The height of the infection tree
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arXiv
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| Format: | Preprint |
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2025
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| author | Kammerer, Emmanuel Kortchemski, Igor Sénizergues, Delphin |
| author_facet | Kammerer, Emmanuel Kortchemski, Igor Sénizergues, Delphin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03526 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The height of the infection tree Kammerer, Emmanuel Kortchemski, Igor Sénizergues, Delphin Probability Primary 60J80, Secondary 05C05, 60G42 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. |
| title | The height of the infection tree |
| topic | Probability Primary 60J80, Secondary 05C05, 60G42 |
| url | https://arxiv.org/abs/2504.03526 |