The threshold limit theorem

Abstract

We will now explore in greater detail the large population limit of the final size distribution for the standard SIR epidemic model E_n,m(λI).We have seen (Section 3.3) that, if the population of susceptibles is large and we introduce a small number of initial infectives, the number of infectious individuals behaves like a branching process in the beginning. If the basic reproduction number R_o= λι is less than or equal to 1, a small outbreak will occur. On the other hand, if R_oexceeds 1, then there is a positive probability that the approximating branching process explodes; this implies, of course, that the branching process approximation will break down after some time. Then it is reasonable to expect that the final epidemic size will satisfy a law of large numbers. This indicates that the asymptotic distribution of the final size actually consists of two parts, one close to zero arid the other concentrated around some deterministic value. In this chapter we sketch the derivation of these results, using the Sellke construction and the beautiful imbedding representation of Scalia-Tomba (1985, 1990). A fluctuation result for the final size, giver a large outbreak, will also be given. In the final section we combine earlier results and indicate the proof of a theorem due to Barbour (1975) on the duration of the (Markovian) standard SIR epidemic.