Extinction and Survival for a Generalized Contact Process with Deterministic Cures

Nancy L. Garcia; Denis A. Luiz; Daniel M. Machado

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arXiv · 2609.29425

Extinction and Survival for a Generalized Contact Process with Deterministic Cures

Nancy L. Garcia·Denis A. Luiz·Daniel M. Machado

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Abstract

We study a generalized contact process on $\mathbb{Z}^d$ parameterized by an infection rate $λ$ and a resetting probability $p \in [0,1]$, modeling deterministic cure times. Once a vertex is infected, its recovery is scheduled exactly one time unit later. Incoming attempts to an already-infected vertex successfully reset its recovery clock with probability $p$, and are ignored otherwise. This unifies spatial versions of classical Type I ($p=0$, non-paralyzable) and Type II ($p=1$, paralyzable) counters. Except in the fully resetting case, deterministic recovery deadlines destroy coordinatewise attractiveness and create a causal shielding effect. Using a first-moment bound on potential causal chains, we prove that the process dies out from finite configurations whenever $λ<1/(2d)$, uniformly in $p\in[0,1]$. The same causal-chain bound yields local convergence to the empty configuration from arbitrary initial states. In dimensions $d \ge 2$, an oriented percolation exploration based on the first infective window establishes global survival for all $p\in[0,1]$ when $λ>-\log(1-p_c^{\mathrm{or}})$. Finally, in the purely non-resetting case $p=0$, we derive a delayed identity for the one-site density and prove that this density remains strictly between zero and one at every finite time.

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BibTeX RIS

Nancy L. Garcia, Denis A. Luiz, Daniel M. Machado. 2026-09-24. Extinction and Survival for a Generalized Contact Process with Deterministic Cures. https://arxiv.org/abs/2609.29425

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