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

TL;DR

This paper studies the extinction phase of a contact process with asymptomatic states, extending previous results by proving exponential decay and analyzing the process for small positive infection rates.

Contribution

It extends the understanding of the extinction phase by proving exponential decay and analyzing the process for small positive infection rates using block construction and perturbation methods.

Findings

Exponential decay of the progeny in the Galton-Watson process.

Extension of the extinction phase to small positive infection rates.

Analysis of the process using block construction and perturbation techniques.

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 $\lambda_1$ when asymptomatic and at rate $\lambda_2$ when symptomatic. Newly infected individuals are always asymptomatic and become symptomatic at rate $\gamma$, 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 $\lambda_1 + \gamma \lambda_2 > 1 + \gamma$, showing in particular that, for all $\gamma > 0$, there is an epidemic for $\lambda_2$ sufficiently large. In contrast, comparing the process with a subcritical Galton-Watson branching process, they proved for the spatial model that, if $\gamma < 1 / (4d - 1)$ and $\lambda_1 = 0$, then there is no epidemic even in the limiting case $\lambda_2 = \infty$. 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 $\lambda_1 > 0$ small.