A growth-fragmentation-isolation process on random recursive trees and contact tracing

Abstract

We consider a random process on recursive trees, with three types of events. Vertices give birth at a constant rate (growth), each edge may be removed independently (fragmentation of the tree) and clusters are frozen with a rate proportional to their size (isolation of connected component). A phase transition occurs when the isolation is able to stop the growth fragmentation process and cause extinction. When the process survives, we characterize its growth and prove that the empirical measure of clusters a.s. converges to a limit law on recursive trees. We exploit the branching structure associated to the size of clusters, which is inherited from the splitting property of random recursive trees. This issue is motivated by the control of epidemics and contact-tracing where clusters correspond to subtrees of infected individuals that can be identified and isolated.

A model for an epidemic with contact tracing and cluster isolation, and a detection paradox

“…We present below a toy model in this framework, which is solvable in the sense that many quantities of interest can be computed explicitly. This model is close to the one introduced recently by Bansaye, Gu, and Yuan [2] as it will be discussed in the final section of this text.…”

“…Bansaye et al investigate the large time asymptotic behavior of the epidemic using different tools, namely they analyze first a deterministic eigenproblem for a growth-fragmentation-isolation equation which is naturally related to their setting; furthermore they also rely on known properties of random recursive trees. They establish results similar to our Theorem 4.1 and Corollary 4.3 in terms of these eigenelements; the statements in [2] are however less precise than ours, as no explicit formulas for the eigenelements are given (only their existence is established).…”

“…This work has been inspired by a recent manuscript by Bansaye et al [2] in which the authors introduced a similar model for epidemics with contact tracing and cluster isolation. The main difference with the present one is that in [2], contaminations are always traceable initially, but traceability gets lost at some fixed rate.…”

“…This work has been inspired by a recent manuscript by Bansaye et al [2] in which the authors introduced a similar model for epidemics with contact tracing and cluster isolation. The main difference with the present one is that in [2], contaminations are always traceable initially, but traceability gets lost at some fixed rate. In other words, edges in the contamination tree are traceable when they first appear, and become untraceable after an exponentially distributed time-laps, independently of the other edges.…”

“…Predicting and controlling the evolution of epidemics has motivated mathematical contributions for a long time and generated a huge literature; let us merely point at the lecture notes [4] and references therein. Models involving contact tracing and isolation, which aims at reducing the transmissibility of infections, have raised a significant interest; see notably among others [1,2,3,5,10,11,12,14]. We present below a toy model in this framework, which is solvable in the sense that many quantities of interest can be computed explicitly.…”