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A spatial host-parasite model with host immunity: Survival and linear spread of parasites on $\mathbb{Z}$

Sascha Franck, Cornelia Pokalyuk

TL;DR

The paper develops SIMI, a spatial host-parasite model on $\mathbb{Z}$ where mobile parasites infect immobile hosts with random immunity and offspring are produced upon successful infection. A renewal-structure approach, built around auxiliary jump times and good-configurations sites, yields a robust analysis of survival and invasion speed, showing linear, ballistic spread conditioned on survival under mild moment conditions on immunity and offspring. The authors prove a law of large numbers for the infection front and provide a lower bound for ballistic growth, while also addressing a polynomial-decay regime in the two-sided setting. The work combines renewal theory, precise tail estimates, and mixing arguments to handle dependencies, offering rigorous constructions (tagged and untagged) and establishing a strong Markov framework for the SIMI process. These results advance understanding of how host immunity shapes spatial parasite invasion and quantify conditions under which parasites spread linearly in time on $\mathbb{Z}$.

Abstract

We introduce a generalized version of the frog model to describe the invasion of a parasite population in a spatially structured immobile host population with host immunity on the integer line. Parasites move according to simple symmetric random walks and try to infect any host they meet. Hosts, however, own an immunity against the parasites that protects them from infection for a random number of attacks. Once a host gets infected, it and the infecting parasite die, and a random number of offspring parasites is generated. We show that the positivity of the survival probability of parasites only depends on the mean offspring and mean height of immunity. Furthermore, we prove through the construction of a renewal structure that given survival of the parasite population parasites invade the host population at linear speed under relatively mild assumptions on the host immunity distribution.

A spatial host-parasite model with host immunity: Survival and linear spread of parasites on $\mathbb{Z}$

TL;DR

The paper develops SIMI, a spatial host-parasite model on where mobile parasites infect immobile hosts with random immunity and offspring are produced upon successful infection. A renewal-structure approach, built around auxiliary jump times and good-configurations sites, yields a robust analysis of survival and invasion speed, showing linear, ballistic spread conditioned on survival under mild moment conditions on immunity and offspring. The authors prove a law of large numbers for the infection front and provide a lower bound for ballistic growth, while also addressing a polynomial-decay regime in the two-sided setting. The work combines renewal theory, precise tail estimates, and mixing arguments to handle dependencies, offering rigorous constructions (tagged and untagged) and establishing a strong Markov framework for the SIMI process. These results advance understanding of how host immunity shapes spatial parasite invasion and quantify conditions under which parasites spread linearly in time on .

Abstract

We introduce a generalized version of the frog model to describe the invasion of a parasite population in a spatially structured immobile host population with host immunity on the integer line. Parasites move according to simple symmetric random walks and try to infect any host they meet. Hosts, however, own an immunity against the parasites that protects them from infection for a random number of attacks. Once a host gets infected, it and the infecting parasite die, and a random number of offspring parasites is generated. We show that the positivity of the survival probability of parasites only depends on the mean offspring and mean height of immunity. Furthermore, we prove through the construction of a renewal structure that given survival of the parasite population parasites invade the host population at linear speed under relatively mild assumptions on the host immunity distribution.

Paper Structure

This paper contains 25 sections, 39 theorems, 480 equations, 1 figure.

Table of Contents

  1. Introduction
  2. (Informal) model definition and main results
  3. The spatial infection model with host immunity
  4. Main results on survival and speed of spread
  5. One-sided host population
  6. Two-sided host population
  7. Good sites and auxiliary jump times
  8. Construction and basic properties of auxiliary jump times and good sites
  9. Auxiliary results
  10. Proofs of auxiliary results
  11. Proof of Lemma \ref{['lem:Kmtail']}
  12. Proof of Lemma \ref{['lem:M_tail']}
  13. Proof of Lemma \ref{['NuMoments']}
  14. Proof of Lemma \ref{['Tdistconv']}
  15. Proof of Lemma \ref{['conditinallyphimixing']}
  16. ...and 10 more sections

Key Result

Theorem 1

There is a probability space $\mathbf{\Omega}^\prime = (\Omega^\prime,\mathcal{F}^\prime,\mathbf{P}^\prime)$ such that for any (possibly random) initial placement of parasites and hosts given by $\zeta = (\ell,r,\eta)$ defined on some probability space $\mathbf{\Omega}^{\prime\prime}= (\Omega^{\prim for some $\theta > 0$, the SIMI with initial configuration $\zeta$ can be constructed as a strong M

Figures (1)

  • Figure 1: $(r_t)_{t\ge0}$ for $\mathbf{P}(I\ge n) = \left(\frac{n-1}{3}+1\right)^{-2.1}$ and $A=4$ a.s.

Theorems & Definitions (103)

  • Theorem 1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Definition 2.7
  • Theorem 2
  • ...and 93 more