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Stochastic SIR model with individual heterogeneity and infection-age dependent infectivity on large non-homogeneous random graphs

Guodong Pang, Étienne Pardoux, Aurélien Velleret

TL;DR

This work analyzes the large-population limit of an individual-based stochastic SIR model with infection-age dependent infectivity on a large non-homogeneous random graph. By tracking infection-age and individual heterogeneity through measure-valued processes, the authors prove a functional law of large numbers that yields a deterministic limit represented by a system of measure-valued equations and a PDE on a graphon. The limiting system admits existence, uniqueness, and regularity results, and can be reformulated as a linear PDE with a boundary condition derived from the susceptible-infected interaction, reflecting the graphon-based connectivity. The framework accommodates general heterogeneity in individuals and connectivity, including continuous state spaces and non-Markovian infection dynamics, providing a tractable limit description for complex epidemic processes on large networks.

Abstract

We study an individual-based stochastic SIR epidemic model with infection-age dependent infectivity on a large random graph, capturing individual heterogeneity and non-homogeneous connectivity. Each individual is associated with particular characteristics (for example, spatial location and age structure), which may not be i.i.d., and represented by a particular node. The connectivities among the individuals are given by a non-homogeneous random graph, whose connecting probabilities may depend on the individual characteristics of the edge. Each individual is associated with a random varying infectivity function, which is also associated with the individual characteristics. We use measure-valued processes to describe the epidemic evolution dynamics, tracking the infection age of all individuals, and their associated characteristics. We consider the epidemic dynamics as the population size grows to infinity under a specific scaling of the connectivity graph related to the convergence to a graphon. In the limit, we obtain a system of measure-valued equations, which can be also represented as a PDE model on graphon, which reflects the heterogeneities in individual characteristics and social connectivity.

Stochastic SIR model with individual heterogeneity and infection-age dependent infectivity on large non-homogeneous random graphs

TL;DR

This work analyzes the large-population limit of an individual-based stochastic SIR model with infection-age dependent infectivity on a large non-homogeneous random graph. By tracking infection-age and individual heterogeneity through measure-valued processes, the authors prove a functional law of large numbers that yields a deterministic limit represented by a system of measure-valued equations and a PDE on a graphon. The limiting system admits existence, uniqueness, and regularity results, and can be reformulated as a linear PDE with a boundary condition derived from the susceptible-infected interaction, reflecting the graphon-based connectivity. The framework accommodates general heterogeneity in individuals and connectivity, including continuous state spaces and non-Markovian infection dynamics, providing a tractable limit description for complex epidemic processes on large networks.

Abstract

We study an individual-based stochastic SIR epidemic model with infection-age dependent infectivity on a large random graph, capturing individual heterogeneity and non-homogeneous connectivity. Each individual is associated with particular characteristics (for example, spatial location and age structure), which may not be i.i.d., and represented by a particular node. The connectivities among the individuals are given by a non-homogeneous random graph, whose connecting probabilities may depend on the individual characteristics of the edge. Each individual is associated with a random varying infectivity function, which is also associated with the individual characteristics. We use measure-valued processes to describe the epidemic evolution dynamics, tracking the infection age of all individuals, and their associated characteristics. We consider the epidemic dynamics as the population size grows to infinity under a specific scaling of the connectivity graph related to the convergence to a graphon. In the limit, we obtain a system of measure-valued equations, which can be also represented as a PDE model on graphon, which reflects the heterogeneities in individual characteristics and social connectivity.

Paper Structure

This paper contains 20 sections, 23 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Notation
  4. Model description
  5. Infection-age dependent infectivity
  6. Construction of the connectivity graph
  7. Epidemic dynamics
  8. Functional law of large numbers
  9. Discussions of the main result
  10. Properties of the limiting system of equations
  11. Existence and uniqueness of solution to the limiting equations
  12. Regularity of $\overline{\mathfrak{F}}(\cdot)$
  13. Alternative representation of the limit as PDEs
  14. Proof of Theorem \ref{['thm_cvg']}, our main result
  15. Convergence of $(\bar{\mu}^{S, N}_{\cdot}, \overline{\mathfrak{F}}^N(\cdot))$
  16. ...and 5 more sections

Key Result

Lemma 3.4

Under Assumption hyp-w, there exists a constant $\omega^*>0$ that is an upper-bound on $\bar{\omega}^N$ uniformly on $\mathbf{\mathbb{X}}^2$ and for $N$ sufficiently large.

Theorems & Definitions (54)

  • Remark 2.1
  • Remark 2.3
  • Remark 2.5
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 3.2
  • Lemma 3.4
  • Theorem 3.6
  • Proposition 4.1
  • ...and 44 more