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Alan Sokal on exponential growth and coronavirus rebound

Alan Sokal writes:

Last week Prime Minister Boris Johnson assured Britons that, come 21 June—at least, if all goes according to plan—we will “re-open everything up to and including nightclubs, and enable large events such as theatre performances.” Life will return to normal, or so he says.

Alas, Johnson is fooling himself, and it takes only a modest understanding of exponential growth to see why. . . .

For the original variant of SARS-Cov-2—the one prevalent throughout the world in early 2020—the evidence shows that, under conditions of normal social interaction, the reproduction number R is somewhere between 2 and 3. Let’s be optimistic and say 2. And the incubation period of the virus—how long it takes, on average, for an infected person to become infectious—is about 5 days. So one infected person will generate two cases after 5 days, then four after 10 days, then eight after 15 days, then 16 after 20 days, then 32 after 25 days, then 64 after one month. Then about 4000 after two months, and about 250,000 after three months, from just one initial case.

This is all approximate, of course, but you get the idea: exponential growth means a very rapid explosion, as anyone who has taken out a loan at 700% interest will sadly attest.

But the B.1.1.7 variant—the one now dominant in the UK—is believed to be about 50% more transmissible. That means an R, under normal social conditions, of somewhere between 3 and 4.5. Let’s again be optimistic and say 3. One infected person will therefore generate about 700 after one month, and about half a million after two months.

But now we have vaccines; let’s factor those in. . . .

[Suppose] we have 85% of the adult population receiving a 70% effective vaccine. Eighty-five percent times 70% equals a 60% reduction in transmission. That brings our assumed R=3 down to about 1.2.

That’s a huge reduction, but 1.2 is still bigger than 1: it means that the epidemic will double every 3 weeks. And this estimate is based on the most optimistic assumptions about the R factor, vaccine efficacy and vaccine uptake. . . .

What could bring R below 1, short of a full-scale lockdown? The answer is straightforward: vaccines, plus all the social distancing measures that we know well . . .

How long will this have to last? Until the number of cases has been reduced to a small enough number that further outbreaks—and there will be many—can be contained by contact tracing and brief local lockdowns. . . .

I think Alan might be too pessimistic here, in that even if the government and businesses officially “return to normal,” enough people will be scared or civic-minded enough to not go to clubs, theatre, etc. Yes, some people will do so, but I expect there will be a lot less circulation of people for awhile. So maybe I should say that I think Alan’s right that life won’t really return to normal anytime soon, but I’m not sure he’s right about Johnson fooling himself. I feel like we could interpret Johnson as saying that people will be free to go out and go to clubs etc., but this freedom will implicitly be relying on the fact that many people won’t actually make use of this freedom, at least not at the rate they’d been doing so before.

86 Comments

  1. Brent Hutto says:

    Looks like a typical bit of Drake Equation-style extrapolation. String together a long chain of assumed values then calculate it all out to a couple digits of precision and you can support whatever surmise you started out with.

    Boris Johnson may end up being correct or completely off base. Alan Sokal may end up being correct or completely off base. If I had to bet at even odds I’d probably put my money on the pessimistic assertion.

    But all those made-up numbers don’t tell us anything about it one way or the other. Why otherwise numerate people find that sort of spurious algebraic claptrap in any way convincing is an eternal mystery to me.

    • Kevin Dick says:

      +1. Notice also how he didn’t factor in naturally acquired immunity. He also didn’t factor in “heterogeneity” of transmission networks and vulnerability.

      The great thing about Drake-style claptrap (I am stealing your excellent phraseology) is that people may not notice when you’ve left out huge factors.

      So the UK is currently at 6.3K per 100K cumulative confirmed cases. At a conservative 33% ascertainment rate, that’s 20% of the population with natural immunity.

      Notice also how he fixated on infections. We don’t care about infections in and of themselves. The vaccines are typically close to _100%_ effective at preventing hospitalization and death. Let’s assume being naturally recovered is the same. If those populations are independent, you have 85% + 20%*15% = 88% of the population that will almost certainly get nothing more than a bad cold if (re)infected.

      Let me get this straight, his solution is to continue applying a hammer to the whole population when it’s only 12% that are vulnerable in any meaningful way and these people _know_ who they are? What was that about it “taking only a modest understanding”?

      • Joshua says:

        > Notice also how he fixated on infections. We don’t care about infections in and of themselves

        I think the biggest error is that doesn’t distinguish between infection and severe disease or death. So yeah, we’re more focused on severe disease and death and theoretically infections without those outcomes isn’t a huge problem.

        But on the flip side of the compounding uncertainty, however, is that there is a chance that a new variant might arrive that’s both more virulent as well as more infectious. That may seem unlikely from estimated virological probabilities, but with uncontrolled infections (we don’t know if/how much the vaccines will prevent infections) it seems to me it would just have to get that much more likely.

      • Carlos Ungil says:

        > The great thing about [Drake-style absurd or nonsensical talk or ideas] is that people may not notice when you’ve left out huge factors.

        That characterisation seems a bit unfair. I’d say that the great thing about explicitly writing down the argument as a combination of factors is that people MAY notice when important factors are left out and check the sensitivity of the conclusions to changes in the assumptions.

        Otherwise you couldn’t point out that the natural immunisation has not been included, we couldn’t discuss what’s the relevant efficacy number regarding propagation, etc.

        If an argument involving numbers is unconvincing because they could be wrong or misleading the same will be true for an argument containing only words. The use of numbers facilitates a productive discussion, as we can see. At least in a numerate-people-oriented forum like this one.

      • dhogaza says:

        “We don’t care about infections in and of themselves.”

        Mutation rate …

        • Kevin Dick says:

          Again, this is not infections in and of themselves. You’re worried about mutations that could produce worse end outcomes such as hospitalization, sequelae, and death.

      • Ben says:

        > He also didn’t factor in “heterogeneity” of transmission networks and vulnerability.

        > The great thing about Drake-style claptrap (I am stealing your excellent phraseology) is that people may not notice when you’ve left out huge factors.

        Alright, so you don’t like handwavy calculations.

        > At a conservative 33% ascertainment rate, that’s 20% of the population with natural immunity.

        > If those populations are independent, you have 85% + 20%*15% = 88% of the population that will almost certainly get nothing more than a bad cold if (re)infected.

        > his solution is to continue applying a hammer to the whole population when it’s only 12% that are vulnerable in any meaningful way

        And then you build a counterargument on a pile of handwavy calculations.

        I don’t think I learned anything from this counterargument. I like the Rahul/Olav critique, and Alan’s response to that.

        That said these handwavy calculations are handwavy so I guess we do expect to be able to beat them at some level. The Rahul/Olav critique sorta seems more useful to understand problems with the argument though.

        • Kevin Dick says:

          The initial part of my comment is against his _argumentation_ not his argument.

          The part about not balancing costs and benefits is my rebuttal to his argument.

  2. Rahul says:

    So his conclusion hinges upon 1.2 > 1.

    But given the multiple levels of approximations involved couldn’t slight changes in his assumptions as well lead to (say) a 0.95 instead of 1.2?

    Uber sensitivity to assumptions?

    • Olav says:

      Yes, that’s exactly right. For example, keeping the other numbers the same and changing the vaccine efficacy to a more realistic (given all the data so far) 0.9 puts the R factor comfortably below 1.

      Given how effective the vaccines seem to be (especially at preventing serious illness), it seems to me there’s plenty of reason for optimism. The biggest X factor is how many people are going to let themselves be vaccinated.

  3. Anoneuoid says:

    Feels just like last Feb to me. There is no way the coming variant wave can be worse than intubate early, infect nursing homes, and OD 70% of hospitalized patients with Ct = 35 on HCQ though.

    And as has been reported in the Pfizer RCT and now observational data from both UK and Israel, the vaccine is about -40% effective for the first week after the first dose. People should be told to quarantine for this immunosuppressed week so they don’t get sick and act as incubators for further strains..

  4. John Williams says:

    “I feel like we could interpret Johnson as saying that people will be free to go out and go to clubs etc., but this freedom will implicitly be relying on the fact that many people won’t actually make use of this freedom,…” That is being very generous to a politician. It seems to me that Johnson is claiming that life will return to normal; that is not consistent with some people being scared sensible.

    • Andrew says:

      John:

      I’m not in Britain and I don’t read the British news media so I’m just guessing on all this. I just think it makes sense to distinguish between the idea that people can be free to go to clubs, theatre, etc., and the idea that life will go back to normal. In his speech, Johnson talks about reclaiming our freedoms and reopening, but that’s consistent with the idea that people vary in their behavior.

      Put it this way: in pre-coronavirus times, suppose that people had some distribution of how often they went out to restaurants, clubs, theatre, etc. If in a few months everything is opened up, there will be some new distribution of attendance. I can only assume this distribution will be shifted down. The point is that this R number depends on behavior, not just on laws.

      • Rahul says:

        One of the framings I don’t like is the use of the word “normal”. If you were to snapshot pre and post industrialization you would see no longer horse riders so much. Did we even want to go back to the “normal”?

        We just need to accept the covid related change and not hanker after the pre-covid “normality”. Not all of it is going to be bad.

        • confused says:

          On the other hand, even the far worse 1918 pandemic, much less 1890, 1957, and 1968, didn’t really have lasting effects on people’s behavior. If this one does, that will IMO be a feature of social media etc. and modern society’s greater risk-aversion.

          Historically, only the really exceptionally deadly things like the Black Death, Plague of Justinian, and New World/Pacific Island smallpox outbreaks (double-digit % fatality rates) seem to have lasting effects on society.

          • Andrew says:

            Confused:

            Yeah, I dunno. I really hate wearing a mask on the subway. It makes the whole experience unpleasant instead of relaxing. So I hope there will be a time in the not so distant future when it won’t be necessary.

            But, to look at it another way . . . someone flew on a plane recently and told me afterward that just about nobody was wearing masks. I was stunned. Trapped in this enclosed space with no masks?? I could hardly imagine.

            • confused says:

              Yeah, but right now everything is influenced by COVID in one way or another, so the awareness/fear of it is constantly reinforced (consciously or not). I’d expect that once that’s no longer the case it will ‘flip back’ pretty quick – but who knows.

              I don’t know. I was germophobic pre-COVID, so I’m probably not a good example. But I will try to live normally post-vaccine+2 weeks.

          • Joshua says:

            confused –

            The relevant question might be whether we modified behaviors more than they did in response to previous pandemics. My thinking is the more behaviors were modified, the largest the level of change once the pandemic eases. Not necessarily the # of deaths.

            • confused says:

              1957 and 1968 basically had no social impact (rather surprisingly, in hindsight, IMO), so definitely more than those.

              1918 … we’ve probably done more, but then they didn’t have tools like telecommuting.

              On the other hand, if modern media/speed of communication has amplified the changes now, it might amplify the change back. I don’t know.

              I expect telecommuting, telehealth visits, etc. to remain much more common & widely accepted than in 2019, but those things were IMO overdue anyway. I find it hard to imagine that actions which imply actual fear of infection will survive long, but who knows…

  5. A.G.McDowell says:

    If you drill through to the actual report you see “Step 4 will begin no earlier than 21 June.” which is not strong evidence of self-deception. “Enable large events” may also involve “The third review will consider the potential role of Covid-status certification in helping venues to open safely” If this includes things like quick result lateral flow testing or (contentious issue here) certificates of vaccination this is not necessarily wishful thinking that the virus will have magically vanished. In fact he does say that “There is therefore no credible route to a Zero Covid Britain or indeed a Zero Covid World”

    (If recent American politics have taught me anything, it is the importance of primary sources: when a politician is quoted saying something silly, go find a transcript to see what they actually said in context)

    • Bob76 says:

      There are strong indications that the UK will adopt a vaccine passport. If they were to do so, one might modify the 0.85 in the calculation above to a significantly larger value—say 0.95. If so, (1-0.95*0.7)*3=1.005. Add in the effects of some slight natural immunity and R drops below 1. Of course, if the new variant has R = 4 or the vaccine is less effective on the variant, then R remains above 1.

      See https://www.bbc.com/news/explainers-55718553. I read the fact that more than 200,000 Brits have signed a petition against such a passport as a sign that there’s a good chance one will be adopted. Brits are not signing petitions that the UK not spend 50% of its GDP to put a Shetland pony on Mars.

      Bob76

  6. Rahul says:

    Is this the same Sokal of the Sokal hoax?

  7. Alan Sokal says:

    I’m grateful for everyone’s critical comments. Let me try to reply to them briefly.

    Brent Hutto: The “long chain of assumed values” comes from published work in medical journals, which I cited in the original article (https://areomagazine.com/2021/03/04/on-covid-boris-johnson-is-fooling-himself-again/); and in each case I chose the _most optimistic_ value.

    Kevin Dick: You are right: in order to keep the arithmetic simple for a popular article, I omitted naturally acquired immunity. But this is not likely to make a big difference: it probably concerns only 20-30% of the population, and (from what I have read, I’m not a virologist) seems to provide _less effective_ immunity than the vaccine.

    Joshua: “Notice also how he fixated on infections. We don’t care about infections in and of themselves.”
    But we *do* care about infections, as they determine the exponential growth rate; if the vaccine prevents disease but not infections or transmission, then this affects only the prefactor — which is important, but far from the whole story. Perhaps, if the vaccine is *extremely* effective in preventing not only serious symptoms but also long-term effects (so-called “long Covid”), then once everyone is vaccinated we won’t care if R > 1: a large fraction of the population will get infected but few will get very sick. (Except the ~~ 10% of the population who refuse to get vaccinated. One could say “serves them right”, but it might still be a big burden on the health system.). If that happens, Covid will, indeed, be like the common cold. But it is far from clear, as yet, whether that will be the case. Until it is, it behooves us to keep R <= 1.

    And I am not suggesting to "continue applying a hammer to the whole population". We'll have to see what is needed to keep R <= 1. If my most optimistic assumptions are valid, then 1.2 is not a whole lot larger than 1, and we'll need only very modest changes from the old "normal" — for instance, continuing to wear masks in indoor public spaces (including public transport).

    Rahul and Olav: You are quite right. If the vaccine(s) are *more* efficacious (including against variants) than currently estimated, and/or if *more* people agree to accept the vaccine than the 85% I assumed — *and* my most optimistic assumptions about the basic reproduction number R_0 turn out to be correct — then R could become <= 1 with no special precautions. We'll just have to see. The point of my article was that Boris Johnson was being reckless in assuming that all this would be the case.

    A.G.McDowell: You are right that Boris Johnson's speech was filled with qualifiers. But all the press — and probably the public too — responded to it by hailing June 21 as a great Liberation Day. It will be very hard for Johnson to back off in May or June and say, "Sorry, I was too optimistic, we won't be able to do what I hoped we would." Moreover, my worry is not so much that things will go bad *before* June 21. It is that things will go well until then, all restrictions will be removed, and then R will creep up above 1, and then — after summer vacations and a return to school and indoor working and crowded public transport — we will have another autumn explosion.

    Andrew and others: You are right that, perhaps, enough of us will continue to wear masks and work from home that the needed social distancing will take place even without any legal obligations. We shall see. But since masks seem to protect other people far more than they protect the wearer, the private incentives for behavior that is *publicly* valuable are weak. It's like making speed limits optional.

    • Joshua says:

      Alan –

      Thanks for the responses. Not that it makes a real difference – you directed a comment to me based on a criticism that I didn’t make (Kevin made it). Just sayin’

    • Anoneuoid says:

      [Suppose] we have 85% of the adult population receiving a 70% effective vaccine. Eighty-five percent times 70% equals a 60% reduction in transmission.

      I wouldn’t assume 70% effective at stopping transmission. That is more like the upper bound soon after vaccination. It depends on age/health, waning, and the variant involved. The latter two work to lower that over time and all the trials excluded the most frail (eg, nursing home residents).

      For Pfizer there was ~2x reduction in neutralizing activity 2 months after second dose and for Moderna there was 2-5x reduction 3 months following the second dose (depending on assay and age):

      https://www.medrxiv.org/content/10.1101/2020.12.09.20245175v1

      https://www.nejm.org/doi/full/10.1056/NEJMc2032195

      And in live virus assays (not pseudovirus) one week after second dose of Pfizer we see:

      These experiments (Extended Data Fig. 9) revealed the following: (1) Convalescent and vaccine sera showed small yet significant reductions (1.7- to 2.5-fold, P < 0.01) in neutralizing activity of B.1.1.7 compared to the WA1/2020 D614G virus (Fig. 5a,e). (2) Sera from both convalescent and vaccinated individuals showed a marked six- to ninefold reduction (P < 0.01) in neutralizing potency against the Wash SA-B.1.351 virus (Fig. 5b,f); and (3) we again observed a smaller decrease (1.7- to 4.5-fold, P < 0.01) in neutralization potency of serum against Wash BR-B.1.1.248, (Fig. 5c,g).

      https://www.nature.com/articles/s41591-021-01294-w

      So after a few months and exposure to new variants neutralizing activity looks like it will be 4-45x lower. About 10x would correspond to a return to what you get 2-3 weeks after the response to the first dose vs the homologous strain.

      I haven’t seen any data on the (most likely slower) waning of the t-cell response though, but that is more related to severity of disease rather than transmission.

      • Anoneuoid says:

        And in the Astrazenca data we see in table 2 that vaccine effectiveness in stopping symptomatic covid dropped to 32% 3-4 months after first dose, while there was basically no influence on asymptomatic positive tests (peaked at 17% 1-3 months after vaccination):

        https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3777268

        From how they report the results (14 days after second dose as a function of time between doses) we can’t say much about waning for that scenario. This vaccine has a problem with antibodies neutralizing the vector (non-covid part) supposedly so unless you wait for those to wane enough before the second dose it won’t do much.

        Anyway, an effectiveness of 70% is probably very optimistic for stopping transmission for more than a month or so after vaccination.

        • Joshua says:

          I don’t think it’s very surprising to experts that antibodies wane, but as to the real world implications of that… do you have evidence of many documented cases of infection post-vaccination?

