The height of the infection tree

TL;DR

This work analyzes the height of the infection tree in a stochastic SIR model on a complete graph with infection rate λ/n and exponential infectious periods. It maps the epidemic infection tree to a uniform attachment tree with freezing, and develops a robust martingale framework for the Laplace transform of the tree's profile to obtain a precise height scaling κ(λ) log n. The height limit exhibits a second-order phase transition at λ_c ≈ 1.8038, with κ(λ) given by a pair of explicit formulas depending on λ, and a Bernoulli survival factor capturing early die-out. The paper combines uniform attachment with freezing, Bienaymé tree couplings, fluid limits, Laplace-transform martingales, and mod-φ convergence to deliver sharp asymptotics for the infection tree’s height and its internal profile, with potential implications for transmission-tree geometry in large epidemics and related frozen recursive-tree models.

Abstract

We are interested in the geometry of the ``infection tree'' in a stochastic SIR (Susceptible-Infectious-Recovered) model, starting with a single infectious individual. This tree is constructed by drawing an edge between two individuals when one infects the other. We focus on the regime where the infectious period before recovery follows an exponential distribution with rate $1$, and infections occur at a rate $λ_{n} \sim \fracλ{n}$ where $n$ is the initial number of healthy individuals with $λ>1$. We show that provided that the infection does not quickly die out, the height of the infection tree is asymptotically $κ(λ) \log n$ as $n \rightarrow \infty$, where $κ(λ)$ is a continuous function in $λ$ that undergoes a second-order phase transition at $λ_{c}\simeq 1.8038$. Our main tools include a connection with the model of uniform attachment trees with freezing and the application of martingale techniques to control profiles of random trees.

Figures (3)

• Figure 1: Simulations of large infection trees for $\lambda=1.1$ (left) and $\lambda=5$ (right). The trees both have $10000$ vertices, and the orange vertices represent the first half of the vertices (in order of appearance). The bold path is the shortest path from the root to the vertex furthest away from the root. In the first case, the orange and blue trees both macroscopically contribute to the length of this path, while in the second case only the orange tree macroscopically contributes.
• Figure 2: In orange, plot of the function appearing in the limit of $\mathsf{Height}(\widehat{\mathcal{T}}^{n})/\log n$, where $\widehat{\mathcal{T}}^{n}$ is the infection tree after the early stages of the epidemic and before the late stages of the epidemic (this is the orange tree in Figure \ref{['fig:ssimus_intro']}). The function $\kappa$ appearing in \ref{['eq:kappa']} is the blue curve for $\lambda \leq \lambda_{c}$ and the orange curve for $\lambda \geq \lambda_{c}$: when $\lambda<\lambda_{c}$, the late stages of the epidemic have an influence on the total height of the infection tree, but not when $\lambda>\lambda_{c}$.
• Figure 3: In blue the Lambert function, in orange the lower bound of RS20 and in green the lower bound of Lemma \ref{['lem:Lambert']}.

Theorems & Definitions (63)

• theorem 1
• Lemma 2
• proof
• proposition 1
• Lemma 3
• proof : Proof of Proposition \ref{['prop:lc']}
• proposition 2
• Lemma 4
• proof
• Lemma 5