Wednesday, October 28, 2020

Herd Immunity, Facts and Numbers

Today, I have a few words to say about herd immunity because there’s very little science in the discussion about it. I also want to briefly comment on the Great Barrington Declaration and on the conversation about it that we are not having.

First things first, herd immunity refers to that stage in the spread of a disease when a sufficient fraction of the population has become immune to the pathogen so that transmission will be suppressed. It does not mean that transmission stops, it means that on the average one infected person gives the disease to less than one new person, so outbreaks die out, instead of increasing.

It’s called “herd immunity” because it was first observed about a century ago in herds of sheep and, in some ways we’re not all that different from sheep.

Now, herd immunity is the only way a disease that is not contained will stop spreading. It can be achieved either by exposure to the live pathogen or by vaccination. However, in the current debate about the pursuit of herd immunity in response to the ongoing COVID outbreak, the term “herd immunity” has specifically been used to refer to herd immunity achieved by exposure to the virus, instead of waiting for a vaccine.

Second things second, when does a population reach herd immunity? The brief answer is, it’s complicated. This should not surprise you because whenever someone claims the answer to a scientific question is simple they either don’t know what they’re talking about, or they’re lying. There is a simple answer to the question when a population reaches herd immunity. But it does not tell the whole story.

This simple answer is that one can calculate the fraction of people who must be immune for herd immunity from the basic reproduction number R_0 as 1- 1/R_0.

Why is that? It’s because, R_0 tells you how many new people one infected person infects on the average. But the ones who will get ill are only those which are not immune. So if 1-1/R_0 is the fraction of people who are immune, then the fraction of people who are not immune is 1/R_0.

This then means that average number of susceptible people that one infected person reaches is R_0 * 1/R_0 which is 1. So, if the fraction of immune people has reached 1 – 1/R_0, then one infected person will on the average only pass on the disease to one other person, meaning at any level of immunity above 1 – 1/R_0, outbreaks will die out.

R_0 for COVID has been estimated with 2 to 3, meaning that the fraction of people who must have had the disease for herd immunity would be around 50 to 70 percent. For comparison, R_0 of the 1918 Spanish influenza has been estimated with 1.4 to 2.8, so that’s comparable to COVID, and R_0 of measles is roughly 12 to 18, with a herd immunity threshold of about 92-95%. Measles is pretty much the most contagious disease known to mankind.

That was the easy answer.

Here’s the more complicated but also more accurate answer. R_0 is not simply a property of the disease. It’s a number that quantifies successful transmission, and therefore depends on what measures people take to protect themselves from infection, such as social distancing, wearing masks, and washing hands. This is why epidemiologists use in their models instead an “effective R” coefficient that can change with time and with people’s habits. Roughly speaking this means that if we would all be very careful and very reasonable, then herd immunity would be easier to achieve.

But that R can change is not the biggest problem with estimating herd immunity. The biggest problem is that the simple estimate I just talked about assumes that everybody is equally likely to meet other people, which is just not the case in reality.

In realistic populations under normal circumstances, some people will have an above average number of contacts, and others below average. Now, people who have many contacts are likely to contribute a lot to the spread of the disease, but they are also likely to be among the first ones to contract the disease, and therefore become immune early on.

This means, if you use information about the mobility patterns, social networks, and population heterogeneity, the herd immunity threshold is lower because the biggest spreaders are the first to stop spreading. Taking this into account, some researchers have estimated the COVID herd immunity threshold to be more like 40% or in some optimistic cases even below 20%.

How reliable are these estimates? To me it looks like these estimates are based on more or less plausible models with little empirical data to back them up. And plausible models are the ones one should be especially careful with.

So what do the data say? Unfortunately, so far not much. The best data on herd immunity so far come from an antibody study in the Brazilian city of Manaus. That’s one of the largest cities in Brazil, with an estimated population of two point one million.

According to data from the state government, there have been about fifty five thousand COVID cases and two thousand seven hundred COVID fatalities in Manaus. These numbers likely underestimate the true number of infected and deceased people because the Brazilians have not been testing a lot. Then again, most countries did not have sufficient testing during the first wave.

