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A contact process with stronger mutations on trees

Fábio Lopes, Alejandro Roldán-Correa

TL;DR

The paper analyzes a spatial pathogen model with stronger mutations on infinite trees, where each new type is eliminated after its progenitor dies and the lineage can mutate with probability $r$ at birth. The authors introduce the process $\\{\\mathcal{G},\\lambda, r\\}$, distinguishing two tree geometries, $\\mathbb{T}_d$ and $\\mathbb{T}_d^+$, and prove a sharp phase transition on directed trees governed by the critical curve $\\lambda_c(d,r) = \left[\frac{\\sqrt{d-1}-\\sqrt{rd}}{d(1-r)-1}\right]^2$, separating extinction and survival. The proofs employ coupling arguments, MGFs, and branching-process techniques (including an Aldous-type lemma) to establish extinction and survival regimes, and then derive corollaries for the full tree geometry by stochastic domination arguments. These results illuminate how stronger mutations can qualitatively alter persistence on tree-like networks, contrasting with non-spatial counterparts. The work advances understanding of mutation-driven persistence under immune-elimination dynamics in spatial settings and provides explicit thresholds for survival on infinite trees.

Abstract

We consider a spatial stochastic model for a pathogen population growing inside a host that attempts to eliminate the pathogens through its immune system. The pathogen population is divided into different types. A pathogen can either reproduce by generating a pathogen of its own type or produce a pathogen of a new type that does not yet exist in the population. Pathogens with living ancestral types are protected against the host's immune system as long as their progenitors are still alive. Each pathogen type without living ancestral types is eliminated by the immune system after a random period, independently of the other types. When a pathogen type is eliminated from the system, all pathogens of this type die simultaneously. In this paper, we determine the conditions on the set of model parameters that dictate the survival or extinction of the pathogen population when the dynamics unfold on graphs with an infinite tree structure.

A contact process with stronger mutations on trees

TL;DR

The paper analyzes a spatial pathogen model with stronger mutations on infinite trees, where each new type is eliminated after its progenitor dies and the lineage can mutate with probability at birth. The authors introduce the process , distinguishing two tree geometries, and , and prove a sharp phase transition on directed trees governed by the critical curve , separating extinction and survival. The proofs employ coupling arguments, MGFs, and branching-process techniques (including an Aldous-type lemma) to establish extinction and survival regimes, and then derive corollaries for the full tree geometry by stochastic domination arguments. These results illuminate how stronger mutations can qualitatively alter persistence on tree-like networks, contrasting with non-spatial counterparts. The work advances understanding of mutation-driven persistence under immune-elimination dynamics in spatial settings and provides explicit thresholds for survival on infinite trees.

Abstract

We consider a spatial stochastic model for a pathogen population growing inside a host that attempts to eliminate the pathogens through its immune system. The pathogen population is divided into different types. A pathogen can either reproduce by generating a pathogen of its own type or produce a pathogen of a new type that does not yet exist in the population. Pathogens with living ancestral types are protected against the host's immune system as long as their progenitors are still alive. Each pathogen type without living ancestral types is eliminated by the immune system after a random period, independently of the other types. When a pathogen type is eliminated from the system, all pathogens of this type die simultaneously. In this paper, we determine the conditions on the set of model parameters that dictate the survival or extinction of the pathogen population when the dynamics unfold on graphs with an infinite tree structure.

Paper Structure

This paper contains 3 sections, 4 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spatial model
  3. Proofs

Key Result

Proposition 2.4

The survival probability in $\{\mathbb{T}_d^+, \lambda, r\}$ is a non-decreasing function of both $\lambda$ and $r$.

Figures (2)

  • Figure 2.1: Model S2 on $\mathbb{T}_4$ vs Model $\{\mathbb{T}_4, \lambda, r\}$. In region (I), both models die out. In region (II), Model S2 dies out, while the behavior of Model $\{\mathbb{T}_4, \lambda, r\}$ remains inconclusive. In region (III), Model S2 dies out, whereas Model $\{\mathbb{T}_4, \lambda, r\}$ survives. In region (IV), Model $\{\mathbb{T}_4, \lambda, r\}$ survives, while the behavior of Model S2 is inconclusive. In region (V), both models survive.
  • Figure 2.2: Monte Carlo simulation outcomes for Model $\{\mathbb{T}_4, \lambda, r\}$. Each point corresponds to the empirical result of 30000 independent realizations for a given pair of parameters $(r,\lambda)$, with $r\in[0.01,1]$ and $\lambda\in[0.01,0.4]$. Colors and symbols represent the number of surviving realizations: pink circles denote total extinction (0 surviving runs), while grey to red symbols indicate increasing survival levels (1--5, 6--9, 10--49, 50--100, 101--499, 500--4999, and $\ge 5000$ surviving realizations). The same theoretical thresholds for survival from Figure \ref{['figura1']} are shown for comparison.

Theorems & Definitions (12)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Remark 2.7
  • proof : Proof of Proposition \ref{['monotonia']}
  • proof : Proof of Theorem \ref{['T:arbol_dirigido']} $(i)$
  • Lemma 3.1
  • ...and 2 more