### 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.

### Calculations

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.

### Conclusions

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.