If you go by the reported numbers, then about two point seven percent of the population in Manaus tested positive for COVID at some point during the outbreak. But the study which used blood donations collected during this time found that about forty-four percent of the population developed antibodies in the first three months of the outbreak.

After that, the infections tapered off without interventions. The researchers estimate the total number of people who eventually developed antibodies with sixty-six percent. The researchers claim that’s a sign for herd immunity. Please check the information below the video for references.

The number from this Brazilian study, about 44 to 66 percent seems consistent with the more pessimistic estimates for the COVID herd immunity threshold. But what it took to get there is not pretty.

2700 dead of about two million that’s more than one in a thousand. Hospitals run out of intensive care units, people were dying in the corridors, the city was scrambling to find ways to bury the dead quickly enough. And that’s even though the population of Manaus is pretty young; just six percent are older than sixty years. For comparison, in the United States, about 20% are above sixty years of age, and older people are more likely to die from the disease.

There are other reasons one cannot really compare Manaus with North America or Europe. Their health care system was working at almost full capacity even before the outbreak, and according to data from the world bank, in the Brazilian state which Manaus belongs to, the state of Amazonas, about 17% of people live below the poverty line. Also, most of the population in Manaus did not follow social distancing rules and few of them wore masks. These factors likely contributed to the rapid spread of the disease.

And I should add that the paper with the antibody study in Manaus has not yet been peer reviewed. There are various reasons why the people who donated blood may not be representative for the population. The authors write they corrected for this, but it remains to be seen what the reviewers think.

You probably want to know now how close we are to reaching herd immunity. The answer is, for all can tell, no one knows. That’s because, even leaving aside that we have no reliable estimates on the herd immunity threshold, we do not how many people have developed immunity to COVID.

In Manaus, the number of people who developed antibodies was more than twenty times higher than the number of those who tested positive. As of date in the United States about eight point five million people tested positive for COVID. The total population is about 330 Million.

This means about 2.5% of Americans have demonstrably contracted the disease, a rate that just by number is similar to the rate in Manaus, though Manaus got there faster with devastating consequences. However, the Americans are almost certainly better at testing and one cannot compare a sparsely populated country, like the United States, with one densely populated city in another country. So, again, it’s complicated.

For the Germans here, in Germany so far about 400,000 people have tested positive. That’s about 0.5 percent of the population.

And then, I should not forget to mention that antibodies are not the only way one can develop immunity. There is also T-cell immunity, that is basically a different defense mechanism of the body. The most relevant difference for the question of herd immunity is that it’s much more difficult to test for T-cell immunity. Which is why there are basically no data on it. But there are pretty reliable data by now showing that immunity to COVID is only temporary, antibody levels fall after a few months, and reinfections are possible, though it remains unclear how common they will be.

So, in summary: Estimates for the COVID herd immunity threshold range from roughly twenty percent to seventy percent, there are pretty much no data to make these estimates more accurate, we have no good data on how many people are presently immune, but we know reinfection is possible after a couple of months.

Let us then talk about the Great Barrington Declaration. The Great Barrington Declaration is not actually Great, it was merely written in place called Great Barrington. The declaration was formulated by three epidemiologists, and according to claims on the website, it has since been signed by more than eleven thousand medical and public health scientists.

The supporters of the declaration disapprove of lockdown measures and instead argue for an approach they call Focused Protection. In their own words:
“The most compassionate approach that balances the risks and benefits of reaching herd immunity, is to allow those who are at minimal risk of death to live their lives normally to build up immunity to the virus through natural infection, while better protecting those who are at highest risk. We call this Focused Protection.”

The reaction by other scientists and the media has been swift and negative. The Guardian called the Barrington Declaration “half baked” “bad science” and “a folly”. A group of scientists writing for The Lancet called it a “dangerous fallacy unsupported by scientific evidence”, the US American infectious disease expert Fauci called it “total nonsense,” and John Barry, writing for the New York Times, went so far to suggest it be called “mass murder” instead of herd immunity. Though they later changed the headline.

