## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 47, Number 1 (2015), 164-181.

### Infection spread in random geometric graphs

#### Abstract

In this paper we study the speed of infection spread and the survival of the
contact process in the random geometric graph
*G* = *G*(*n*, *r*_{n}, *f*) of
*n* nodes independently distributed in
*S* = [-½, ½]^{2} according to a certain density
*f*(·). In the first part of the paper we assume that infection
spreads from one node to another at unit rate and that infected nodes stay in
the same state forever. We provide an explicit lower bound on the speed of
infection spread and prove that infection spreads in *G* with speed at
least *D*_{1}*nr*_{n}^{2}. In the
second part of the paper we consider the contact process
ξ_{t} on *G* where infection spreads at rate
λ > 0 from one node to another and each node independently
recovers at unit rate. We prove that, for every λ > 0, with high
probability, the contact process on *G* survives for an exponentially long
time; there exist positive constants *c*_{1} and
*c*_{2} such that, with probability at least
1 - *c*_{1} / *n*^{4}, the contact process
ξ^{1}_{t} starting with all nodes infected
survives up to time
*t*_{n} = exp(*c*_{2}*n*/log*n*) for
all *n*.

#### Article information

**Source**

Adv. in Appl. Probab., Volume 47, Number 1 (2015), 164-181.

**Dates**

First available in Project Euclid: 31 March 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1427814586

**Digital Object Identifier**

doi:10.1239/aap/1427814586

**Mathematical Reviews number (MathSciNet)**

MR3327320

**Zentralblatt MATH identifier**

1337.60242

**Subjects**

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60C05: Combinatorial probability 62E10: Characterization and structure theory

**Keywords**

Random geometric graph speed of infection spread survival time of contact process

#### Citation

Ganesan, Ghurumuruhan. Infection spread in random geometric graphs. Adv. in Appl. Probab. 47 (2015), no. 1, 164--181. doi:10.1239/aap/1427814586. https://projecteuclid.org/euclid.aap/1427814586