On the extinction phase of the contact process with an asymptomatic state

TL;DR

The study analyzes an asymptomatic variant of the spatial contact process, where infections occur at rates $λ_1$ (asymptomatic) and $λ_2$ (symptomatic), with asymptomatics converting to symptomatic at rate $γ$. It proves that, for $γ<1/(4d-1)$, extinction persists even when $λ_2=\infty$ if $λ_1$ is small enough, by establishing exponential decay of the Galton-Watson progeny and embedding a block construction that yields domination by a high-parameter oriented percolation process; a perturbation argument then extends extinction to small positive $λ_1$. The main contributions are a rigorous exponential decay bound for the branching progeny, a detailed block-percolation framework, and a perturbative extension of extinction from $λ_1=0$ to small $λ_1>0$, clarifying how local interactions counteract mean-field epidemic behavior. These results advance understanding of extinction regimes in spatial epidemic models with asymptomatic states and provide precise thresholds for parameter regions ensuring disease die-out.

Abstract

The contact process with an asymptomatic state, introduced in [Belhadji, Lanchier and Mercer, Stochastic Process. Appl., 176:104417, 2024], is a natural variant of the basic contact process that distinguishes between asymptomatic (state 1) and symptomatic (state 2) individuals. Infected individuals infect their healthy neighbors at rate $λ_1$ when asymptomatic and at rate $λ_2$ when symptomatic. Newly infected individuals are always asymptomatic and become symptomatic at rate $γ$, and infected individuals recover at rate one regardless of whether they are asymptomatic or symptomatic. Belhadji, Lanchier and Mercer proved that, in the mean-field approximation, there is an epidemic if and only if $λ_1 + γλ_2 > 1 + γ$, showing in particular that, for all $γ> 0$, there is an epidemic for $λ_2$ sufficiently large. In contrast, comparing the process with a subcritical Galton-Watson branching process, they proved for the spatial model that, if $γ< 1 / (4d - 1)$ and $λ_1 = 0$, then there is no epidemic even in the limiting case $λ_2 = \infty$. In this paper, we prove an exponential decay of the progeny of the Galton-Watson branching process, and use a block construction and a perturbation argument, to extend the extinction phase of the process to $λ_1 > 0$ small.

Theorems & Definitions (11)

• Theorem 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10