Some of the criticism focused on the people who wrote the declaration, or who they might have been supported by. These are ad hominem attacks that just distract from the science, so I don’t want to get into this.

The central element of the criticism is that the Barrington Declaration is vague on how the “Focused Protection” is supposed to work. This is a valid criticism. The declaration left it unclear just to how identify those at risk and how to keep them efficiently apart from the rest of the population, which is certainly difficult to achieve. But of course if no one is thinking about how to do it, there will be no plan for how to do it.

Why am I telling you this? Because I think all these commentators missed the point of the Barrington Declaration. Let us take this quote from an opinion piece in the Guardian in which three public health scientists commented on the idea of focused protection:
“It’s time to stop asking the question “is this sound science?” We know it is not.”
It’s right that arguing for focused protection is not sound science, but that is not because it’s not sound, it’s because it’s not science. It’s a value decision.

The authors of the Great Barrington Declaration point out, entirely correctly, that we are in a situation where we have only bad options. Lockdown measures are bad, pursuing natural herd immunity is also bad.

The question is, which is worse, and just what do you mean by “worse”. This is the decision that politicians are facing now and it is not obvious what is the best strategy. This decision must be supported by data for the consequences of each possible path of action. So we need to discuss not only how many people die from COVID and what the long-term health problems may be, but also how lockdowns, social distancing, and economic distress affect health and health care. We need proper risk estimates with uncertainties. We do not need scientists who proclaim that science tells us what’s the right thing to do.

I hope that this brief survey of the literature on herd immunity was helpful for you.

I have a video upcoming later today with astrophysicist (!) Niayesh Afshordi from Perimeter Institute about his new pandemic model (!!), so stay tuned. He will also join the Thursday chat at 5pm CET. Note that this is the awkward week of the year when the NYC-Berlin time shift is only 5 hours, so that's noon Eastern Time.


  1. It's not clear to me what whether r0 converges to 0 or to 1, where the virus smolders through the population in a slow burn. That's where the other virii that we know about are, right? So do we know anything about what makes the difference between a virus that becomes endemic, and one that disappears? As far as I can tell, there's only 2 strategies: hold of the deaths until there's a vaccine, or suppress the virus until there's a vaccine. Unless, of course, we don't get a vaccine that works...

  2. Great article! Thanks for bringing some rationality to this discussion.

  3. If COVID-19 is allowed to replicate unconstrained throughout the world’s population as is occurring currently in the U.S., the probability is greatly increased that the virus will mutate more vigorously and expansively than if the virus is tightly localized and controlled. A virus population that has mutated widely and prolifically makes production of a single vaccine increasingly more problematic.

    Massive genetic study shows coronavirus mutating and potentially evolving amid rapid U.S. spread

    The largest U.S. genetic study of the virus, conducted in Houston, shows one viral strain outdistancing all of its competitors, and many potentially important mutations.

  4. The Covid-19 virus is attacking the individual organism (human) but also the complex system of all those organisms. The responses need, by necessity, be both different yet complementary. Science plays a greater role in the former but the latter requires a full systems response. This has been lacking to date.

  5. Thanks, this is a very well balanced article IMHO. I especially loved your conclusion and the fact that we shall weight pros and cons of political decisions, backed by numbers and sound analysis, whitout claiming that a (political) decision is "backed by science" whereas other decision are not - btw, this argument closely resembles the "God is with us" argument that usually preceded any major disaster in history.

  6. Unsere Herdgang ist sehr effektiv gegen Erkältungen :-)

  7. Good article as always ;)
    I think we can add the long-term consequences for the infected (estimated 10-40%) that add to the human cost.

    The Guardian piece is not that good at making its point; the best *scientific* argument against focused protection is that is very hard to protect a group when there is widespread community infection. So it's a complimentary measure at best, not the solution.

    Moreover, as the article correctly points out, the declaration is the same "magnified minority" tactic used countless time against global warming.

  8. There is this professor from Stockholm University, Sweden, that appearantly is giving EAD classes on epidemiologic mathematics. From what I see, he made simlilar remarks about herd immunity (I think he quoted that same article, that sabine shows in the video).
    Here is his channel, if anyone is interested in having a more detailed idea about these mathematical models.

