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 benefits from partial vaccination (the reason for this is that people can be ordered geographically, so many people will only get a chance to infect people that were already infected. Therefore treating each new person as an independent source, as in Lumifer's expected newly infected number of people model, will give wrong answers).
*Of course the chance is only zero within the model, the actual chance of an epidemic spread (or anything, for that matter) cannot be 0.
I think percolation theory concerns itself with a different question: is there a path from starting point to the "edge" of the graph, as the size of the graph is taken to infinity. It is easy to see that it is possible to hit infinity while infecting an arbitrarily small fraction of the population.
But there are crazy universality and duality results for random graphs, so there's probably some way to map an epidemic model to a percolation model without losing anything important?
Another month, another rationality quotes thread. The rules are: