POISSON APPROXIMATION FOR THE FINAL-STATE OF A GENERALIZED EPIDEMIC PROCESS

A so-called generalized epidemic model is considered that describes th e spread of an infectious disease of the SIR type with any specified d istribution for the infectious period. The statistic under study is th e number of susceptibles who ultimately survive the disease. In a pion eering paper, Daniels established for a particular case that when the population is large, this variable may have a Poisson-like behavior. T his 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 equival ent Markovian representation of the model and the use of a suitable co upling via a random walk.