  9. One problem with isolating vulnerable people, predominantly the old, is that isolation kills vulnerable people too.

    There is also the legal, ethical, and logistic questions of how we can force old people to remain isolated?

  10. Mutation is really the ticking time bomb. Do we know enough about this virus to rule out the possibility that it could mutate into a form that is both very easily transmitted and also quite deadly? There's evidence that even in its current form or forms it attacks many organs. I'm curious about the long term health of survivors. It will take time to see exactly what we are dealing with.

  11. One big piece of actual science missing from the discussion of the Great Barrington declaration is what is the ratio of people in the “focused protection” from the general population and more importantly how does that unfluence the necessary percentage of immune people to achieve herd immunity.

  12. Hi Sabine,

    It is good to see that you have entered this discussion.

    Francois Balloux (UCL) , Wes Pedgen ( Carnegie Mellon) , and Maria Chikina (University of Pittsburgh), Stefan Baral (Johns Hopkins), among others, have been discussing these scientific and moral issues openly and with rigorous thinking since March.

    A few additional things to think about are:

    1) What is the a priori probability (assumption) that measures to severely control viral spread will be sustainable in the long run before a safe and effective vaccine can be developed?

    Wes and Maria argue quite effectively that if measures are introduced that cannot be sustained, then more people can die.

    2) Age-targeted mitigations and Increased financial and social help for workers that care for the elderly (so they do not have to have 2 jobs!) reflect the underlying differences in contact patterns between different age/population groups , which can be exploited to reduce the number of infections among the most vulnerable to death.

    3) The assumption that there is some level of immunity which reduces both symptoms and infectiousness is likely sound. However naturally obtained immunity and immunity via vaccine are not 100%. If immunity is fleeting (6-12 months) , requires booster vaccinations, does not work as well in the elderly, this virus will be a natural fixture of our life like 200+ other respiratory viruses. This relates to the sustainability of measures and what kind of adaptations to life we are collectively willing to endure. Ideally this should be made democratically, not by a perpetual State of Exception by the sovereign executive.

    Many challenges both scientifically, philosophically, politically, and morally. One thing to remember is that we cannot use fear to extinguish art, love, the quest for truth in sciences, and political community. This would be both sad, and short lived. Also, the backlash would be horrific.


  13. The problem with pursuing herd immunity is that it may not be possible because immunity to this virus may be transient. There are already almost 300 reported cases of repeat infection. There is at least one report of the repeat infection being more severe than the initial infection. We may see much more of this when we get into the second quarter of next year.

    I am not optimistic going forward. This virus should eventually evolve into our fifth common cold coronavirus, but getting there without a vaccine is going to be extremely painful.

  14. Excellent analysis. I am disappointed that no models or quantitative analyses have been generated to estimate the costs of lockdowns and distancing. As you point out in your all too brief final paragraph, the ideal policy cannot be determined with only one-half of the equation in place.

    The US and other leading countries should study these costs and develop a tiered system (a bit like DEFCONS) whereby progressively harsher measures are taken based on prevailing R-naught and lethality rates. Come to think of it, we deserve an explanation as to why this was not done long ago.

  15. Really disappointed in this article. It pretends that policy choices haven't already been made; e.g., not requiring face masks when in public, in so many US states. The naivete about the real world evident in this article is well....


    Here is this nice lady from Germany, telling the world, we don't yet have to listen to the scientists.

    In the meantime:

    A University of Kansas study released this week backs up what health officials have been telling us for months: Masks do work by significantly slowing the spread of COVID-19.

    “We found a 50% reduction in the spread of COVID-19 in counties that had a mask mandate compared to those without,” Donna K. Ginther, director of the Institute for Policy and Social Research at KU, said in video presenting the study’s findings.

    “Masks, it is important to note, do not eliminate COVID, but they significantly slow the spread of the disease — at least here in Kansas,” she said.

    The study’s release comes at a time that Kansas Gov. Laura Kelly is trying for a second time to get a statewide mask mandate implemented. Kelly had issued a statewide mask mandate July 2, but under state law Kansas’ 105 counties were allowed to opt out -- and most did.


