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.

Tuesday, October 20, 2020

My Brief Talk About My Father

I was recently asked to record something for the ceremony inducting my father, and a number of other people, into the New Jersey Hall of Fame. The whole ceremony is now webbed. My talk starts at 1:04:19. Some here may find it of interest.

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.

Wednesday, October 14, 2020

Cryptography vs. Big Brother: How Math Became a Weapon Against Tyranny

Reason has now run the first two of four episodes in a video about encryption, the Cypherpunks, and related issues. So far they are very good, and I thought people reading this blog might find them of interest.

Part one:

Part two YouTube link:

Facebook link:



Tuesday, October 13, 2020

My First Political Donation?

 Probably not, but I can't remember any others. 

The Arkansas senate contest is a two person race between the Republican incumbent and the Libertarian candidate, the Democratic candidate having for some reason dropped out. Judging by his web page, the Libertarian candidate, Ricky Dale Harrington, is in favor of things all or most of which I am in favor of — and he is polling at 38%. I don't think he is likely to win but it isn't impossible, and a senate with 49 Republicans and one Libertarian plus a Republican VP to break ties, or 50 and 1 plus a Democratic VP, is a tempting vision, possibly the best even marginally plausible outcome I can imagine for this election.

Friday, October 09, 2020

Can Anyone Do Arithmetic?

My current minor irritation is Gavin Newsom, governor of California. He keeps saying things that imply that California needs to reduce its CO2 output in order to reduce global warming in order to prevent future forest fires.

I think it dubious that warming, about a degree C in the U.S. (I don't have figures for California) over the past century plus, has anything to do with the forest fires, but put that aside. Total global warming since it started c. 1913 has been about 1.4°C. The IPCC believes the rate has increased — suppose it is now three degrees a century. The population of California is about 1/200th of the world population, the GNP about 1/40th of the world GNP. I don't have data on CO2, so will guess it is 1/50th of world CO2 output. If California cuts that in half, it reduces world CO2 output by about 1%. Over a century, that might reduce warming by something like .03°c. All calculations are back of the envelope approximations and I know that not all functions in the world are linear, but that's enough to give us some idea of the scale of the effect.

Actually, I'm not so much irritated at Newsom — I expect politicians to be demagogues. I'm irritated at the rest of the world, all of the media and all of the people who take blatant nonsense seriously either because it is says what they want to hear or because they cannot be bothered to do arithmetic.

Trump is a demagogue too, but at least people call him on it.

Sunday, October 04, 2020

Free Audiobooks. Sort of. makes available vouchers for free audiobooks as part of the promotion process. I have such vouchers for The Machinery of Freedom and Future Imperfect.  I will give them to anyone willing to listen to a book and agree that, if he likes it, he will put a favorable review up somewhere that others can see it — a blog, Facebook, Amazon, ...  . 

I have vouchers for both the UK and the US marketplace. If you are interested, give me  contact information either here, via my email, or via FB.