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Herd Immunity for Smallpox

A population has "herd immunity" when the immune population is high enough so that if an infection is introduced, it dies out without building up. You can see this by sliding the initial immune fraction until the red graph does not increase in the illustration for smallpox.
The limit represents the fraction of the population left susceptible after an epidemic. You can see this by sliding the final time until is effectively zero.
This Demonstration shows the spread of smallpox in a large fixed population with contact number , infectious period , and daily contact rate, .

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Diseases like measles, mumps, rubella, and polio are referred to as "S-I-R" diseases because you are born susceptible (S) to the disease, can become infectious (I), and when you recover you are "removed" (R) and can neither transmit nor catch the disease again. These diseases are well-modeled by the differential equations:
where is the fraction of the population that is susceptible, is the fraction that is infectious, and the parameters and depend on the particular disease (and can be measured from real epidemics). The model assumes there are no births or deaths (), which is a reasonable simplification for a short-term epidemic.
The contact number of a disease can be measured by testing the susceptible fraction before and after an epidemic, . The number is the reciprocal of the number of days a person is infectious, and .
A population has "herd immunity" when the immune population is high enough so that if an infection is introduced, it dies out without building up. It is easy to show mathematically that herd immunity happens when . This quantity is about for measles, for rubella, and for polio. This shows why people no longer have polio epidemics, but still have outbreaks of measles. About 5% of vaccinations do not confer immunity, so measles requires 99% vaccination.
The limit represents the fraction of the population left susceptible after an epidemic. You can see this by sliding the final time until is effectively zero. In the smallpox illustration, an initial population with 60% immunity is left with only 5% unaffected, or 55% affected. Notice that when only 5% of the population is immune initially, less than 1% remain unaffected, or 94% are affected. Native Americans suffered large epidemics when European settlers introduced smallpox into their 100% susceptible populations.
More details on this model are in K. Stroyan's "Using Calculus to Model Epidemics," Calculus: The Language of Change, 2nd ed., San Diego, CA: Academic Press, 1998 pp. 23–43, which can be viewed on his website. Wikipedia has an interesting entry on smallpox vaccination.
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