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Limit theorems for a random graph epidemic model. (English) Zbl 0928.92023

Summary: We consider a simple stochastic discrete-time epidemic model in a large closed homogeneous population that is not necessarily homogeneously mixing. Rather, each individual has a fixed circle of acquaintances and the epidemic spreads along this social network. In case the number of initially infective individuals stays small, a branching process approximation for the number of infectives is in force.
Moreover, we provide a deterministic approximation of the bivariate process of susceptible and infective individuals, valid when the number of initially infective individuals is large. These results are used in order to derive the basic reproduction number and the asymptotic final epidemic size of the process. The model is described in the framework of random graphs.

MSC:

92D30 Epidemiology
05C80 Random graphs (graph-theoretic aspects)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J85 Applications of branching processes

Keywords:

degree sequence
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References:

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