A so-called generalized epidemic model is considered that describes the spread of an infectious disease of the SIR type with any specified distribution for the infectious period. The statistic under study is the number of susceptibles who ultimately survive the disease. In a pioneering paper, Daniels established for a particular case that when the population is large, this variable may have a Poisson-like behavior. This result was discussed later by several authors. In the present work, a necessary and sufficient condition is derived that guarantees the validity of such a Poisson approximation for the generalized epidemic. The proof relies on two key ideas, namely, the building of an equivalent Markovian representation of the model and the use of a suitable coupling via a random walk.
"Poisson Approximation for the Final State of a Generalized Epidemic Process." Ann. Probab. 23 (3) 1139 - 1162, July, 1995. https://doi.org/10.1214/aop/1176988177