    Yeah, they opted out after a nice rational calculation of the costs of lockdowns and distancing and economic distress. Or they weren't going to listen to the science anyway, and now they've got their talking points from this scientist in Germany.

    1. I have no idea what you want. I didn't say one should not listen to scientists and I certainly didn't say anything against wearing masks. Are you sure you are actually commenting on my video and not on someone else's?

    2. It’s odd to me how the discussion reverts to masks upon the mere suggestion that the costs of lockdowns and distancing are raised as an issue. As if anyone serious on the subject is actually objecting to masks or that masks are at issue. They’re not. It’s a distraction - a way for people who have no response to the question of balancing costs to change the subject.

  16. This article is like - let's go to the Vatican, consider the costs and benefits of God, and decide what to do next.

    1. I have yet to find a way to parse your comment in a way that makes any sense.

  17. Immunity is temporary, so herd immunity is probably an illusion. This will be endemic and to the extent there is herd immunity it will be as we have herd immunity to the common cold.

    I had a check up, and I have new heart arrhythmia that may have come from the Covid. This is an endothelial disease and this has an impact on organ functions because arteries and arterioles are compromised. We can expect a growth in the number of cases of congestive heart failure, renal disease, strokes and COPD related illnesses. We may be only seeing a dress rehearsal for what is to come.

  18. Going to the Vatican is like going to the Hotel Idlehour for a change and a rest: the bellboy gets the change and the hotel gets the rest. (free after Mad Magazine)

  19. Any strategy has to consider the local conditions like capacity of hospitals; the death rate will skyrocket suddenly if we can't treat new cases at all.

    Ultimately balancing lock downs with the infection rate is going to be putting a price on human life; implicitly deciding that a life saved, or permanent health damage saved, is only worth so many $.

    We do that now; I know, with insurance companies and governments deciding if safety regulations are worth the expense, so this is just another example.

    But that's what we are doing; asking whether 230K dead in the USA (and maybe a million permanently damaged) has been worth the $ cost of the shutdown.

    Perhaps being more explicit about that, putting an empirical price tag on lives at various ages and states of health, would make the policy decisions easier.

  20. If you want "natural herd immunity" with normal life and opened economy without any restrictions or precautions, then you should assume R0=3 or even higher.
    "Superspreaders" is not the special kind of people, it is class of environmental conditions which make ordinary people "superspreaders".
    In Hong Kong there are "bars, weddings, religious sites and restaurants" ( ). Sure any people can visit bars, weddings and religious sites.

  21. There is nice article on economy

    In general, countries which have managed to suppress COVID19 have experienced less economic damage. There is no need to choose between protecting the economy or protecting health.
    Choosing to suppress the virus protects both.

    1. No one says you have to choose between protecting the economy or protecting health, the question is where is the optimal balance. Of course this solution will always suppress the virus.

  22. One or two weeks after the preprint on the supposed herd immunity in Manaus, the city began to experience its second wave.

  23. I ran some numbers on what it would take to achieve herd immunity in the US. My estimate was about 10,000,000 deaths. I sent the numbers in a letter to be published in our local paper. I titled the letter "Herd Stupidity" which is what it really is.

    To all of you over there who are wondering what the hell we are doing over here I can tell you that running the numbers on our upcoming election we are all about to see a historically unique American election that will be a massive repudiation of Trump, his supporters, and enablers.

    1. That number seems too high using even the most pessimistic estimates on herd immunity. I get pessimistically something like 3.5 million deaths.

  24. "Roughly speaking this means that if we would all be very careful and very reasonable, then herd immunity would be easier to achieve."
    How come? If we all be very careful (social distancing etc.) what we control is the spreading, not herd immunity. What am I missing here?

    1. As I said, R_0 is not a property of the disease alone. It also depends on how efficiently we infect each other. If we, say, all stopped talking entirely, R_0 would be lower, hence herd immunity would be easier to achieve. Not that I think that's a reasonable course of action.

    2. Just piling on: I think it is fair to say that R_0 is also subject to the prevalence of social distancing, hospital overloading, crowded working conditions, and (in the USA at least) the cost of health care, and thus in general, poverty.

