[Paper Review] A shape theorem for an epidemic model in dimension $d\ge 3$

Enrique D. Andjel, Nicolas Chabot+1 more|arXiv (Cornell University)|Oct 4, 2011

Stochastic processes and statistical mechanics17 references3 citations

TL;DR

This paper establishes a shape theorem for the set of infected or immune individuals in a stochastic SIR epidemic model on $ℤ^d$ for $d \geq 3$, using dynamic renormalization techniques to handle locally dependent percolation with random infection durations. The key result shows that, conditional on non-extinction, the infected region grows ballistically and converges to a deterministic limiting shape, extending prior results from $d=2$ to higher dimensions with general infection time distributions.

ABSTRACT

We prove a shape theorem for the set of infected individuals in a spatial epidemic model with 3 states (susceptible-infected-recovered) on ${\mathbb Z}^d,d\ge 3$, when there is no extinction of the infection. For this, we derive percolation estimates (using dynamic renormalization techniques) for a locally dependent random graph in correspondence with the epidemic model.

Motivation & Objective

• To establish a shape theorem for the spatial spread of infection in a stochastic SIR model on $\mathbb{Z}^d$ for $d \geq 3$, extending prior results limited to $d=2$.
• To address the challenge of random infection durations (positive with positive probability), which invalidates prior comparisons to independent percolation models used in lower dimensions.
• To develop and apply dynamic renormalization techniques to handle locally dependent percolation structures arising from the epidemic dynamics.
• To prove that the set of infected or immune individuals grows ballistically and converges to a deterministic limiting shape when the infection does not go extinct.

Proposed method

• Adapt dynamic renormalization techniques from Grimmett and Marstrand (1990) to analyze first-passage percolation with locally dependent edge weights corresponding to the epidemic model.
• Use a locally dependent random graph representation of the epidemic process, where edges represent successful infection transmissions with time-dependent weights.
• Introduce random neighborhoods based on percolation on slabs to replace circuit-based methods that fail in $d \geq 3$, enabling control of open paths in high dimensions.
• Apply the FKG inequality and sub-exponential estimates (instead of exponential ones) to establish positive lower bounds on path existence probabilities across slabs.
• Leverage results from non-oriented percolation in $d \geq 3$ to control the geometry of open clusters and derive shape convergence.
• Use moment conditions of order $d$ on infection durations only to localize infected (but not yet immune) individuals within the asymptotic shape.

Experimental results

Research questions

• RQ1 Does the set of infected or immune individuals in a spatial SIR model on $\mathbb{Z}^d$ for $d \geq 3$ converge to a deterministic limiting shape when the infection survives?
• RQ2 Can dynamic renormalization techniques be adapted to handle locally dependent percolation models with random infection times in high dimensions?
• RQ3 How can the absence of circuit-based path control in $d \geq 3$ be overcome to prove a shape theorem?
• RQ4 What role do moment conditions on infection durations play in localizing infected individuals within the asymptotic shape?
• RQ5 Can the shape theorem be established without assuming deterministic infection durations, as in earlier works?

Key findings

• The paper proves a shape theorem for the SIR epidemic model on $\mathbb{Z}^d$ for $d \geq 3$ under non-extinction, showing that the infected region grows ballistically and converges to a deterministic limiting shape.
• The authors establish the existence of a positive lower bound $\delta_1 > 0$ on the probability of open paths across slabs of thickness $6N$, enabling the construction of long-range connections in the percolation model.
• Sub-exponential estimates are derived for the percolation model, sufficient for the shape theorem, replacing the exponential estimates used in earlier works with deterministic infection times.
• The random neighborhoods introduced by Chabot (1998) are shown to remain valid and effective in the current setting with random infection durations.
• The moment condition of order $d$ on infection durations is sufficient to localize infected individuals within the asymptotic shape, consistent with results in $d=2$.
• The proof relies on dynamic renormalization and slab-based percolation arguments, avoiding circuit-based methods that fail in $d \geq 3$.