# Phase transition for a non-attractive infection process in heterogeneous environment

## TL;DR

This paper analyzes a complex infection process in a heterogeneous environment, demonstrating conditions under which the infection either persists or dies out, based on a global parameter, using coupling with a Markov chain.

## Contribution

It introduces a non-attractive three-state contact process model and proves the existence of both survival and extinction regimes depending on a parameter.

## Key findings

- Infection dies out for large q.
- Infection survives for small q.
- Coupling with a Markov chain reveals drift properties.

## Abstract

We consider a non-attractive three state contact process on $\mathbb Z$ and prove that there exists a regime of survival as well as a regime of extinction. In more detail, the process can be regarded as an infection process in a dynamic environment, where non-infected sites are either healthy or passive. Infected sites can recover only if they have a healthy site nearby, whereas non-infected sites may become infected only if there is no healthy and at least one infected site nearby. The transition probabilities are governed by a global parameter $q$: for large $q$, the infection dies out, and for small enough $q$, we observe its survival. The result is obtained by a coupling to a discrete time Markov chain, using its drift properties in the respective regimes.