          -snip-

          Overall, the results demonstrate that CD4+ and CD8+ T cell responses in convalescent COVID-19 subjects or COVID-19 mRNA vaccinees are not substantially affected by mutations found in the SARS-CoV-2 variants.

          https://www.biorxiv.org/content/10.1101/2021.02.27.433180v1

          • Joshua says:

            And further, of those who do get infected post vaccination, how many will then transmit the infection forward:

            -snip-
            Here, analyzing positive SARS-CoV-2 test results following inoculation with the BNT162b2 mRNA vaccine, we find that the viral load is reduced 4-fold for infections occurring 12-28 days after the first dose of vaccine. These reduced viral loads hint to lower infectiousness, further contributing to vaccine impact on virus spread.

            https://www.medrxiv.org/content/10.1101/2021.02.06.21251283v1

            • Anoneuoid says:

              Your first paper is about 14 days after second dose and second 12-30 days. These are the least interesting intervals that have been cherrypicked to show maximal effectiveness. Neither tell us about waning.

              All the stuff to worry about is in days 1-14 after first dose and then months later after waning. And for a proper study they need to compare the same timepoints after infection/vaccination. Not peak vaccine response vs after waning convalescent.

              do you have evidence of many documented cases of infection post-vaccination?

              There have been multiple clinical trials now. The most recent one reported 50% effective at preventing symptomatic covid but only looked up to a month after vaccination in healthy people. Once again that is peak effectiveness. Waned preexisting antibodies appeared to not be effective at all:

              Two other trials, partially or wholly conducted in South Africa and contemporaneous with circulation of the B.1.351 variant, have recently reported efficacy results. In the South African arm (N=6576) of a large multi-national phase 3 study evaluating the efficacy of a single dose of Ad26.COV2.S, preliminary vaccine efficacy against moderate to severe Covid-19 was reported to be 56%, with 95% of cases reportedly due to the B.1.351 variant; however, vaccine efficacy against all-severity Covid-19 specific to the B.1.351 variant has not yet been reported, precluding a direct comparison.23 In the second trial, ChAdOx1-nCoV19 was evaluated in a phase 2 trial (N=2026) in South Africa, in a study population resembling ours with predominantly mild to moderate Covid-19, and reported vaccine efficacy of 22% (95% CI: -50 to 60) overall, and 10% (95% CI: -77 to 55) against the B.1.351 variant, with the latter comprising 95% of cases.24 Our study was subject to certain limitations. The efficacy results are preliminary (median follow-up of 66 and 45 days following first and second doses, respectively), and are limited in scope to the primary endpoint and subgroups of the primary endpoint, as well as post-hoc analysis of B.1.351 variant sequencing data; therefore, caution is warranted in the interpretation of our results, particularly in the PLWH cohort, which represents a relatively small fraction of the study population. Importantly, at the time of analysis, the study had captured almost exclusively mild to moderate Covid-19 endpoints in a predominantly young, healthy population; consequently, we have not as yet been able to report on vaccine efficacy against severe Covid-19

              https://www.medrxiv.org/content/10.1101/2021.02.25.21252477v1

              • Joshua says:

                >… have been cherrypicked to show maximal effectiveness.

                Lol. OK. I tried to have a reasonable convo. Clearly not possible and in-kind responses are clearly something Andrew won’t want.

              • Anoneuoid says:

                You didn’t cherrypick, the people running these studies did. There is no excuse for leaving out details of what happened days 1-14 in nearly every single vaccine study.

                Likewise there is no excuse for reporting “effectivenes” at the peak as if it is not going to wane sharply from there then continue waning more slowly thereafter.

                I can tell from reading this blog that many highly educated people are very misinformed about the true state of affairs.

              • Joshua says:

                The problem for me wasn’t that you were certain that I cherry-picked.

  8. Alan Sokal says:

    I should stress that this article concerned the situation in the UK, where the Astra Zeneca vaccine is predominant. If the Pfizer and Moderna vaccines turn out to be more effective in preventing infection and transmission than Astra Zeneca, then the R factor in the US might be lower than my estimate.

    I also have a question that perhaps the medical statisticians here can answer: Why did the clinical trials focus so single-mindedly on measuring severe disease, but didn’t bother (except as an afterhtought, and sometimes unsystematically) to measure infection? As I have argued, we *do* care also about infection and transmission, as it determines the exponential growth rate. Why didn’t the clinical trials give everyone a PCR test once a week? I know it would have increased the cost of the trial: let’s say 50,000 people x 20 PCR tests/person x $100/test = $100 million. But that is utterly negligible compared to the total economic cost of the pandemic. Right now we’re in a situation where the effectiveness of the vaccines in preventing infection (and thus, presumably, transmission) has to be estimated from fragmentary data that may also be subject to systematic error. For instance, the Astra Zeneca preprint (https://poseidon01.ssrn.com/delivery.php?ID=462120099114112080088112074002119067050047004027052090098060117024014020039090045045029013096013127067071126008085107108101085096068094084004086009125116050034053002001018096111097084120124104022020037127000007048065016094029122085088100104029125003082010122084029014015116094088096124080088#page=14) gives (without comment!) the bizarre result that one dose reduces PCR positivity by 67% (95%CI 49%, 78%) while two doses reduces it by only 54% (95%CI 45%, 62%). Maybe that is just statistical fluctuation? Systematic error due to confounding variables? (Some people got a half-strength first dose, and some got a full dose.) Who knows?

    • dhogaza says:

      “[Suppose] we have 85% of the adult population receiving a 70% effective vaccine. Eighty-five percent times 70% equals a 60% reduction in transmission. That brings our assumed R=3 down to about 1.2.”

      That 70% effectiveness in reducing transmission is based on a vaccinated person’s likelihood of transmitting covid to an unvaccinated person (if the vaccinated person becomes infected, of course).

      The effectiveness will be much higher for the case where an infected vaccinated person is in contact with another uninfected but vaccinated person. So the higher the percentage of vaccinated people in the population, the better things get.

      Ditto for the case where someone has some degree of naturally acquired immunity due to having had covid earlier.

      Of course, variants that partially escape vaccine-induced immunity screw things up somewhat …

      • dhogaza says:

        “That 70% effectiveness in reducing transmission is based on a vaccinated person’s likelihood of transmitting covid to an unvaccinated person (if the vaccinated person becomes infected, of course).”

        Sorry, a vaccinated person becoming infected when exposed to an unvaccinated person who is infected.

        • Alan Sokal says:

          Reading your comment, at first I worried that I had forgotten to *square* the ineffectiveness of the vaccine, i.e. 30% x 30% = 9% instead of 30%. But after reflection, I think that I got it right after all. The point is that the vaccine reduces by 70% one’s chance of getting infected (or at least getting infected enough to have a positive PCR test) when exposed to the virus; that means, if everybody is vaccinated, that the R factor is 30% of what it otherwise would be. If only a fraction of the population is vaccinated, then the correct calculation is the one I gave.

          Of course, this oversimplifies things quite a lot, because there is not in reality a simple binary classification of people as “infected” or “not infected”, with some fixed R factor for the former and R=0 for the latter. Rather, each infected person has a different viral load, and the R factor for transmitting the infection is presumably an increasing (but otherwise largely unknown) function of that viral load; furthermore, the viral load of the recipient is presumably an increasing (but otherwise largely unknown) function of the viral load of the transmitter. And the vaccine presumably acts to reduce the viral load that one would otherwise get. Depending on the details of this process, the resulting R factor could be either larger or smaller than my simple calculation predicts.

    • Nick Adams says:

      PCR tests detect the presence of viral RNA which does not equate with infectivity. Infectivity is better measured by viral culture which is much more expensive and laborious (but has been done in a few small studies). A negative PCR excludes infectivity at the time it is done but not a few days hence if it happens to be done in the very early stages of infection (prior to viral shedding from the lung), thus a second test is needed a few days later. A positive PCR can persist for a few weeks after the infectious period when viral RNA fragments are present but not viable visions.

    • Anoneuoid says:

      the bizarre result that one dose reduces PCR positivity by 67% (95%CI 49%, 78%) while two doses reduces it by only 54% (95%CI 45%, 62%).

      You are comparing “Any PCR+ 22 to 90 days after one dose” to “Any PCR+ > 14 days after second dose”. This is not the same thing.

  9. jrkrideau says:

    @ John Williams
    It seems to me that Johnson is claiming that life will return to normal

    Boris will claim anything that pops into his mind. It need not have any, even tenuous, relation to reality. That tunnel to Ireland the Isle of Man is a great example.

    I suspect Sokal’s projections are a worst case scenario but Boris is unlikely to even understand the word.

    • Alan Sokal says:

      > I suspect Sokal’s projections are a worst case scenario …

      Well, I tried to make it a *best* case scenario (see my original article in Areo Magazine for the details). But of course, it could happen that vaccine uptake is even higher than the 85% I optimistically predicted, and it could happen that Astra Zeneca’s efficacy at preventing transmission is higher than the 70% I optimistically assumed. We shall have to wait and see. My point is simply that, until we have better evidence on this, we should avoid doing anything that is likely to push R above 1 for an extended period.

  10. xyu says:

    I personally don’t like this pessimistic view.

    Yes, the pandemic may not be over by June, and new variants may come up that can escape vaccines and new surges may appear, but I think if we can successfully bring the hospitalization and mortality down to a significantly low level (e.g., at seasonal flu level), I did not see why we should not open up the society fully, maybe still with masks and social distancing, like most east Asian countries do.

    After almost all people get vaccinated, we can safely treat the COVID-19 as a true “big flu.”

    • Joshua says:

      xyu –

      > did not see why we should not open up the society fully, maybe still with masks and social distancing, like most east Asian countries do.

      The problem there is that most Asian countries have functional contact tracing, sufficient resources devoted to effective isolation, and the % of citizens who think wearing a mask in public spaces is some huge infringement on their “freedoms” is relatively small.

  11. xyu says:

    Many theoreticians are very naive and insensitive to politics and real life situations. Millions of people are without a job or underemployed, a lot of people are stressed and depressed, and many small business and even large business went under. For many people, COVID-19 is the least thing to worry about.

    • Andrew says:

      Xyu:

      Polls show that most people are worried about covid. Worry about covid is not just coming from naive theoreticians; it’s coming from people at all levels of society. We’ve discussed this in some previous blog posts. It would be convenient to think that anti-covid measures are only favored by the elites, but that’s not what the polls show.

      • xyu says:

        Andrew:

        Sorry, I was planning to take back the previous comment because later I felt that comment may be interpreted as personal attack, which is not what I meant. If somebody felt offended, I am really sorry for that.

        Back to the poll issues, I think those vulnerable people are hard to reach, and their views are not commonly reflected in the poll results. But I know there are some ways to get estimates (e.g., multilevel post stratification), but again, I still do not feel confident about the results about those neglected people.

    • Joshua says:

      xyu –

      > Many theoreticians are very naive and insensitive to politics and real life situations

      It’s seems pretty implausible to me that ” many theoriticians” are insensitive to the real life suffering that is resulting from the pandemic. Attributing insensitivity to suffering to people, by reverse engineering from your disagreement with them, might be sub-optimal.

      > For many people, COVID-19 is the least thing to worry about.

      You seem to think there is some clear way to disaggregate the effects of the pandemic itself from the effects of NPIs directed at mitigating the effects of the pandemic.

      I haven’t seen any explanations for how Pelle can so that. Maybe you have a good explanation?

      • xyu says:

        sorry for the unintended interpretation of attributing persons. This is not what I meant.

        About 81% of infected people were asymptomatic or with mild symptoms, and most of these people were young (i.e., working) people. I understand that it is inappropriate to consider risk and benefit from the individual point of view during the pandemic, but for those who have to work onsite to support their families, the risk of infection and benefit of earning money will be evaluated in their minds.

        This does not mean people disagree that NPI mitigates the pandemic. It is just different view of the impact of pandemic.

        BTW, I am not supporting quick and reckless opening up like that of MS or TX.

        • Joshua says:

          xyu –

          > About 81% of infected people were asymptomatic or with mild symptoms, and most of these people were young (i.e., working) people.

          I think a problem with that figure exists, in that the cohort of people without jobs or unemployed may well be on average at higher risk for more serious outcomes from infection (more likely minorities, with comorbidities, etc.). In which case, that 81% number wouldn’t be representative (although I don’t know by how much).

          > but for those who have to work onsite to support their families, the risk of infection and benefit of earning money will be evaluated in their minds.

          No doubt. But I don’t think very many of them are likely to evaluate their own risk independently of the risk to their families and communities should they get infected.

          There does seem to be some evidence that those who have lost jobs due to the pandemic are more likely on average to support mitigation interventions. Of course, much depends on the specifics of the interventions – for example whether they contain stimulus or extended unemployment benefits. But the benefits like stimulus payments likely wouldn’t be coming of there weren’t also interventions such as mandates for some businesses to close.

          One thing I’ve seen is that some folks arguing on behalf of those hit the hardest by the pandemic aren’t actually listening to what those who have been hit the hardest are actually saying about what they think would be in their best interests.

          Instead, I think some of what we’re seeing is people from those segments of society that have long had the most agency and influence, a minoritory in terms of numbers, getting upset that they aren’t getting to dictate the policies they prefer – and justifying their advocacy by asserting that they’re arguing on behalf of those who lack agency.

          I’m not putting you in that category, just suggesting that the full context is quite complex.

          • xyu says:

            I fully agree with you that things are more complicated than epidemic models and estimates. Decision making in politics is hard, and becomes harder because of politicians and interest groups.

  12. paul alper says:

    Do any of these numbers/percentages/fractions take into account that the public is reading about them and therefore adjusting its behavior, thus destroying the assumptions/conclusions? Reminds me of the recent election polling in that the publishing of the polls may have altered what the polling results contended.

  13. Kevin Bryan says:

    These calculations do not account for natural immunity, which is 30% of the British population. Nor for the fact that conditional on having the vaccine, the personal risk of hospitalization and death is on par with a normal flu year even with widespread covid transmission. At that point, why would anyone not just live their normal life?

    • Andrew says:

      Kevin:

      People have a range of attitudes. I take it from your comment that you will just live your normal life. Others are still scared, and others have just gotten out of the habit of going to clubs, theatre, etc. Lots of people will continue to wear masks. And lots of people will avoid tightly packed crowds, even after they’ve received their shots. I expect it will take awhile before the aggregate patterns of proximity and transmission will return to what they were in January 2020.

      • Kevin Bryan says:

        I agree there will be variance in behavior, but there is not variance in risk. I think many people do not understand the actual statistics here. It is literally true that conditional on a vaccine, your risk of hospitalization and death from Covid is no more than the flu, and even that is only if Covid is spreading in a completely unmitigated way and in comparison to the flu conditional on flu vaccination at its normal rate. We know what people did to avoid the flu, and what public policy did to prevent it – effectively nothing.

        It strikes me that many people, out of reasonable concern from the tragedy of the pandemic, are thinking of the vaccine as just another potential factor to mitigate risk. But it isn’t. It is the final tool in our toolkit to return to normal life. There is nothing coming after the vaccine, and the world will not stop living forever.

  14. Roy says:

    A point I keep on making is that I believe there are now 7 countries that have brought COVID cases to zero, and life really has pretty much returned to normal. We should be spending more time looking at what these countries did right, and actually worked. Depending on how you look at it, the “sample size” on these are quite large (yes i know there is a clustering effect by country).

    • Joshua says:

      Not that I really disagree with the larger point that practices in other countries are useful information, but getting back to “normal” in a country where there were never very many infections, and where a large segment of the public doesn’t think that wearing a mask in public is an unacceptable infringement on “freedom,” would be a very different process then us getting back to “normal.”

      Maybe that’s basically what you meant by “clustering effect?”

      • Roy says:

        Only responding to “clustering effect”. If we assume complete independence of individuals in these countries, then the sample size is in the millions. But there is likely a “country effect” (all the things you listed for example). So the effective sample size is not 7, nor is it the total number of people in the country. This is where hierarchical models can be very useful.

        As for the seven countries, you don’t have to agree with anything he says, but look at https://twitter.com/yaneerbaryam – he does a good job of tracking Asia, South America and Africa, which you don’t see too much elsewhere.

    • Rahul says:

      Which seven countries?

    • Kevin says:

      What is normal? There are zero developed countries without either lockdowns or completely closed borders. Even those countries with “zero” cases have to lock down society every time there is a single unknown origin case (see Auckland and Melbourne recently). A country where it is impossible to leave or come in is not “normal” in any reasonable sense of the word.

      What exactly is the endgame here? Keep the border closed forever? Widespread vaccination plus acceptance of low-level circulation of Covid just as we accept low-level circulation of the flu and pneumonia and many other diseases is the only reasonable outcome, and it’s where the world is going to be literally by this summer.

      • Joshua says:

        Kevin –

        > Widespread vaccination plus acceptance of low-level circulation of Covid just as we accept low-level circulation of the flu and pneumonia and many other diseases is the only reasonable outcome,…

        What gives you the view that you determine what the “reasonable outcome[s]” are for a country like New Zealand or Australia? Seems to me that the citizens of those countries are the ones who make that determination.

        >… and it’s where the world is going to be literally by this summer

        As of now, it seems that Australia will be limiting its borders for the rest of this year.

        • Howard Edwards says:

          >>… and it’s where the world is going to be literally by this summer

          >As of now, it seems that Australia will be limiting its borders for the rest of this year.

          Which is consistent since this summer begins in December (in Auckland and Melbourne).

  15. Dzhaughn says:

    I think Sokal’s argument is bogus. He doesn’t know.

    But if you want to be pessimistic, look at Israel’s numbers, per the WHO. While Israel’s case rate is about 50% below it’s January peak, while the basically unvaccinated USA’s rate is 25% of what it was then, and the UKs is 10%. (The EU is pretty steady, but at a rate well below Israel’s.) The USA vaccination progam is just a drop in the bucket, but Israel is nearly done. As an amateur, I would have expected more.

    • confused says:

      IMO, though, what matters post-vaccine is not cases but hospitalizations and deaths. We don’t count cases for any of the regular circulating respiratory viruses…

      (And these don’t mutate into deadly strains either – even flu pandemics appear to come from animal strains, not mutations within the human-circulating flu population, so I am not sure why we should expect COVID variants to matter once COVID goes endemic.)

      • Anoneuoid says:

        There are a number of reasons you don’t really want immunity that does not stop transmission:

        Immunity elicited by direct vaccination or by maternal vaccination prolongs host survival but does not prevent infection, viral replication or transmission, thus extending the infectious periods of strains otherwise too lethal to persist. Our data show that anti-disease vaccines that do not prevent transmission can create conditions that promote the emergence of pathogen strains that cause more severe disease in unvaccinated hosts.

        https://pubmed.ncbi.nlm.nih.gov/26214839/

        This strange phenomenon went down in history as the «Hoskins paradox», a classical example of OAS. In practice, it refers to the propensity of the human immune system to exploit the immunological memory B and T cells, selected on the basis of a previous contact with a specific epitope, when a new, slightly different, version of the original antigen is encountered, in order to gain time in the attempt to fight the infection. However, in this way, the immune system gets entrapped inside the first response against the antigenic determinant, becoming unable to mount potentially more effective responses during subsequent infections from the mutated pathogen (Fig. 1 ). OAS has been reported not only in relation to influenza virus, but also to dengue virus and human immunodeficiency virus (HIV) [4], [5]

        https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7204740/

        • confused says:

          “Immunity that does not stop transmission” is I think pretty much what we have for lots of ‘normal’ viruses, though.

          I am unconvinced that there is anything ‘special’ about COVID except that we have tons of the population immunologically-naive to it, unlike the other circulating human coronaviruses.

          And, specifically, *elderly* people immunologically-naive to it. Five years from now, people will be born naive to it, but they will be either vaccinated or infected before reaching high-risk ages.

          A disease’s impact when it is new doesn’t necessarily correlate to its impact in a population it’s endemic in.

          I don’t think new strains will be an issue post-endemic for the same reason they aren’t for the four circulating human coronaviruses, parainfluenza, etc. Influenza pandemics seem to come from animal strains – not mutation within humans.

          Indeed, is there any case of a pandemic/epidemic being started from a new strain of a known circulating pathogen evolving within humans, vs. a zoonosis or contact between two previously separated human populations (e.g. smallpox in the New World/measles in Polynesia)?

          • Anoneuoid says:

            There may not have been anything special about the virus to start, but never before has a virus experienced so close to a monoculture of selective pressure in the middle of a pandemic.

            So it will be special now.

            • confused says:

              vaccination has been rolled out during pandemics before, certainly in 2009 (though that was far less serious), and I believe 1968 and 1957 had flu vaccines too?

              I think there is a lot of “recency bias” here – the crisis *we* are living through *right now* is special and unique.

              (I am also pretty skeptical of the virus having as much ‘evolutionary room’ as tends to be implied. The four circulating human coronaviruses don’t evolve to become dangerous, measles doesn’t evolve to evade vaccine immunity, etc. I think there are pretty strong practical limits/evolutionary constraints on viruses.)

      • Dzhaughn says:

        It’s true that severity matters. But the death data is not really favorable either. Don’t know about hospitalization there.

        • confused says:

          If reporting delays are anything like the US’s, it’s much too early to see the effects in death data; we at least are still reporting December-January deaths.

          And if it’s 3-4 weeks between infection and death, the effect should become obvious only 3-4 weeks after vaccination became common among high-risk populations *plus* the reporting lag.

  16. confused says:

    I am not sure I agree with the implication that the only factors affecting transmission/R are vaccines and behavior.

    There’s immunity from natural infection, and COVID does seem to look like a seasonal pattern in high latitudes (which the UK is) — though very oddly, not in the southern tier of US states, which are still mostly temperate-zone.

    Also – do we really entirely know why past respiratory pandemics faded out after 1-2 years, even before vaccines? It’s not exactly herd immunity, as those viruses didn’t die out entirely – just ceased to be pandemic…

  17. Johnny A says:

    The effects of infections after “85% of the adult population [has received] a 70% effective vaccine” are asymmetric, no? Most of these infections will arise in people who chose not to get vaccinated. I care a lot less about their welfare than I do of the welfare of those who got vaccinated. I realize there are still externalities and uncertainties here; in particular, that continued infections will impose a burden on healthcare workers and the healthcare system, and that new variants may arise as the virus circulates.

    Post-widespread-vaccination, I am going to make my decision about whether to go dancing (think swing or ballroom dancing, where you are very close to your partner’s face) based on the risks & externalities that are generated for people who have been vaccinated. If you didn’t get vaccinated, I don’t care about the risks you continue to face.

    • Joshua says:

      > Most of these infections will arise in people who chose not to get vaccinated.

      This makes me think of another point. Maybe vaccinated and non-vaccinated will cluster in groups. Hence, the overall average has limited utility.

      • confused says:

        I am sure they will, because some social groups/subcultures are taking COVID much more seriously than others.

        Also, social groups stratify by age

        Everything about COVID has been really clustered, actually. I think that’s made the “political/social” side of things notably worse, because two people in the same *general* area can be seeing very different realities.

        If I didn’t follow the statistics, I’d probably be pretty susceptible to believing “it’s massively overblown”, since everyone I *personally* know who has had COVID got over it without much trouble.

        But really, this isn’t unexpected – my age and social circle means that I know very few people in the really high-risk age group. Also, I think my social circle has a higher proportion of people taking COVID at least semi-seriously than the state of Texas overall (unless this is one of those “very loud minority” things and the ‘COVID is a conspiracy’ people are actually a lot less numerous than I’m thinking).

  18. I don’t think that using basic exponential models (or even basic SEIR models) is defensible at this point. We’ve seen that they are poor approximations to reality and that “exponential” growth tends to stop quickly for reasons that are still poorly understood.

    People were worried about the possibility of the atomic bomb igniting the entire atmosphere during the Manhattan project. After all, some simplistic models back then suggested that it was something to worry about. Nowadays physicists don’t worry about this happening because (I assume and really hope) they know more about the relevant physics today after investigating the question further. People who use unbounded exponential growth or SEIR models are like people who still use the old models that suggest an atom bomb will ignite the atmosphere – stuck in a simplistic model not borne out by reality. Plenty of places over the world are “open” but exponential growth isn’t hitting them like the simplistic models would suggest. I can kind of excuse non-epidemiologists for fixating over exponential growth a year ago – so did I, and most of us really had no idea about epidemics – but to still do it now is just not ok.

