## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 25, Number 2 (2015), 663-674.

### On the expected total number of infections for virus spread on a finite network

Antar Bandyopadhyay and Farkhondeh Sajadi

#### Abstract

In this work we consider a simple SIR infection spread model on a finite population of $n$ agents represented by a finite graph $G$. Starting with a fixed set of initial infected vertices the infection spreads in discrete time steps, where each infected vertex tries to infect its neighbors with a fixed probability $\beta\in(0,1)$, independently of others. It is assumed that each infected vertex dies out after an unit time and the process continues till all infected vertices die out. This model was first studied by [*Ann. Appl. Probab.* **18** (2008) 359–378]. In this work we find a simple lower bound on the expected number of ever infected vertices using *breath-first search* algorithm and show that it asymptotically performs better for a fairly large class of graphs than the upper bounds obtained in [*Ann. Appl. Probab.* **18** (2008) 359–378]. As a by product we also derive the asymptotic value of the expected number of the ever infected vertices when the underlying graph is the random $r$-regular graph and $\beta<\frac{1}{r-1}$.

#### Article information

**Source**

Ann. Appl. Probab. Volume 25, Number 2 (2015), 663-674.

**Dates**

First available in Project Euclid: 19 February 2015

**Permanent link to this document**

http://projecteuclid.org/euclid.aoap/1424355127

**Digital Object Identifier**

doi:10.1214/14-AAP1007

**Mathematical Reviews number (MathSciNet)**

MR3313752

**Zentralblatt MATH identifier**

1322.60206

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C80: Random graphs [See also 60B20]

Secondary: 60J85: Applications of branching processes [See also 92Dxx] 90B15: Network models, stochastic

**Keywords**

Breadth-first search local weak convergence percolation on finite graphs random $r$-regular graphs susceptible infected removed model virus spread

#### Citation

Bandyopadhyay, Antar; Sajadi, Farkhondeh. On the expected total number of infections for virus spread on a finite network. Ann. Appl. Probab. 25 (2015), no. 2, 663--674. doi:10.1214/14-AAP1007. http://projecteuclid.org/euclid.aoap/1424355127.