The extinction time of a subcritical branching process related to the SIR epidemic on a random graph

TL;DR

This paper derives an exponential tail approximation and a Gumbel limit law for the extinction time of a subcritical multitype branching process modeling an SIR epidemic on a random graph with unbounded degrees, allowing for countably many types.

Contribution

It extends previous models by allowing countably many types and only requiring a second moment condition for the offspring distribution.

Findings

Exponential tail approximation for extinction time.

Gumbel limit law for large populations.

Applicable to models with unbounded degrees.

Abstract

This note gives 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 infinity). We only require a second moment for the offspring-type distribution featuring in our model.