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