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Spatially dense stochastic epidemic models with infection-age dependent infectivity

Guodong Pang, Etienne Pardoux

TL;DR

The paper develops a non-Markovian, spatially dense epidemic model where both the patch count and population diverge, introducing infection-age dependent infectivity and graphon-like spatial coupling via a kernel $\beta(x,y)$. It proves a functional law of large numbers (FLLN) that yields time-space integral equations for the limiting dynamics and derives a corresponding PDE for the infection-age density across space, connecting to classical Diekmann–Thieme and Kendall models in appropriate limits. The results place stochastic dynamics on graphon into a PDE framework, enabling analysis of spatially heterogeneous epidemics with general infectious period distributions and nonlocal infection interactions. The methodology combines measure-valued process limits, weak convergence in time-space function spaces, and detailed handling of age-structured infection processes, with implications for understanding geographically dispersed disease spread and non-Markovian epidemic dynamics on graphon-like networks.

Abstract

We study an individual-based stochastic spatial epidemic model where the number of locations and the number of individuals at each location both grow to infinity. Each individual is associated with a random infection-age dependent infectivity function. Individuals are infected through interactions across the locations with heterogeneous effects. The epidemic dynamics can be described using a time-space representation for the the total force of infection, the number of susceptible individuals, the number of infected individuals that are infected at each time and have been infected for a certain amount of time, as well as the number of recovered individuals. We prove a functional law of large numbers for these time-space processes, and in the limit, we obtain a set of time-space integral equations. We then derive the PDE models from the limiting time-space integral equations, in particular, the density (with respect to the infection age) of the time-age-space integral equation for the number of infected individuals tracking the age of infection satisfies a linear PDE in time and age with an integral boundary condition. These integral equation and PDE limits can be regarded as dynamics on graphon under certain conditions.

Spatially dense stochastic epidemic models with infection-age dependent infectivity

TL;DR

The paper develops a non-Markovian, spatially dense epidemic model where both the patch count and population diverge, introducing infection-age dependent infectivity and graphon-like spatial coupling via a kernel . It proves a functional law of large numbers (FLLN) that yields time-space integral equations for the limiting dynamics and derives a corresponding PDE for the infection-age density across space, connecting to classical Diekmann–Thieme and Kendall models in appropriate limits. The results place stochastic dynamics on graphon into a PDE framework, enabling analysis of spatially heterogeneous epidemics with general infectious period distributions and nonlocal infection interactions. The methodology combines measure-valued process limits, weak convergence in time-space function spaces, and detailed handling of age-structured infection processes, with implications for understanding geographically dispersed disease spread and non-Markovian epidemic dynamics on graphon-like networks.

Abstract

We study an individual-based stochastic spatial epidemic model where the number of locations and the number of individuals at each location both grow to infinity. Each individual is associated with a random infection-age dependent infectivity function. Individuals are infected through interactions across the locations with heterogeneous effects. The epidemic dynamics can be described using a time-space representation for the the total force of infection, the number of susceptible individuals, the number of infected individuals that are infected at each time and have been infected for a certain amount of time, as well as the number of recovered individuals. We prove a functional law of large numbers for these time-space processes, and in the limit, we obtain a set of time-space integral equations. We then derive the PDE models from the limiting time-space integral equations, in particular, the density (with respect to the infection age) of the time-age-space integral equation for the number of infected individuals tracking the age of infection satisfies a linear PDE in time and age with an integral boundary condition. These integral equation and PDE limits can be regarded as dynamics on graphon under certain conditions.
Paper Structure (10 sections, 19 theorems, 253 equations)

This paper contains 10 sections, 19 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Notation
  4. Model and FLLN
  5. Model Description
  6. FLLN
  7. PDE Models
  8. Some technical preliminaries
  9. Proof of the Convergence of $\bar{S}^N(t,x)$ and $\bar{\mathfrak{F}}^N(t,x)$
  10. Proof for the Convergence of $\bar{{\mathfrak{I}}}^N(t,\mathfrak{a},x)$

Key Result

Lemma 2.1

Under the assumption in eqn-initial-L1conv, $\|\bar{{\mathfrak{I}}}^N(0,\mathfrak{a}, \cdot) - \bar{{\mathfrak{I}}}(0,\mathfrak{a}, \cdot)\|_{1} \to 0$ in probability as $N\to\infty$, where $\bar{{\mathfrak{I}}}(0, d \mathfrak{a}, x) = \bar{{\mathfrak{T}}}(0,d \mathfrak{a}, x) F^c(\mathfrak{a})$.

Theorems & Definitions (39)

  • Lemma 2.1
  • proof
  • Remark 2.1
  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Corollary 3.1
  • ...and 29 more