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?
The main question of percolation theory, whether there exists a path from a fixed origin to the "edge" of the graph, is equivalently a statement about the size of the largest connected cluster in a random graph. This can be intuitively seen as the statement: 'If there is no path to the edge, then the origin (and any place that you can reach from the origin, traveling along paths) must be surrounded by a non-crossable boundary'. So without such a path your origin lies in an isolated island. By the randomness of the graph this statement applies to ...
Another month, another rationality quotes thread. The rules are: