TY - GEN

T1 - Applications of branching processes to the final size of SIR epidemics

AU - Ball, Frank

AU - Neal, Peter John

PY - 2010/1/22

Y1 - 2010/1/22

N2 - This paper considers applications of branching processes to a model for the spread of an SIR (susceptible → infective → 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 λ 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.

AB - This paper considers applications of branching processes to a model for the spread of an SIR (susceptible → infective → 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 λ 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.

KW - SIR epidemicsw

KW - final outcome of epidemic

KW - susceptibility set

KW - coupling

KW - total progeny of branching process

KW - central limit theorems

KW - random graphs

U2 - 10.1007/978-3-642-11156-3_15

DO - 10.1007/978-3-642-11156-3_15

M3 - Conference contribution/Paper

SN - 9783642111549

SP - 209

EP - 225

BT - Workshop on Branching Processes and Their Applications

A2 - González Velasco, Miguel

A2 - Puerto, Inés M.

A2 - Martínez , Rodrigo

A2 - Molina, Manuel

A2 - Mota, Manuel

A2 - Ramos, Alfonso

PB - Springer

T2 - Workshop on Branching Processes and Their Applications

Y2 - 20 April 2009 through 23 April 2009

ER -