    Also, I hate the SEIR model because the implied social graph is the complete graph and this is simply ridiculous. My conjecture here, formulated far too strong to be likely to be true as stated, would be that the herd immunity threshold on a non-complete graph is always lower than the one obtained by the SEIR model.

    • Anoneuoid says:

      SIR models were originally developed for measles in NYC circa 1910, iirc. At least it was only in one city. Almost all adults were already immune, and once recovered you stayed immune. So it was a much simpler scenario driven by susceptibles born into the population.

      Still you can use such models to learn general “laws”, maybe.

    • confused says:

      >>(I assume and really hope) they know more about the relevant physics today after investigating the question further

      Yeah, that’s definitely not possible. (Not that I think it was terribly plausible even with 1945 knowledge, but they really didn’t know much about what they were doing at all.)

      I think the idea was some sort of nuclear reaction in the atmosphere. One certainly can’t “ignite the atmosphere” chemically in any self-sustaining way (otherwise it would have reacted long before humans came along).

      I think there are now limits on exponential growth because we have a significant % infected – there isn’t room for more than one or two doublings many places (the US is probably well over 25% infected by now, and when you count in vaccinations as well…) It’s the early part of an epidemic that is more exponential.

      • >I think there are now limits on exponential growth because we have a significant % infected

        Again: this kind of basic epidemiology 101 model is just too simplistic to account for the wide range of data we have seen so far. Something seems to usually stop exponential growth far before any (naively calculated) immunity effects kick in. Whether it is people modifying their behaviour in the face of rising case numbers, or a high variance in how many people one infected person infects, or some otherwise strange factor (remember “epidemiological dark matter”?) I do not know. What I do know, is that whenever some place loosens restrictions there is an outcry about how “exponential growth” now means complete carnage in a short time will certainly occur. This never materializes; and I don’t see why introducing the word “variants” should change anything – if you’re predicing carnage already without variants and it doesn’t materialize then I don’t see why you should be trusted to predict carnage with variants.

        People like to pretend that the government is in control and nobody is socializing now so that “normal social conditions” will suddenly mean everyone socializes after not socializing. Truth is that people break the rules (something I am quite happy about) and have been breaking the rules for a long time – we still don’t seem to have exponential growth. How can you claim to be sure that R is at least 3 under “normal” social conditions?

        • confused says:

          Historically respiratory pandemics have ended in ~1-2 years, yeah, and I am not sure “herd immunity” is exactly the cause; the pandemic flu viruses don’t disappear (transmission going to 0), they become seasonal flu viruses, I believe.

          We didn’t have a lot of genetic science in past pandemics, so we can’t compare the variants observed now to their equivalents in say 1918, 1957, or 1968.

          ‘Breaking the rules’ goes both ways, too; TX (my state) is removing mask mandates this week, but lots of people will keep wearing masks. And people last March were being careful before any mandates were in place (and many businesses, etc. went to work-from-home well before any official requirements existed).

          I think one thing that has been underemphasized all along is the difference between the question of the efficacy of *individual actions* vs *overall measures*.

          IE, it’s one thing to say that masks of a particular type worn a particular way are X% effective at reducing transmission. But that doesn’t necessarily say anything about the effect of a mask mandate on the course of the pandemic… unless you know how a mask mandate will change actual mask-wearing behavior.

        • confused says:

          There’s also the factor that COVID problems tend to ‘cluster’ because risk is very unevenly distributed, so even if the total number of deaths in a state is high, it may not *look* like ‘carnage’ to the majority of people.

          I live in TX, which has had well over 40,000 deaths. However, the relatively few people I know who’ve had COVID were universally mild cases.

          But this actually isn’t surprising – I’m fairly young (31) and know few people in the really high-risk age group. (My friends are close to my age, people I know through work are below retirement age, my parents’ generation are all under 65…)

          If I were 90 and lived in a long-term-care facility, I’d have a very different picture…

    • Alan Sokal says:

      > I hate the SEIR model because the implied social graph is the complete graph and this is simply ridiculous. My conjecture here, formulated far too strong to be likely to be true as stated, would be that the herd immunity threshold on a non-complete graph is always lower than the one obtained by the SEIR model.

      This is a very important point, and I am grateful to you for raising it. (I am also somewhat embarrassed at having overlooked it: in the past, I worked quite a bit on percolation-type models, which play an important role in statistical physics.)

      Indeed, the percolation threshold for any graph is bounded below by the “mean-field” value. Thus, for instance, for bond percolation on the square lattice Z^2, the percolation threshold is p_c = 1/2 — and since each site on the square lattice has 4 neighbors, this corresponds to a critical R factor R_c = 2, far above the R_c = 1 found on the complete graph (and in the SIR model). See G. Grimmett, Percolation (Springer-Verlag, 1989, 2nd ed 1999), eq. (1.13) and proof of Theorem 1.10.

      So the question is: How much does this matter in practice for the propagation of an epidemic in a human population? My guess is that, at least for a large city — particularly one reliant on public transport — the social-interaction network is vastly denser than the square lattice Z^2, and much closer in behavior to a complete graph. I think that some epidemiologists are using mobile-phone data to estimate the actual interconnectivity of modern populations, both at the city level and more widely.
      https://www.nytimes.com/interactive/2020/03/22/world/coronavirus-spread.html
      https://www.nature.com/articles/s41467-020-18190-5
      https://advances.sciencemag.org/content/6/23/eabc0764

      Maybe there are some specialists here who can comment on this.

      • thanks for responding Alan! The analogy to percolation is quite nice but I do think that there are still some subtle points that play a role above and beyond what a naive percolation model would suggest. Firstly, in this case you care about how fast the virus percolates not just about the percolations threshold. I’m not sure there is any “exponential growth” on the lattice even if herd-immunity treshold is relatively high. Most importantly, however, one must appreciate that the social graph only matters during the time in which you are infected; something as simple as “avoid people while you feel sick” which is followed even among “rule-breakers” means the relevant portion of the social graph is people you meet while infectious but not yet sick. Even in a city, this is far from the complete graph, and Z_2 is also a poor approximation because it is far too uniform. Since you certainly know more percolation theory than I do: can we bound the percolation threshold somehow using some measure of heterogeneity?

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