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A functional central limit theorem for SI processes on configuration model graphs

Part of: Limit theorems

Published online by Cambridge University Press:  06 September 2022

Wasiur R. Khudabukhsh ,
Casper Woroszylo ,
Grzegorz A. Rempała  and
Heinz Koeppl
Wasiur R. Khudabukhsh*
Affiliation:
University of Nottingham
Casper Woroszylo*
Affiliation:
BHP Billiton
Grzegorz A. Rempała*
Affiliation:
The Ohio State University
Heinz Koeppl*
Affiliation:
Technische Universität Darmstadt
*
*Postal address: School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
**Postal address: 480 Queen Street, Level 12, Brisbane QLD 4000, Australia.
***Postal address: Mathematical Biosciences Institute, The Ohio State University, Jennings Hall 3rd Floor, 1735 Neil Ave., Columbus, OH 43210, United States of America.
****Postal address: Bioinspired Communication Systems, Technische Universität Darmstadt, Rundeturmstrasse 12, 64283 Darmstadt, Germany.
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Abstract

We study a stochastic compartmental susceptible–infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval. We split the population of graph vertices into two compartments, namely, S and I, denoting susceptible and infected vertices, respectively. In addition to the sizes of these two compartments, we keep track of the counts of SI-edges (those connecting a susceptible and an infected vertex) and SS-edges (those connecting two susceptible vertices). We describe the dynamical process in terms of these counts and present a functional central limit theorem (FCLT) for them as the number of vertices in the random graph grows to infinity. The FCLT asserts that the counts, when appropriately scaled, converge weakly to a continuous Gaussian vector semimartingale process in the space of vector-valued càdlàg functions endowed with the Skorokhod topology. We discuss applications of the FCLT in percolation theory and in modelling the spread of computer viruses. We also provide simulation results illustrating the FCLT for some common degree distributions.

Keywords

SI processfunctional CLTconfiguration modelrandom graphsscaling limit

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60F05: Central limit and other weak theorems 92D30: Epidemiology
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 3 , September 2022 , pp. 880 - 912
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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