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A RECENT POSTING in the AAAS Member Community Digest discusses the relationship between a disease’s herd immunity and its R_{0}, “R-naught.”

Indeed, it turns out that, to be successful, the herd immunity percentage must be greater than 1 – 1/R_{0}.

Recognizing this gave me a better understanding about these two epidemiological terms. And verifying the relationship turns out to be reasonably straightforward algebra.

**Herd Immunity.** The human body is a marvelous device: Surviving a disease imparts future immunity to it, perhaps only briefly, perhaps for an extended period. A vaccine imparts the same immunity, with a similar caveat about duration.

Herd immunity occurs when enough of the herd has either survived a disease or been vaccinated against it. Also, note, as the Digest contributor wrote, “Once a population reaches herd immunity status, the infections don’t automatically stop. This is the point at which the infection rate begins to decrease.”

Eventually, the diminishing rate of infection reaches a point where even susceptible individuals have only insignificant likelihood of infection.

Herd immunity is measured as a percentage of the population and depends on the contagious rate of the disease. For example, protection against measles, which is highly contagious, requires a herd immunity of at least 95 percent. By contrast, polio, being less contagious, has a lower herd-immunity threshold, 80–85 percent.

**R-naught.** R_{0} is a measure of an infectious disease’s average rate of transmission. For example, suppose Covid-19’s R_{0} = 5.7 (see CDC data, July 2020). This implies that an infected individual is likely to infect nearly six other people.

Thus, R_{0} greater than 1 leads to an outbreak. R_{0} = 1 describes a disease that will persist in a population, but without an epidemic. If R_{0} is less than 1, the disease will decline and eventually be no longer significant.

**Let’s Talk Algebra.** The online magazine plus.maths.org is offered by Britain’s University of Cambridge.

I’ll use their notation here; I’ll follow U.S. practice of spelling it “naught.” The Cambridge researchers note that R_{0}, the basic reproduction number, is “the average number of people an infected person goes on to infect, given that *everyone in the population is susceptible.*” They define R to be the “effective reproduction number, the average number an infected person goes on to infect in a population where *some people are immune.”* That is, **R = sR_{0},** where

Now suppose a disease has an R_{0} greater than 1 and, thus, its epidemic threatens. However, researchers note, “if the *effective* reproduction number R is less than 1, the disease will eventually fizzle out.”

So, to achieve herd immunity, we need **R < 1**.

Equivalently, ** sR_{0} < 1**.

Rearranging algebraically, ** s < 1/R_{0}**.

“On other words,” researchers say, “we need to get the proportion of susceptible people in the population to less than 1/R_{0}.”

Given *s* represents susceptible people, then **1 – s represents the immune**, i.e., a state of herd immunity.

Hence, this **threshold of herd immunity = 1 – 1/R _{0}**. Q.E.D. (

**Covid-19 Implications.** For example, given an uncontrolled R_{0} = 5.7, then the minimum for herd immunity would require 1 – 1/5.7 or 82 percent of the population to have been infected. However, if intervention methods such as social distancing and masks reduce R_{0} to 1.5, say, then herd immunity would come at 1 – 1/1.5 = 33 percent. ds

© Dennis Simanaitis, SimanaitisSays.com, 2020

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