      As in our meat-packing plants, which became Covid-19 hotspots early on because the very low-wage workers, living on the poverty line, had no viable financial choice other than attending crowded working places even after they KNEW they were dangerous; their choice was work or starve and go homeless. Thus R_0 increases.

      People that would be bankrupted by paying thousands of dollars for a hospital stay try to grit their teeth and tough it out, thus increasing the infection rate amongst their family and neighbors.

      Eight people living in a two-bedroom house with one bathroom cannot effectively self-isolate when ill, thus increasing R_0.

    3. Incidentally, same issue with the meat packaging plants in Germany. Curious, that, no?

    4. Miguel Cunha: Suppose, hypothetically, we reduced the transmission rate to zero. Within two months, anybody infected by Covid-19 would be over it; either by recovering or dying. Presumably they would no longer be vectors for it, and Covid-19 would die out. The rest of the population, that never had it, would no longer get it.

      Herd immunity means the transmission rate is so low that infections decrease exponentially. If R_0 is 0.9, then the total size of the outbreak is 10 people before the virus dies out.

      Getting R_0 under 1 gets the virus under control, makes it die out, and once it is gone we can do away with the masks and social distancing. Low enough and we wouldn't even need a vaccine.

      The lower R_0 is, the faster that can happen. It doesn't have to be JUST masks and social distancing, it can also be early diagnosis, free isolation, health care and financial support of the infected, and free contact tracing, all of these can contribute to a lower transmission rate.

    5. Sabine: Hm. I was under the impression that Germany had a stronger social safety net than that, and stronger work-place safety regulators than that.

      Are meat packagers more often immigrants with fewer protections? Or am I just mistaken about my impressions, from afar?

    6. Dr Castaldo,

      In all honesty I haven't really followed this, I just remember lots of hypotheses from poor employment safety to poor air circulation to cold air conserving viruses. Or maybe all of the above, or none. I guess at this point it's all speculation.

    7. Dr Hossenfelder: Well, I guess insufficient space between employees is a problem in every work environment, from college classrooms to meat packing plants; packing us in tight, and ignoring disease transmission has always been a way to save money and maximize building utility. Buildings are expensive!

  25. how much biology have you studies ?

    1. Me? I studied math and physics, not biology. Having said that, there's no biology in pandemic models, if you ever looked at one.

    2. Of course there is biology in pandemic models! These equations encode at least three biological statements: that this virus is a replicator; that those who don't die become immune; that this immunity last forever. Unfortunately the latter is almost certainly false, which is why we never ever reached herd immunity for a cold-like virus, and which is one of the two reasons why these models are considered “dangerous fallacy unsupported by scientific evidence”.

    3. There is no biology in that because each of these mechanisms is parameterized with numbers that simply derived from data. Doesn't matter where there comes from. You could multiply rocks and it would work the same. Also, it is utterly trivial to include a finite-time immunity window.

  26. As long as we want a model to represent some specific reality, we need to make sure our equations captures this reality specifically (instead of *some* possible reality). Also, you're 100% right it is trivial to include a finite-time immunity windows in these models. My point is: doing this change the result. In other words: the models you advertised in this post and the next conclude we can reach herd immunity because they are not based on the correct biological assumptions (as far as we know, e.g. unless this virus behave very differently than any of its cousins).

  27. Way back in January I caught some kind of flu. I was miserable on day one, and the next two days was ejecting virus particles with nose blowing, and sneezing. But that was it. I did wonder if I might have caught covid-19 from one of the Mandarin speaking Chinese patrons of the casinos I visited frequently, who unknowingly was carrying the virus. But, since mid-January I haven't had a single cold; a stretch of almost 10 months. I do wonder if being outdoors a lot either working or exercising might have enhanced my immune system. Luckily, out of a circle of relatives and friends totaling about 40 or 50 people, I haven't heard of any of them coming down with this bug. Or if any of them did it must have been mild. Nonetheless our huge multi-generation Thanksgiving and Christmas gatherings won't be held this year, to be replaced by smaller gatherings.


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