Each transition is modeled as a Poisson process with a variable rate parameter.
Specifically, the firing rate of the process is equal to its rate constant times the product of its input population counts.
(EDIT: this is only true to a first approximation, for large population sizes. The refinement is described in comment 14.)
Suppose in our example, that the transition rate for infection was 2.1, and the transition rate for recovery was 7.9.
Then in state \\(x = (5,2,1)\\):
* The firing rate for infection would be \\(2.1 \cdot S \cdot I = 10.2\\).
* The firing rate for recovery would be \\(7.9 \cdot I = 15.8\\).
And to say that the firing rate for a process is \\(z\\) means the following: in an infinitesimal time interval \\(dt\\), the probability of the process firing is \\(z * dt\\).
I.e., in the limit as dt goes to 0, Prob(firing)/dt = z.