Rates of convergence for the three state contact process in one dimension

TL;DR

This paper investigates the convergence rates of a modified three-state contact process in one dimension, analyzing conditions for epidemic emergence and decay, and establishing bounds on infection spread velocities.

Contribution

It introduces a modified contact process model and derives new results on convergence rates, epidemic thresholds, and velocity bounds in one-dimensional settings.

Findings

Exponential decay of infected regions when $ul ext{ }< ext{ } ext{critical value}$

Existence of epidemic spread when $ul ext{ }> ext{ } ext{critical value}$

Velocity ratio bound of $ul ext{ }/ ext{ } ul$ for infection spread

Abstract

The basic contact process with parameter $\mu$ altered so that infections of sites that have not been previously infected occur at rate proportional to $\lambda$ instead is considered. Emergence of an infinite epidemic starting out from a single infected site is not possible for $\mu$ less than the contact process' critical value, whereas it is possible for $\mu$ greater than that value. In the former case the space and time infected regions are shown to decay exponentially; in the latter case and for $\lambda$ greater than $\mu$, the ratio of the endmost infected site's velocity to that of the contact process is shown to be at most $\lambda / \mu$.