Non-linear branching processes and Crump-Mode-Jagers processes with interaction

TL;DR

This work develops a mean-field framework for CMJ branching processes with interaction by thinning births according to a state-dependent probability $C(t, \mu^N_t)$, where $\mu^N_t$ is the empirical age distribution. The authors prove a law of large numbers for the local tree structure, showing convergence to a non-linear CMJ tree described by a time-inhomogeneous renewal equation for $u_t(a)$. They establish well-posedness of the limit, construct a coupling with an i.i.d. family of non-linear CMJ trees via a CMJ process with immigration, and use propagation-of-chaos techniques to derive corollaries: convergence of the age distribution and a description of infection chains as time-inhomogeneous Markov processes. The approach provides a general methodology for analyzing interacting population processes and has direct interpretations in epidemiology, where contact rates depend on the evolving states of the population. The results extend classical CMJ theory by integrating interaction through pruning and illuminate how macroscopic limits emerge from coupled genealogies.

Abstract

We consider a class of Crump-Mode-Jagers processes with interaction, constructed by removing a newly born offspring with a probability that depends on the age structure of the population at its birth time. We prove a law of large numbers for the tree structure of the process in a local topology, and show how this result condenses several other limit theorems (convergence of the empirical age distribution, of ancestral lineages). Beyond this specific example, our work illustrates a more general principle that we formalise. As in standard propagation of chaos, the trees generated by typical individuals become independent as the number of individuals goes to infinity. This allows us to express the distribution of the local tree structure around a typical individual in terms of a time-inhomogeneous branching process, which we call a non-linear branching process.

Figures (2)

• Figure 1: Illustration of the CMJ process with immigration. The left panel corresponds to the CMJ tree of potential immigration times, which is started from a unique ancestor. At each potential immigration time, a new particle might be added to the right panel (represented by an arrow) and starts to reproduce as a CMJ process. The process $(I^\eta(t))_{t \ge 0}$ counts the number of particle on the right.
• Figure 2: Illustration of the coupling, starting from a single particle. Non-fictitious individuals in the coupling (with types $(1,1)$, $(1,0)$, and $(0,1)$) are represented by dark lines, whereas fictitious individuals (with types $\dag$ and $*$) by blue lines. The birth times of individuals on the left (with types $(1,1)$ and $\dag$) correspond to the potential immigration times in the CMJ process with immigration. The tree of $(1,1)$ particles is thinned, but it is complemented with $*$ particles so that it is distributed as CMJ tree. Whenever a particle is immigrated and added to the right of the picture, it produces a tree which we complement with $\dag$ particles so that it is also distributed as a CMJ tree.

Theorems & Definitions (18)

• Proposition 1
• Theorem 1
• Corollary 1
• Corollary 2
• Example 1
• Example 2
• Proposition 2
• proof
• proof : Proof of Proposition \ref{['prop:nonLinearExistence']}
• Lemma 1