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**An SIR epidemic model on a population with random network and household structure, and several types of individuals.**
*(English)*
Zbl 1236.92043

Summary: We consider a stochastic SIR (susceptible \(\rightarrow \) infective \(\rightarrow \) removed) epidemic model with several types of individuals. Infectious individuals can make infectious contacts on two levels, within their own ‘household’ and with their neighbours in a random graph representing additional social contacts. This random graph is an extension of the well-known configuration model to allow for several types of individuals. We give a strong approximation theorem which leads to a threshold theorem for the epidemic model and a method for calculating the probability of a major outbreak given few initial infectives. A multitype analogue of a theorem of F. Ball, D. Sirl and P. Trapman [ibid. 41, No. 3, 765–796 (2009; Zbl 1176.92042)] heuristically motivates a method for calculating the expected size of such a major outbreak. We also consider vaccination and give some short numerical illustrations of our results.

### MSC:

92D30 | Epidemiology |

05C80 | Random graphs (graph-theoretic aspects) |

60J85 | Applications of branching processes |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |

05C90 | Applications of graph theory |