Herd immunity is a sliding scale.
You can treat herd immunity as a sliding scale, but you can treat it as a hard threshold as well.
In the hard threshold sense it means that if you infect a random individual in the immune herd, the disease does not spread. It might infect a few other people, but it will not spread throughout the entire (non-immunized) herd, it will die out locally without any need for a quarantine.
Mathematically, you need a model that describes how the disease spreads in a given population. Plug in the numbers and calculate the expected number of people infected by a sick person. If it's greater than 1, the disease will spread, if it's less then 1, the disease will die out locally and the herd is immune.
The spreading of deseases sounds like it would be modeled quite well using Percolation Theory, although on the applications page there is mention but no explanation of epidemic spread.
The interesting thing about percolation theory is that in that model both DanielLC and Lumifer would be right: there is a hard cutoff above which there is zero* chance of spreading, and below that cutoff the chance of spreading slowly increases. So if this model is accurate there is both a hard cutoff point where the general population no longer has to worry as well as global...
Another month, another rationality quotes thread. The rules are: