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# COVID-19 ALGEBRA

A RECENT POSTING in the AAAS Member Community Digest discusses the relationship between a disease’s herd immunity and its R0, “R-naught.”

Indeed, it turns out that, to be successful, the herd immunity percentage must be greater than 1 – 1/R0.

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. R0 is a measure of an infectious disease’s average rate of transmission. For example, suppose Covid-19’s R0 = 5.7 (see CDC data, July 2020). This implies that an infected individual is likely to infect nearly six other people.

Thus, R0 greater than 1 leads to an outbreak. R0 = 1 describes a disease that will persist in a population, but without an epidemic. If R0 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 R0, 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 = sR0, where s is the percentage of susceptible.

Now suppose a disease has an R0 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, sR0 < 1.

Rearranging algebraically, s < 1/R0.

“On other words,” researchers say, “we need to get the proportion of susceptible people in the population to less than 1/R0.”

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/R0. Q.E.D. (Quod Erat Demonstrandum, “that which was to be demonstrated.”)

Covid-19 Implications. For example, given an uncontrolled R0 = 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 R0 to 1.5, say, then herd immunity would come at 1 – 1/1.5 = 33 percent. ds