Herd Immunity for "S-I-R" Diseases

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:

Smallpox_1.gif

where s is the fraction of the population that is susceptible, i is the fraction infectious and the parameters a and b depend on the particular disease (and can be measured from real epidemics.)  The model assumes there are no births or deaths (s+i+r=1), which is a reasonable simpification for a short-term epidemic.

The "contact number" c of a disease can be measured by testing the susceptible fraction before and after an epidemic, Smallpox_2.gif.  The number b is 1/the number of days a person is infectious and a =b c.

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 s[0]<1/c=b/a.  This quantity  b/a is about 0.06 for measles, 0.14 for rubella, and 0.22 for polio.  This shows why we no longer have polio epidemics, but still have outbreaks of measles. About 5% of vaccinations do not confir immunity, so measles requires 99% vaccination.

The limit Smallpox_3.gif represents the fraction of the population left after an epidemic.  You can see this by sliding the final time until i[t] 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 un-affected, 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 Chapter 2 of the text Calculus: The Language of Change, on this website.  Wikipedia has an interesting entry on Smallpox vaccination.

Spikey Created with Wolfram Mathematica 8.0
Herd Immunity for Smallpox from the Wolfram Demonstrations Project by Keith Stroyan