Herd Immunity. Vaccination Programs

What is herd immunity? Well, everyone knows what immune means. Immune comes from the Latin word immunis, which means free from service, as in not having to go fight for the Roman army against the Visigoths. With Barbarians no longer at the gate, nowadays immunity usually refers to the status of being free from the possibility of acquiring a particular infectious disease. You can become immune either by having had a disease before or by receiving a vaccine.

Now, suppose you have a herd of livestock, and want to keep them free from hoof and mouth disease. After immunizing all but one of them, you wonder why on earth you're working so hard to track down the last calf. After all, even if this last calf were somehow infected, infection couldn't spread to the rest of the herd because all the others are vaccinated. In effect, the herd is immune, even though there is an individual calf that's not.

In fact, why even vaccinate the last 2? After all, if your herd is big enough, all the others are vaccinated, so no infected individuals will ever come into contact with the last calf. Then you get to thinking maybe you could get away with only vaccinating 90% or so of the herd maybe this would be enough that even if a calf were to get infected, most of the calves they come into contact with would be immune and an epidemic couldn't really get started.

Now you know that for an epidemic to get started, the R₀ has to be greater than 1. Remember that R₀ is the number of secondary cases that result when you introduce an infective into a totally susceptible population. What if you introduce a single infective into a herd that has been mostly vaccinated (so it is not a totally susceptible population)? How many secondary cases result? Well, even though the herd is no longer completely susceptible, the same principle can be applied. The average number of infectious cases caused by an infectious case introduced into a community in which a large number of individuals have been vaccinated can be called R^v₀. (We've got to call it something, after all!) This is equal to R₀ times the chance that an individual that you meet is susceptible. With a perfect vaccine and completely random mixing, this is just the chance that an individual wasn't vaccinated (1 minus the coverage rate, which for lack of imagination we'll call C):
 
R^v₀ = R₀ × (1-C)  

So now the question is what percentage of the herd do we have to vaccinate to prevent an epidemic from happening?

An epidemic won't take off as long as R^v₀ < 1. This is the same as saying that
 
1 < R₀ × (1-C)
 
or, using just a little bit of algebra,  

C < 1-1/R₀
 
Thus, the herd is in some sense protected, as long as we've vaccinated the proportion

C < 1-1/R₀.

For example, if the R₀ of a disease in a community is 2, then a coverage rate of just 50% will be enough to ensure that an epidemic won't really take hold. If the R₀ is 10, then 90% coverage would be necessary.

Bottom line: you can eradicate a disease by vaccination even if the entire population is not vaccinated.

Does this really work? Yes and no. Smallpox was clearly eradicated without vaccinating the entire population. Likewise, polio was eradicated from the Western Hemisphere without 100% coverage. Exact calculations though are difficult for several reasons, including the fact that there's a lot of heterogeneity in the world (what's good enough for one community may not be for another), and the fact that vaccines aren't always perfect.