We give an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to ∞). We only require a second moment for the offspring-type distribution featuring in our model.
"The extinction time of a subcritical branching process related to the SIR epidemic on a random graph." J. Appl. Probab. 52 (4) 1195 - 1201, December 2015. https://doi.org/10.1239/jap/1450802763