## 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

Zbl 1176.92042
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