Extinction and Survival for a Generalized Contact Process with Deterministic Cures
Abstract
We study a generalized contact process on ℤd parameterized by an infection rate λ and a resetting probability p ∈ [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∈[0,1]. The same causal-chain bound yields local convergence to the empty configuration from arbitrary initial states. In dimensions d ≥ 2, an oriented percolation exploration based on the first infective window establishes global survival for all p∈[0,1] when λ>−log(1−pcor). 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
@misc{garcia2026extinction,
title = {{Extinction and Survival for a Generalized Contact Process with Deterministic Cures}},
author = {Nancy L. Garcia and Denis A. Luiz and Daniel M. Machado},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29425},
eprint = {2609.29425},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29425},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Extinction and Survival for a Generalized Contact Process with Deterministic Cures
AU - Nancy L. Garcia
AU - Denis A. Luiz
AU - Daniel M. Machado
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29425
UR - https://arxiv.org/abs/2609.29425
AB - We study a generalized contact process on ℤ^d parameterized by an infection rate λ and a resetting probability p ∈ [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∈[0,1]. The same causal-chain bound yields local convergence to the empty configuration from arbitrary initial states. In dimensions d ≥ 2, an oriented percolation exploration based on the first infective window establishes global survival for all p∈[0,1] when λ>−log(1−p_c^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.
ER -