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Threshold behaviour and final outcome of an epidemic on a random network with household structure. (English) Zbl 1176.92042

Summary: We consider a stochastic SIR (susceptible\(\rightarrow \)infective\(\rightarrow \)removed) epidemic model in which individuals may make infectious contacts in two ways, both within ‘households’ (which for ease of exposition are assumed to have equal size) and along the edges of a random graph describing additional social contacts. Heuristically motivated branching process approximations are described, which lead to a threshold parameter for the model and methods for calculating the probability of a major outbreak, given few initial infectives, and the expected proportion of the population who are ultimately infected by such a major outbreak. These approximate results are shown to be exact as the number of households tends to infinity by proving associated limit theorems. Moreover, simulation studies indicate that these asymptotic results provide good approximations for modestly sized finite populations. The extension to unequal-sized households is discussed briefly.

MSC:

92D30 Epidemiology
60K35 Interacting random processes; statistical mechanics type models; percolation theory
05C80 Random graphs (graph-theoretic aspects)
60J85 Applications of branching processes
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
65C20 Probabilistic models, generic numerical methods in probability and statistics
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