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Renewal contact process with dormancy

Noemi Kurt, Michel Reitmeier, András Tóbiás

TL;DR

The paper analyzes a renewal contact process with dormancy, where dormancy wake-up times follow a renewal process with heavy tails. It establishes a sharp dichotomy: when no infection can pass between dormant sites and wake-ups have heavy tails with exponent $\alpha\in(0,1)$, infection growth is at most logarithmic (or sublinear under a gap condition), while allowing infections between dormant individuals yields positive survival probability on finite graphs, with explicit cardinality thresholds for survival vs. extinction. The authors deploy a combination of the graphical construction, Dynkin–Lamperti type arguments for renewal times, and percolation-based couplings (including an iterated site percolation on a macroscopic grid) to obtain sublinear/logarithmic growth bounds and finite-graph survival/extinction criteria. These results illuminate how heavy-tailed dormancy can slow or block the spread of infection and identify parameter regimes where nontrivial phase behavior persists in non-Markovian settings, with potential implications for ecological and epidemiological modelling.

Abstract

We consider the contact process with dormancy, where wake-up times follow a renewal process. Without infection between dormant individuals, we show that the process under certain conditions grows at most logarithmically. On the other hand, if infections between dormant individuals are possible, the process survives with positive probability even on finite graphs.

Renewal contact process with dormancy

TL;DR

The paper analyzes a renewal contact process with dormancy, where dormancy wake-up times follow a renewal process with heavy tails. It establishes a sharp dichotomy: when no infection can pass between dormant sites and wake-ups have heavy tails with exponent , infection growth is at most logarithmic (or sublinear under a gap condition), while allowing infections between dormant individuals yields positive survival probability on finite graphs, with explicit cardinality thresholds for survival vs. extinction. The authors deploy a combination of the graphical construction, Dynkin–Lamperti type arguments for renewal times, and percolation-based couplings (including an iterated site percolation on a macroscopic grid) to obtain sublinear/logarithmic growth bounds and finite-graph survival/extinction criteria. These results illuminate how heavy-tailed dormancy can slow or block the spread of infection and identify parameter regimes where nontrivial phase behavior persists in non-Markovian settings, with potential implications for ecological and epidemiological modelling.

Abstract

We consider the contact process with dormancy, where wake-up times follow a renewal process. Without infection between dormant individuals, we show that the process under certain conditions grows at most logarithmically. On the other hand, if infections between dormant individuals are possible, the process survives with positive probability even on finite graphs.

Paper Structure

This paper contains 6 sections, 6 theorems, 53 equations.

Table of Contents

  1. Introduction and motivation
  2. Model and main results
  3. Logarithmic growth
  4. Finite graphs
  5. Survival on finite graphs
  6. Extinction on finite graphs

Key Result

Theorem 2.2

Let $G=(\mathbb{Z}^d,\mathbb{E}^d)$ be the Euclidean lattice, $\lambda_{dd}=0$ and assume $\varrho$ satisfies condition (S), then the process $r_t$ has at most logarithmic growth, i.e. for every $\varepsilon>0$ we have Moreover, if $\varrho$ satisfies (G) then $r_t$ growth at most sub-linearly, i.e. for every $\varepsilon\in(0,\varepsilon^*)$ we have

Theorems & Definitions (16)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 3.1: Dynkin--Lamperti theorem
  • Lemma 3.2
  • proof
  • ...and 6 more