Earlier quoted context omitted.
And, to give an example we've probably all become familiar with, this this is equivalent to how a disease becomes an epidemic as the R value crosses 1.
When you say R, are you referring to R_0? Because R_0 is not a mathematical predictor of percolation.
To sketch here:
Consider R the reproductive number of a disease (or R_e, the effective reproductive number, to be specific). This depends on both the inherent contagiousness of the disease, but also on the behavior of the population. If people choose to have fewer contacts (or bars are closed) then R decreases.
Let's say covid is spreading. We ask people to limit their contacts, and see if this stops spread.
We can think of this as trying to remove edges from the contact graph that the disease spreads on.
The contact graph becoming connected or unconnected as we remove edges is clearly percolation.
(Subject to some modeling assumptions about edges being removed at random, but these assumptions are common to both Erdos renyi graph models and SIR style compartmental models).
Make sense?