Sunday, October 18, 2020

The Cost of Getting to Herd Immunity: A Back of the Envelope Calculation

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, in lives and money, of such a policy would be, so I decided to do one.


The numbers are very uncertain, for at least two reasons. One is that we do not know what percentage of the population must be immune to reach herd immunity. 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. 

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 asymptomatic and not detected. I am using infection mortality calculations by John Ioannidis along with an estimate from the CDC a while back that the number infected in the U.S. is about ten times the number of known cases. The two are roughly consistent, at least for Santa Clara Country where I live, which happens to provide quite detailed information on mortality. Using those assumptions, and assuming a policy that protects everyone 70 or over, I get:

7.7 million known cases so far, implying 77 million infections which is 23% of a population of 331 million

Required to reach herd immunity, an additional 37% or 122 million infections

Infection mortality rate for people under 70, Ionidas data for Santa Clara County, .07%.

.0007x122 million = 85,000 deaths.

The mortality figures assume adequate hospital space, so the next question is how long the process would take if done at a rate that does not overload the hospitals. To calculate that, I use the following numbers , based on a web search:

Total staffed hospital beds: 924,000 

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

 The following are much rougher numbers, also based on webbed information.

Time in hospital, non-ICU, two weeks

Time in hospital, ICU cases, 1 week ICU + 1 week non-ICU

Since I am assuming that only a tenth of infections show up as known cases, 122 million infections imply 12 million cases. According to webbed information, 20% of cases end up requiring hospital care, of which 42% go to the ICU. From my assumptions, I get:

2.4 million hospitalized, of which 1 million are in the ICU. So total non-ICU load is 3.8 million non-ICU patient weeks, 1 million ICU patient weeks. If we assume that half of both sorts of beds are being used for non-Covid patients, that implies that we could provide the non-ICU beds in a little over 8 weeks, but that the ICU beds would take 32 weeks. We should allow about another six months (guesswork — I haven't done calculations) for the infection rate to get low enough so it's safe to end quarantine.

[Correction: This assumes that the ratio of hospitalization to infections is independent of age. If we instead assume that it changes with age in proportion to mortality, that lowers my hospitalization figures by about a factor of three, so time until herd immunity is only about 11 weeks. 

I have now worked out the numbers on that assumption and, if my calculations are correct, if you end quarantine at 31 weeks, deaths in the next week due to Covid should be one or zero.]

This analysis assumes that we can control the rate of infection in the younger than seventy population, probably by varying the strength of the sort of restrictions that have been used — limits on large meetings, restaurant seating, and the like. 

What about the cost of older people quarantining? Currently, about 30 million people are seventy or over. Almost all of them are retired, so quarantining does not reduce their income. It does increase some costs, and it makes life substantially less pleasant. Figure pecuniary cost, mostly the cost of having groceries delivered instead of shopping for them, of $10/week. Assume ten percent of the people are not retired and so require an income subsidy of $20,000/year. Run the program for a full year, to allow enough time after herd immunity is reached for the infection to almost disappear, and the total monetary cost is about $76 billion

Final conclusion, based on lots of very uncertain assumptions — this is a back-of-the-envelope calculation:

Cost in lives: 85,000

Cost in money: $76 billion

Time until we can go back to normal for everyone: 1 year or until mass vaccination, whichever is shorter.

Compare that to the current policy. The U.S. death rate is about 5000/week, so it will take 17 weeks of it to kill 85,000 people. Nobody, with the possible exception of President Trump, believes that we will have mass vaccination that soon. So on these figures the herd immunity costs fewer lives, fewer dollars — current subsidies have been measured in trillions — and much fewer restriction.

It does not follow that we should do it, because there is a lot of uncertainty in my calculations. I am accepting John Ioannidis' calculations for mortality, which are controversial. I am ignoring costs such as the problem of separately housing elderly people and younger people who currently live with them. I'm using beds as the relevant measure of hospital capacity, rather than medical personnel. I am assuming that there is no way of substantially expanding ICU capacity, even with considerable excess capacity in non-ICU beds. I am using hospitalization and ICU figures based on current experience, although that experience is heavily weighted towards older patients likely to have more serious cases. I am ignoring tweaks to improve the program, such as identifying the most at risk people under seventy and having them quarantine too, thus bringing down the mortality rate of those not quarantining.

My conclusion is not that we should do it — I don't know enough. It is that the proposal is not absurd, might be an improvement.

Throughout my calculations I have assumed that the quarantining of the elderly is perfectly successful, which is unlikely. My model is for the government to encourage and subsidize self quarantining, not require it — any elderly people who want to risk infection, with a probability of death if infected at about 5%,  are free to do so, and some will. In the worse possible case, where all of the elderly choose to break quarantine and all of them get infected, that would be an additional 1.5 million deaths. [Correction — that was using the case mortality ratio rather than the infection mortality ratio. If I assume the same ratio of cases to infection I have been using, which is probably wrong for the older population, the figure drops to 150,000 deaths. I'm not sure where between those numbers is realistic.] How many it actually will be will depend on how many choose to break quarantine and how many of those get infected.

One more question. Suppose we had followed this policy from the beginning. Using the same assumptions, I get:

Total deaths: 139,000

Time to herd immunity: about a year and a half — or until mass vaccination, whichever is shorter.

Monetary cost: $120 billion, assuming no mass vaccination.

Under our current policy, total deaths are 219,000 so far, likely to run to something close to 400,000 by the time we have mass vaccination. Total monetary cost is hard to estimate but at least an order of magnitude bigger. Total non-monetary human cost probably much larger as well.


Benjamin Cole said...

Well done. Sensible, practical.

Jim Ancona said...

As I understand it, your death estimate of 85,000 only includes people under age 70. Are you assuming that no one over age 70 will get the disease once this protect the elderly strategy goes into effect? Perhaps you or the GB Declaration folks could explain in detail what that strategy is. Otherwise this feels very much like an "Assume a can opener" solution.

Something like 40 to 60 percent of COVID deaths in the US are tied to nursing homes and other long-term care facilities. I don't know how much experience you have with nursing homes or other facilities for the elderly. In my limited experience, the workers there are mostly young and poorly educated, working for near the minimum wage. Many are immigrants. They are not well-trained in infection control or use of PPE. My understanding is that many work part-time at multiple facilities in order to make ends meet, meaning that it's easy for the disease to spread between facilities. Because they're part-time, they don't get sick leave, and thus have an incentive to go to work when ill and even to conceal their illness from their employers in order to keep getting paid.

If we had proven strategies for protecting the elderly in such facilities, the strategy for everyone else wouldn't matter nearly so much, because the rest of us have the ability to take responsibility for our own safety, while people in institutions can't. But in my opinion, anyone arguing for a herd immunity with protecting the elderly strategy should have a credible explanation of how the "protect the elderly" part will work in order to be taken seriously.

Mitch said...

Is wearing an N95 mask considered to be a proven strategy for the elderly (and everyone else) to protect themselves? These seem to be becoming available and don't require the goodwill of others for airborne mitigation.

Michael Wolf said...

The most important thing isn't how close to the mark your back of the envelope calculation is, or isn't. It seems at least a reasonable first take, on which to build.

The important Q is why, 7 months since unprecedented lockdowns started, is a blog post on David Friedman's blog the only serious attempt to even begin to make such a calculation? Your back-of-the-envelope estimates are more than we've gotten in the way of a cost/benefit analysis than any government or public health official has given us in 7 months.

Which is criminally insane.

David Friedman said...

@Jim Ancona:

I discussed that in the post:

"Throughout my calculations I have assumed that the quarantining of the elderly is perfectly successful, which is unlikely. My model is for the government to encourage and subsidize self quarantining, not require it — any elderly people who want to risk infection, with a probability of death if infected at about 5%, are free to do so, and some will. In the worse possible case, where all of the elderly choose to break quarantine and all of them get infected, that would be an additional 1.5 million deaths. How many it actually will be will depend on how many choose to break quarantine and how many of those get infected."

Jared said...

What are your thoughts on Robin Hanson's idea of variolation along with your strategy? I feel like it would cut the cost and death rate down by a lot from your current plan.

David Friedman said...


If Robin is right about variolation, it would cut the death rate a lot. But I'm trying to focus on a single issue.

A1987dM said...

Another thing you're neglecting is the (monetary and non-monetary) cost of people under 70 being infected. I'd be willing to pay several hundred dollars to avoid getting COVID as badly as my girlfriend (in her 30s) got --- and for people who unlike me cannot work from home, you'd have to add the lost productivity from the time they cannot work (one month and a half in the case of my girlfriend).

Anonymous said...

David, two issues I see - 1) what is the likelihood that one infection grants long-term immunity? I think very low, as humans don't develop long-term immunity to any other coronavirus. Why would you suspect this one is different?

2) Herd immunity wouldn't mean the end of the virus, it would just mean the end of the out of control phase. A long-term simmering disease akin to measles or mumps wouldn't be much fun either.

- Bobboccio

gdp said...

With respect, "Herd Immunity" does not mean what you think it means: It is the condition for an epidemic to NOT START, not the condition for an epidemic to end.

The most basic model of epidemics is the so-called "SIR Model", in which the rate of growth of the infected subpopulation is proportional to the fraction of the population that is "susceptible" (S) times the fraction of the population that is "infected" (I), minus the rate at which infected patients transition from the "Infected" to "Recovered" state which is proportional to the "Infected" population. Note that because of the product term S*I in the growth-rate, the dynamics of an epidemic are nonlinear.

When a negligible fraction of the population is initially "Infected", then if less than 1/R0 of the population is initially "Susceptible" (where R0 is the "basic reproduction number" of the disease organism, a measure of how "infectious" it is), then the growth rate of the "Infected" subpopulation is negative, and small "clusters" of infection "fizzle out"; this is so-called "Herd Immunity".

However, once an epidemic is already underway, the nonlinear dynamics of an epidemic are quite different: The fraction of infected persons grows roughly exponentially until it peaks, then exhibits a much longer decay. Final exit from the epidemic does not occur until a much more stringent condition is satisfied, which to a first approximation is that the remaining susceptible populations is exponentially small on the order of 1/exp(R0). For details see the discussion in:

Best current estimate of R0 at this time is that R0 for SARS-CoV-2 is at least 2.4, and quite possibly in excess of 3. That means that the most optimistic estimate is that this epidemic does not end until at least 91% of the population has had CoViD-19 and either recovered or died. --- i.e., basically everyone.
So you need to revise your numbers, by A LOT.

The whole point of "social distancing" is to try to reduce the "effective R0" of the epidemic. R0 can be factored as {number of contacts per unit time}{probability of transmission per contact}{duration of the infectious period}. By decreasing the number persons one contacts each day ("social distancing") and the likelihood of transmission ("physical distancing", masks and gloves, frequent hand-washing), the effective R0 can be reduced, terminating the current "wave" earlier with fewer dead, while not overwhelming the hospital system.

But what is being falsely called "Herd Immunity" is NOT an effective strategy, because it underestimates the number of people who will get sick and th number of people who will die by an exponential factor. It is tantamount to saying "Let essentially the entire population get sick, and accept that a huge number of people will die".

David Friedman said...


As you can see if you look at the current version of the post, I have worked out the numbers for how long after herd immunity is reached you have to wait before the number infected is low enough so that, when you drop quarantine, the death rate from Covid is below one per week.

David Friedman said...

"That means that the most optimistic estimate is that this epidemic does not end until at least 91% of the population has had CoViD-19 and either recovered or died. --- i.e., basically everyone."(that's ninety-one percent — for some reason Firefox on my machine shows it that way in the original post but -1 in my response)

That is not correct. If R0 is 3, then once more than a third of the population is recovered and immune you are had herd immunity, which means that each infected person infects no more than one other. Further, R0 at the beginning of the pandemic is higher than R0 later, even with no precautions, because the population is not homogeneous, and the people more likely to get the disease are dropping out of the relevant population as they die or recover, which lowers R0. So if R0 starts at 3, it will be lower by the time herd immunity is an issue.

I am happy to have people criticize my analysis — that's why I posted it. But you first have to read it and try to understand it.

"because it underestimates the number of people who will get sick and the number of people who will die by an exponential factor."

"Exponential" does not mean "large." "Underestimates by an exponential factor" is gibberish. .5**N is exponential in N — 1/2,1/4,1/8, ...

Bloggophereo said...

David, love your analyses and your blog, please take my comments in that manner.

You addressed my comment about the meaning of herd immunity, but not about whether people gain immunity after getting sick. If we don't, any plan to get people sick now to avoid them getting sick later seems a little premature.

When discussing your post later, I had another critique... how would one go about speeding people along to herd immunity? The NHL cancelled their season before lockdowns and while I was still working in the office. The gov't wasn't the one who told them to take precautions. Would you have people ordered back to work? Ordered to stop taking precautions? I am not sure how you or anyone could intentionally speed along the populace to herd immunity. Just announce that our great nation wants herd immunity ASAP so please go lick a door knob immediately?

- Bobboccio

Ernie said...

The CDC planning estimates are here:

The CDC updated the scenarios 9/20/2020.
Notably, it provides a range of IFR (Infection Fatality Ratio) by age group. It also provides a range of Infections to Confirmed Case ratio.

Less back of the envelope but still highly uncertain. No cost benefit estimates. but one could make those estimates based on the various scenarios.

David Friedman said...


If people don't become immune for a substantial length of time after getting the disease, then neither herd immunity nor a vaccine is likely to work. But I think the evidence is overwhelming that almost everyone does get immune for at least six months, because millions of people have had the disease and there have been only a handful of cases that appear to be reinfections. If the latest version of my calculations is correct — and it may not be, of course — then in about eight months the disease is essentially eliminated. You then have the problem of keeping it from reappearing, but hopefully that can be controlled by testing, or revaccinating as needed, or other methods.

Bloggophereo said...

Well David, I hope you are right and people gain sterilizing immunity after infection. If not though, we still might find success with a vaccine with a periodic booster.

- Bobboccio

Mr Bitcoin said...

Has Sweden achieved herd immunity?

I think NY is close to herd immunity