

Putting Covid-19 mortality into context

[Cross-posted with Statistics and Biodemography Research Group blog.]

The age-specific estimates of fatality rates for Covid-19 produced by Riou et al. in Bern have gotten a lot of attention:

0-910-1920-2930-3940-4950-5960-6970-7980+Total
.094.22.911.84.013469818016
Estimated fatality in deaths per thousand cases (symptomatic and asymptomatic)

These numbers looked somewhat familiar to me, having just lectured a course on life tables and survival analysis. Recent one-year mortality rates in the UK are in the table below:

0-910-1920-2930-3940-4950-5960-6970-7980-89
.012.17.43.801.84.2102885
One-year mortality probabilities in the UK, in deaths per thousand population. Neonatal mortality has been excluded from the 0-9 class, and the over-80 class has been cut off at 89.

Depending on how you look at it, the Covid-19 mortality is shifted by a decade, or about double the usual one-year mortality probability for an average UK resident (corresponding to the fact that mortality rates double about every 9 years). If you accept the estimates that around half of the population in most of the world will eventually be infected, and if these mortality rates remain unchanged, this means that effectively everyone will get a double dose of mortality risk this year. Somewhat lower (as may be seen in the plots below) for the younger folk, whereas the over-50s get more like a triple dose.

The coronavirus spectre

This article about the effect of the coronavirus pandemic on air travel mentions social-media criticism of millennials (of course!) for ignoring public health advice by taking advantage of lowered airfares for inessential travel. It occurred to me, though, that the well-publicised observation that the virus seems hardly to affect children and young people at all may create different incentives for different age groups.

And that reminded me of The Subtle Knife, book 2 of Phillip Pullman’s fantasy trilogy His Dark Materials about Oxford scholars (and children) exploring the multiverse. A significant portion of that book is set in a parallel world that has been overtaken by “spectres” that attack and devour the minds of adults, but leave children unharmed. So children run wild and the few remaining adults are in hiding.

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