Wednesday, October 21, 2020

Mortality from the Herd Immunity Strategy, a BOTE Estimate: Second Try

I have now redone my calculations, using the CDC data that a commenter on my previous post pointed me at. The results are less optimistic. [Some further revisions have now been included]

Recently, three epidemiologists came out with a public statement arguing for a policy of reaching herd immunity by protecting old people from Covid while letting it spread through the younger population. The proposal has been supported by some, fiercely criticized by others. I have not seen any calculation of what the costs of such a policy would be, so I decided to do one.

My Model

Everyone seventy or over is quarantined, kept from contact with anyone who might carry the virus. The virus is permitted to spread through the rest of the population, controlled only to the extent of not overloading the hospital system. Since this is a simple model, I assume we do it perfectly. The result is an infection rate that just fills available hospital beds, kept down to that if necessary by the sorts of restrictions we are familiar with. Eventually the unquarantined population reaches herd immunity, meaning that each infected person passes the infection to no more than one other person, at which point the number of infected persons starts to decline. When it gets low enough so that we can end quarantine without producing a significant number of deaths, we do so. All of my calculations are for the U.S.

The Numbers

My main source is the CDC’s COVID-19 Pandemic Planning Scenarios. Where figures are given for different age groups, I try to estimate the average for under 70’s.


Ratio of infections to case counts: 11

Median days of hospitalization for those not admitted to the ICU:3.5

Median days of hospitalization for those admitted to the ICU: 12

Percentage of those hospitalized admitted to the ICU: 30%

Infection Fatality Ratio under 70: .0015

Infection Fatality Ratio 70+ .054


Early calculations assumed, implausibly, that everyone was equally likely to catch the disease, and concluded that herd immunity required about 80% immune. Dropping that assumption lowers the number, since as the more at risk people get infected, die, or recover, the average vulnerability of the population falls. By how much it lowers it is not known. In my calculations I assume that 60% does it. That is the point at which the disease just reproduces itself. As more people get infected and either die or become immune, the number infected starts to go down.

The second problem is that, while we have reasonable estimates of how many people die, we do not know how many have been infected, since many infections are not detected. I am using the estimate of 11 from the CDC, but they report a range of possible values from 6 to 24.


These numbers let me calculate mortality from the model:

Cases so far: 8.35 million

Infections so far: 8.35x11=92 million

U.S. population: 328 million

Required for herd immunity: 197 million

Additional infections required: 197 million – 92 million = 105 million

Resulting mortality: 105 million x .0015 = 158,000

This is not the total mortality resulting from my model, since herd immunity is only the point at which, without precautions, infections stop increasing.

Suppose we want to maintain quarantine until we reach the point where dropping it will result in no more than ten Covid deaths/week. If N is the number of infected individuals at that point and Ro is 2, meaning that if nobody was immune or taking special precautions, each person would pass on Covid to 2 others over a contagious period of about two weeks, then the number who get Covid in the next week will be N x (fraction of the population not immune) =N[.09 (the people just leaving quarantine)x236/328 (fraction of them not immune)] +N/total population = .065N +N/328,000,000. The number who die will be that times the infection mortality rate for people 70+, since at this point most of the under 70’s will be immune. Ignoring the second term, which is tiny, we have .054x.065N = .0035N =10. So N = 10/.0035 = 2900. 

So if we maintain quarantine until there are only 2900 cases, dropping quarantine will result in about ten deaths a week from Covid.

Timing Calculations

How long does the process take before it is safe to end quarantine?

Numbers from various online sources:

Total staffed hospital beds: 924,000  

ICU beds, "medical surgical" or "other ICU" (not counting neonatal ICU, burn care, etc.): 63,000

I assume that half of the beds can be used for Covid patients.

Percent of cases requiring medical care: 20%

From the CDC figures, 14% go to regular beds for an average of 3.5 days.

6% go to the ICU and have an average hospital stay of 12 days.

The CDC page does not say how much of that time is in the ICU, but I found another source that reported a median length of stay in the ICU, for studies outside of China, of 7 days. That source gave a median for total hospital stay outside of China of 5 days, which is higher than the CDC figure, so I take its ICU length of stay figure as a high estimate and use it. That implies that cases that go to the ICU consume 7 days of ICU care plus 5 days of ordinary care.


It follows that each case consumes, on average, .42 days of ICU care and .79 days of regular care. So 462,000 regular beds can handle 585,000 cases a day but 31,500 ICU beds can only handle 75,000 cases a day, making the ICU beds the bottleneck. The number of infections is 11 times the number of cases, so the hospital system can handle the result of 825,000 infections a day.


The herd immunity figure I have been using so far is for the whole population, including those in quarantine. About 9% of the population are 70 or over, so the not-quarantined population is 91%. They reach herd immunity with 96 million infections. At the maximum the ICU beds can handle, that takes 116 days or about 17 weeks. At that point the number of infections starts to decline and the ICU beds are no longer at capacity.


The hospitals are handling 5.775 million infections/week, or about 1.9% of the not-quarantined population. Using a spreadsheet, I calculated that by week 43, the number of infections would be down to 2900. At that point 44% of the non-quarantined population have been infected, so total mortality is .0015 x .44 x 298,000,000 = 196,000.


The current death rate from Covid is about 750/day. Suppose we assume that mass vaccination sufficient to reduce that to near zero will occur in six months, which seems if anything a bit pessimistic. At the current death rate, that results in about 135,000 deaths. Since I am assuming mass vaccination by week 26, I ought to cut off my model at that point as well. That drops the number infected to 43%, reducing mortality to 192,000.It follows that if the numbers in my model are correct, we are probably better off not following the model, at least as judged by number of deaths.

What Might Change the Conclusion?

