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Brownian snails with removal die out in one dimension

Ivailo Hartarsky, Lyuben Lichev

Abstract

Brownian snails with removal is a spatial epidemic model defined as follows. Initially, a homogeneous Poisson process of susceptible particles on $\mathbb R^d$ with intensity $λ>0$ is deposited and a single infected one is added at the origin. Each particle performs an independent standard Brownian motion. Each susceptible particle is infected immediately when it is within distance 1 from an infected particle. Each infected particle is removed at rate $α>0$, and removed particles remain such forever. Answering a question of Grimmett and Li, we prove that in one dimension, for all values of $λ$ and $α$, the infection almost surely dies out.

Brownian snails with removal die out in one dimension

Abstract

Brownian snails with removal is a spatial epidemic model defined as follows. Initially, a homogeneous Poisson process of susceptible particles on with intensity is deposited and a single infected one is added at the origin. Each particle performs an independent standard Brownian motion. Each susceptible particle is infected immediately when it is within distance 1 from an infected particle. Each infected particle is removed at rate , and removed particles remain such forever. Answering a question of Grimmett and Li, we prove that in one dimension, for all values of and , the infection almost surely dies out.
Paper Structure (10 sections, 2 theorems, 23 equations)

This paper contains 10 sections, 2 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. The model
  3. The result
  4. Proof of Theorem \ref{['thm:main']}
  5. Linearly localised infection
  6. Linear number of infections
  7. Frequently bounded number of infections
  8. Too many infections
  9. Many chances to extinguish infection
  10. At most linear growth

Key Result

Theorem 1

There exists $c > 0$ such that, for all sufficiently large $T > 0$, In particular, almost surely there exists $T>0$ such that $I(t)=0$ for all $t\ge T$.

Theorems & Definitions (4)

  • Theorem 1
  • Remark 2: Extensions
  • Proposition 3
  • Remark 4