Herd Immunity

Herd immunity is cool.  It's like, science, but it's also an intuitive theory that you can understand without performing Bayesian estimations.

So, epidemiologists like to study the spread of disease.  They noticed that some diseases spread more easily than others.  (Forgive me for channeling Contagion.)  By looking at how fast diseases spread in a population, you can estimate how many people the average sick individual infects--this number is called R₀ (pronounced "R-naught" by normal people and "Arrg-naut" by pirates).  That means that, on average, a sick person is capable of infecting R₀ people during their illness.  R₀ is different for every microbe, based on the microbe's properties.  The more and easier ways a microbe can be spread, the higher the R₀.  For instance, if an illness can only be spread by a child sneezing directly into your mouth from less than six inches away, it will probably have a lower R₀ than one that will spread to anyone touching a doorknob after the infected person for the next 12 hours.

Kate Winslet explaining R-nought in Contagion.

People hypothesized that if a proportion of the population were immune to a disease, transmission and then incidence would decrease.  If R₀ is the number of people on average infected by an individual, it makes sense that if (R₀-1)/R₀ of the population were immune, the effective transmission rate would decrease to less than one person per sick person, and incidence of the disease would decrease as a result.  So if you on average infected four people with a cold (R₀=4), then if at least (4-1)/4 or 3/4 or 75% of the population were immune, you would transmit your cold to fewer than one person on average.

With no one in a population immune and the R0>1, infections occur exponentially.  

This is usually what happens when an epidemic occurs--a large percentage of the population is susceptible to a disease, so the disease can spread rapidly through the population.  One example is the flu--because the virus mutates every year, the entire population is susceptible to it every year.  Its R₀ is slightly above 1.

With 50% of the population immune, or [(R0-1)/R0=(2-1)/2], the effective R0 becomes 1.   

Only one person is infected by each sick person, and the transmission rate drops.  The incidence (number of people with a disease at any one time) of the disease also drops--instead of having eight sick people by the fourth generation, there is only one sick person.  The disease will still persist in the population at a stable level.  This is usually what happens when an endemic disease is present--the percentage of immune people in the population prevent the disease from spreading exponentially, but a susceptible population continuously transmits the disease so it persists in the population.  Many childhood diseases act this way--while adults have often achieved immunity by, well, getting the disease as children, newer members of humanity continue to enter the world non immune and susceptible to infection.  Adults who escaped infection as children are often protected by the immunity of other adults until they become schoolteachers or start hanging out with their own children.

With slightly more than 50% of the population immune, R0 is less than 1, and herd immunity is achieved. 

If slightly less than one person is infected by each sick person, transmission will eventually stop and the incidence of the disease will decrease until it reaches zero:  basically, the disease will die out of the population.  This doesn't mean that transmission can't happen at all--a few unlucky susceptible people can still be infected before the disease reaches a host that does not pass it on--but it does mean that it will fail to become either epidemic or endemic.  The population (herd) as a whole will not support the disease.

One way to achieve herd immunity is through vaccinations.  If you need >(R₀-1)/R₀ of the population immune, and a vaccine successfully immunizes E proportion of people who receive it, then the amount of people who need to be vaccinated is greater than:

Ta-da!

So, if a vaccine immunizes 90% of the people who receive it, and R₀ is 2, then the number of people who need to be vaccinated to achieve herd immunity is ((2-1)/2)/.9=0.55, or >55% of the population.

Or, if you'd prefer a real-world example to my hypothetical illness, measles has an R₀=12-18.  If both an initial and booster dose of the MMR vaccine is given, the effectiveness of the vaccine is 99%.  So,
((12-1)/12)/.99 or ((18-1)/18)/.99 = 92-95% of the population needs to be fully vaccinated to achieve herd immunity against measles.

Fine, P., Eames, K., Heymann, D. "'Herd Immunity': A Rough Guide," Vaccines. (2011) 52:911-916