If my model and my assumptions are correct we are better off not following the model, at least measured by mortality. Many of the assumptions are uncertain, however, and the difference between the results of the two strategies is not all that large, which raises the question of whether there are plausible changes in either the model or the parameters that would reverse that conclusion.

Tweaking the Model

One possibility would be to include in the quarantine people under seventy who were for one reason or another at unusually high risk, thus bringing down the mortality rate for those not in the quarantine.

What Might Change the Conclusion?

The mortality figures are not very sensitive to the assumptions that went into my calculation of how long the process would take, although the timing is. The three parameters that could substantially alter the result are the ratio of infections to cases, the mortality rate estimates, and the requirement for herd immunity.

The CDC gives ranges for the first two. At the high end of the range for the ratio of infections to costs, we are almost at herd immunity already, so the mortality costs of the model would be much less. At the low end of the mortality rate estimates, total mortality is about half as great, reversing the conclusion. The same would be true of any substantial reduction in the requirement for herd immunity.

The conclusion is also, of course, sensitive to the assumptions about the alternative to the model. If death rates rise significantly or if mass vaccination takes longer than I assume, that might raise the mortality from the present strategy above that of the model.

As should be obvious, my conclusions are uncertain, both because I am working with a simplified model and because many of the relevant parameters are uncertain. And I am ignoring lots of practical issues associated with mass quarantines. But a back-of-the-envelope calculation is still better than nothing.

Commenters are invited to try to duplicate my calculations and see if I have made any mistakes — I have found and corrected several in the past day.


Mark P said...

Dr. Friedman - Have you considered using Sweden as a control for your model? They have, generally, followed the advice offered in the Great Barrington Declaration throughout the entire pandemic (schools never closed, no masks, very light business regulation if any, etc). They do criticize themselves for not more effectively isolating the elderly at the beginning, so their actual numbers are going to be notably worse than a model that assumes perfect protection of the elderly.

The WHO just published a survey of the known seroprevalence studies (, and it computes an IFR of .57% (comparable to the CDC aggregate) and seroprevalence of 6.3% in Sweden in late May. That IFR is comparable to the CDC estimates in aggregate.

Seems that if your model is to be predictive, it should be able to explain the trajectory in Sweden ( They have had no appreciable death count since the end of July. And for your modeling purposes, they never had excess pressure on their health care system. They did prepare for overflow capacity at the peak, but my understanding is they never had to use it.

Mark P said...

Just noticed it is easy to find the breakdown of deaths in Sweden by age (e.g. As a rough correction of the actual mortality data in Sweden for your model, it seems you could subtract out the 70+ deaths (assume they were protected and survived).

Looks like that site gives 5269 70+. That leaves 653 deaths for your model to account for, from the start of March until now.

Anonymous said...

You assume that we are capable of quarantine. That is the basic problem.

If we simply quarantined everyone everyone who tested positive, or even just everyone with obvious symptoms, we would be in much better shape. This is a much smaller population than those over 70.

Mark P said...

Dr. Friedman - I see that Sweden just today removed their recommendations specifically for those 70+; they now default to the recommendations for everybody (

That means your model of isolating the elderly will start to diverge from the Sweden experience, as those 70+ will feather into the broader society more. But it still seems that your model should be applicable to the ~650 <70 who died since the pandemic started.

Michael Wolf said...

Unless I am completely misunderstanding it, the link to the study Mark P posted gives an IFR of 0.27%, not 0.57%.

"Across 51 locations, the median COVID-19 infection fatality rate was 0.27% (corrected 0.23%)." - directly from his link to the Ioannidis serological study.

The CDC currently estimates the following:
0-19: 0.003%
20-49: 0.02%
50-69: 0.5%
70+: 5%


Mark P said...

Michael Wolf - You are reading that right; the median IFR in the paper is .23%. The survey does include a study from Sweden, however, so that is the number I quoted. Given that rate aligns roughly with the CDC estimate, it would imply Sweden is a reasonable control for the model.

One other note: Sweden publishes their ICU utilization ( That should provide good data to check the model assumptions/inputs.

Anonymous said...

Dear Professor Friedman,
Please, would you read this comment by chemist S. Fowkes?

It seems to me that your calculations could be very different if there was a cure, even if it was unpopular with the Medical Doctors. Having a real cure –for example, vtiamn C– would render irrelevant both strategies, namely the "quarantine until mass vaccination which may take years" and the "focused protection which may kill people" which Professor M. Kulldorff has been advocating for since March.

Many times in the past have Doctors become obsessed with a purported cure (leeches, mercury) to the exclusion of anything else. Pride seems to be a problem sometimes. Unlike many other things, vitamin C seems to be harmless. Just because of that I think trying out vitamin C as a cure should be considered appropriate.

I propose that everyone pretends that the compound "sodium ascorbate" is a new synthetic drug discovered by accident, and the State grants a nice patent on it, after coming up with a nice, catchy new name. If this small lie meant to boost the revenues from selling the cure, is what we need to save many, many, many lives, I would vote for it right away.

Is it possible to save millions by corrupting the Law a tiny little bit?

Anonymous said...

The link did not appear.

Bloggophereo said...

David, Off-Topic, but have you expressed your thoughts anywhere on what should be done, if anything, to help the economy in general, and especially impacted industries (travel, restaurants) in particular? Would love to hear your opinion here.

- Bobboccio

David Friedman said...


One useful thing would be a mechanism by which individuals could easily show either that they had had the disease and were now cured and presumptively immune, or that they had recently tested negative. Both groups, especially the first, could then be selectively employed in jobs where not being contagious was especially important, such as working with an old folks home, with a salary bonus. One possible result would be to give young adults, for whom the risk is very low, an incentive to get infected in order to get such a job. That would be a more selective version of the herd immunity approach.