Applications of branching processes to the final size of SIR epidemics

Abstract

This paper considers applications of branching processes to a model for the spread of an SIR (susceptible $\to$ infective $\to$ removed) epidemic among a closed, homogeneously mixing population, consisting initially of m infective and n susceptible individuals. Each infective remains infectious for a period sampled independently from an arbitrary but specified distribution, during which he/she contacts susceptible individuals independently with rate $n^{-1}\lambda$ for each susceptible. The well-known approximation of the early stages of this epidemic model by a branching process is outlined. The main thrust of the paper is to use branching processes to obtain, when the infectious period is constant, new and probabilistically direct proofs of central limit theorems for the size of an epidemic which becomes established. Two asymptotic situations are considered: (i) many initial infectives, where m and n both become large, for which establishment is asymptotically certain; and (ii) few initial infectives, where m is held fixed and only n becomes large, for which asymptotically establishment is not certain and may not be possible. The model with constant infectious periods is closely related to the Erdös-Réenyi random graph and our methodology provides an alternative proof of the central limit theorem for the size of the giant component in that graph.

Mathematics Subject Classification (2000): 92D30, 60J80, 60F05, 05C80