The probability that a susceptible will escape \(R_0 I\) contacts on a population of size \(N\) is \((1-1/N)^{R_0 I}\) \(\approx \exp(-R_0 I/N)\). Thus, the expected number infected is approximately \(S (1-\exp(-R_0 I/N))\). I guess this doesn’t lend itself well to Poisson, but does lend itself to a cloglog binomial. Should investigate.