A growth-fragmentation-isolation process on random recursive trees and contact tracing
The evolution of random recursive trees (RRT) through growth, fragmentation and isolation is studied to model the outbreak of pandemic. An RRT is to represent a finite set of patients connected by edges if the infector-infectee information is retrievable in the contact tracing records. Growth means a RRT infects a new individual; fragmentation happens when the information of who infected a patient is lost, resulting in a RRT split into two subtrees; isolation occurs if a patient in an RRT is detected so we put every patient in the RRT to self-isolation. By assigning occurrence rates to growth, fragmentation and isolation, we show when the number of (non-isolated) patients goes to infinity or to zero, and the existence of a phase transition, and give the convergence speed upon survival of pandemic. The main tool is a theorem for non-conservative semigroups and the martingale approach applied to a Markov branching process with infinite types.
This is a joint work with Vincent Bansaye (Ecole Polytechnique) and Chenlin Gu (Tsinghua University).