## Journal of Applied Probability

- J. Appl. Probab.
- Volume 52, Number 4 (2015), 1195-1201.

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

#### Abstract

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.

#### Article information

**Source**

J. Appl. Probab., Volume 52, Number 4 (2015), 1195-1201.

**Dates**

First available in Project Euclid: 22 December 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1450802763

**Digital Object Identifier**

doi:10.1239/jap/1450802763

**Mathematical Reviews number (MathSciNet)**

MR3439182

**Zentralblatt MATH identifier**

1342.60150

**Subjects**

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92D30: Epidemiology

Secondary: 05C80: Random graphs [See also 60B20] 60J28: Applications of continuous-time Markov processes on discrete state spaces

**Keywords**

Multitype branching process exponential tail approximation Gumbel SIR epidemic

#### Citation

Windridge, Peter. The extinction time of a subcritical branching process related to the SIR epidemic on a random graph. J. Appl. Probab. 52 (2015), no. 4, 1195--1201. doi:10.1239/jap/1450802763. https://projecteuclid.org/euclid.jap/1450802763