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Derivation of a spatial replicator system with environmental heterogeneity from a co-colonization SIS model with N strains and P patches

Sten Madec, Erida Gjini

TL;DR

This work derives a spatial replicator equation for $N$ strains in a multi-patch SIS coinfection model across $P$ patches by exploiting $\varepsilon$-quasi neutrality and $\varepsilon$-slow migration. A novel Perron–Frobenius-based reduction of Metzler matrices yields an ab initio slow-fast framework that produces a discrete-space replicator with patch-specific payoff matrices $\Lambda_p$ and a migration operator $\mathcal{M}$. The reduced dynamics, $\frac{d}{d\tau} z_p^i = \Theta_p z_p^i[(\Lambda_p\mathbf{z}_p)_i - \mathbf{z}_p^T\Lambda_p\mathbf{z}_p] + d(\mathcal{M}\mathbf{z}^i)_p$, encapsulate how environmental heterogeneity shapes interpatch interactions and strain frequencies via $m_{pk}=d_{pk}(\omega_p^*\cdot X_k^*)$. The approach connects to continuous spatial reductions and demonstrates a commutativity between reduction and discretization, offering a tractable framework to analyze spatial effects on multi-strain, coinfection dynamics with potential extensions to broader patch-structured epidemiological systems.

Abstract

The interplay between local and regional processes in the dynamics of ecological communities remains a challenge to model, analyze and predict. This is especially notable in infectious diseases with multiple strains, where several layers of heterogeneity can interact, including strain biological traits and environmental heterogeneity among locations where disease can spread. Motivated by this challenge, here we study a Susceptible-Infected-Susceptible (SIS) model with co-colonization and multiple interacting strains where hosts move between a set of inter-connected patches. Under strain similarity and slow migration rate, we derive a fast-slow approximation of the global metacommunity dynamics, resulting in a spatial replicator system for N strains across P patches. In contrast to a discretization approach on the spatial slow-fast PDE originally derived in(Le and Madec, 2023), here the slow-fast reduction is managed ab-initio by a new approach using strongly the Perron-Frobenius Theorem for Metzler matrices, which simplifies and clarifies the structure of the co-colonization system.

Derivation of a spatial replicator system with environmental heterogeneity from a co-colonization SIS model with N strains and P patches

TL;DR

This work derives a spatial replicator equation for strains in a multi-patch SIS coinfection model across patches by exploiting -quasi neutrality and -slow migration. A novel Perron–Frobenius-based reduction of Metzler matrices yields an ab initio slow-fast framework that produces a discrete-space replicator with patch-specific payoff matrices and a migration operator . The reduced dynamics, , encapsulate how environmental heterogeneity shapes interpatch interactions and strain frequencies via . The approach connects to continuous spatial reductions and demonstrates a commutativity between reduction and discretization, offering a tractable framework to analyze spatial effects on multi-strain, coinfection dynamics with potential extensions to broader patch-structured epidemiological systems.

Abstract

The interplay between local and regional processes in the dynamics of ecological communities remains a challenge to model, analyze and predict. This is especially notable in infectious diseases with multiple strains, where several layers of heterogeneity can interact, including strain biological traits and environmental heterogeneity among locations where disease can spread. Motivated by this challenge, here we study a Susceptible-Infected-Susceptible (SIS) model with co-colonization and multiple interacting strains where hosts move between a set of inter-connected patches. Under strain similarity and slow migration rate, we derive a fast-slow approximation of the global metacommunity dynamics, resulting in a spatial replicator system for N strains across P patches. In contrast to a discretization approach on the spatial slow-fast PDE originally derived in(Le and Madec, 2023), here the slow-fast reduction is managed ab-initio by a new approach using strongly the Perron-Frobenius Theorem for Metzler matrices, which simplifies and clarifies the structure of the co-colonization system.

Paper Structure

This paper contains 22 sections, 7 theorems, 61 equations, 1 table.

Table of Contents

  1. Introduction
  2. Model and assumptions
  3. Discrete space and notations
  4. SIS with coinfection (SIDS) in space
  5. Main assumptions
  6. Strain neutrality and no migration: $\varepsilon=0$.
  7. One strain model without migration
  8. $N-$strain neutral model and $\varepsilon=0$
  9. First, aggregated variables
  10. Second: the whole attractor
  11. Third: back to the original variables
  12. Quasi-Neutrality among strains
  13. The aggregated variables
  14. The reduction
  15. Patch heterogeneity and the matrix $\mathcal{M}$.
  16. ...and 7 more sections

Key Result

Proposition 1

[Invariant set]For each patch $p\in\llbracket1,P\rrbracket$, denote the total sum: and $\mathbf{\Sigma}=(\Sigma_p)_{p\in\llbracket1,P\rrbracket}$. Assume that assumption MatrixD-(i) and (iii) holds true. Then the set is positively invariant under mainsys. If moreover the irreducibility assumption MatrixD-(ii) holds true and $\delta>0$, then any component $\text{x}(t)$ of the solution $\text{X}(t

Theorems & Definitions (12)

  • Proposition 1
  • proof
  • Lemma 1
  • Theorem 1: Dynamics under strain neutrality and no migration
  • proof
  • Lemma 2
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • ...and 2 more