# Mitigation Multiplies

I hesitate to join the ranks of the abundant newly-minted amateur epidemiologists out there. But here goes.

I’ve heard many people claim that a population can’t reach herd immunity to *any* disease until 60-70% of the population has been infected. This is a grave overgeneralization; as this primer explains, the critical value for herd immunity directly depends on how contagious the disease is:

The critical value, or threshold, in a given population, is the point where the disease reaches an endemic steady state, which means that the infection level is neither growing nor declining exponentially. This threshold can be calculated by taking

R_{0}, the basic reproduction number, or the average number of new infections caused by each case in an entirely susceptible population that is homogeneous, or well-mixed, meaning each individual can come into contact with every other susceptible individual in the population,^{[9]}^{[29]}^{[42]}multiplying it byS, the proportion of the population who are susceptible to infection, and setting this product to be equal to 1:

Thus, since measles is ultra-contagious, you don’t reach herd immunity until over 90% of the population is immune. The critical value for flu, in contrast, is barely one-third of that. Estimates of the critical value for COVID-19 range from 29-74%. Check out this table.

The standard way of boosting immunity, of course, is vaccination. As vaccination goes up, *S* goes down – and society moves toward herd immunity. Yet in principle, *anything* that reduces susceptibility to contagion has the same effect. If people wash their hands more, that effectively reduces *S*. If people socialize less, than reduces *S*. If people stop shaking hands, that reduces *S.* If people live in a low-density area, that reduces *S. *

Critically, all of these defensive factors *multiply*. If washing your hands more, socializing less, ending handshakes, and living in West Virginia each reduces *S* by 2%, then doing all four reduces *S* to .98^4=92.2% of its original value. It’s the same as the logic of contraception. Suppose you use birth control pills, condoms, withdrawal, *and* 50% abstinence. Your probability of pregnancy equals your base probability, multiplied by the failure rates of all four methods.

Some of my friends overrate *R*_{0}, imagining that getting *R*_{0}<1 prevents full infection of the population. But with a finite population (i.e., any actual population), you can infect *everyone* with an *R*_{0 }far below 1. Suppose your society has 15 people, and 10 are already sick. Then even with *R*_{0}=.34, all 15 people eventually catch the disease. After all, the infinite sum of 10 + .34*10 + .34^2*10+…=10/(1-.34)>15.

That’s the bad news. The good news is that infection and vaccination are only the two best-known ways to effectively shrink *S*. An endless list of behaviors are not only on the shelf, but are already in play. Paid voluntary human experimentation would help us find the low-hanging behavioral fruit in a matter of weeks, but at least almost everyone today bases their precautions on germ theory rather than witchcraft.

Additional good news: Fatality rates vary dramatically by age and even more dramatically by health status. So once you multiply the effect of all forms of mitigation to figure out when we’ll hit herd immunity, you *shouldn’t* mechanically multiply that by the *average* infection mortality rate to estimate total deaths. As Dan Klein puts it:

Let’s look at Wikipedia’s definition of herd immunity. It is based on a threshold condition where the still-susceptible proportion of the population multiplied by

R0equals 1.What is

R0? It is “the average number of new infections caused by each case in an entirely susceptible populationthat is homogeneous, or well-mixed” (Wikipedia, italics added).Well-mixed is what we strive to prevent! We strive to separate the vulnerable from the rest of the population.

I’m not so optimistic about the “we” part. But *you* can definitely do so. If you’re vulnerable, take high precautions. If you’re not vulnerable, give the vulnerable their space.

P.S. Yes, you could just as easily say that behavior affects *R*_{0 }rather than *S*, but I thought my way was a bit clearer.

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