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SPDE for stochastic SIR epidemic models with infection-age dependent infectivity

Guodong Pang, Etienne Pardoux

TL;DR

The paper addresses fluctuations around PDE limits for a non-Markovian SIR epidemic with infection-age dependent infectivity by formulating the dynamics as a measure-valued process driven by two independent Poisson random measures. It derives a functional law of large numbers leading to a linear PDE with a boundary determined by nonlinear Volterra-type integral equations and a functional central limit theorem yielding a linear SPDE driven by Gaussian noises, with an explicit solution. Uniqueness of the PDE and SPDE limits is established under regularity assumptions on the infectious duration distribution, and the results provide a rigorous framework for understanding the stochastic fluctuations around non-Markovian epidemic PDE limits, including explicit expressions for the limiting processes and their covariances. This work extends SPDE-based fluctuation analysis to non-Markovian epidemic models and lays groundwork for studying fluctuations around age-structured epidemic limits in more complex settings.

Abstract

We study the stochastic SIR epidemic model with infection-age dependent infectivity for which a measure-valued process is used to describe the ages of infection for each individual. We establish a functional law of large numbers (FLLN) and a functional central limit theorem (FCLT) for the properly scaled measure-valued processes together with the other epidemic processes to describe the evolution dynamics. In the FLLN, assuming that the hazard rate function of the infection periods is bounded and the ages at time 0 of the infections of the initially infected individuals are bounded, we obtain a PDE limit for the LLN-scaled measure-valued process, for which we characterize its solution explicitly. The PDE is linear with a boundary condition given by the unique solution to a set of Volterra-type nonlinear integral equations. In the FCLT, we obtain an SPDE for the CLT-scaled measure-valued process, driven by two independent white noises coming from the infection and recovery processes. The SPDE is also linear and coupled with the solution to a system of stochastic Volterra-type linear integral equations driven by three independent Gaussian noises, one from the random infection functions in addition to the two white noises mentioned above. The solution to the SPDE can be also explicitly characterized, given this auxiliary process. The uniqueness of the SPDE solution is established under stronger assumptions (density and its derivative being locally bounded) on the distribution function of an infectious duration.

SPDE for stochastic SIR epidemic models with infection-age dependent infectivity

TL;DR

The paper addresses fluctuations around PDE limits for a non-Markovian SIR epidemic with infection-age dependent infectivity by formulating the dynamics as a measure-valued process driven by two independent Poisson random measures. It derives a functional law of large numbers leading to a linear PDE with a boundary determined by nonlinear Volterra-type integral equations and a functional central limit theorem yielding a linear SPDE driven by Gaussian noises, with an explicit solution. Uniqueness of the PDE and SPDE limits is established under regularity assumptions on the infectious duration distribution, and the results provide a rigorous framework for understanding the stochastic fluctuations around non-Markovian epidemic PDE limits, including explicit expressions for the limiting processes and their covariances. This work extends SPDE-based fluctuation analysis to non-Markovian epidemic models and lays groundwork for studying fluctuations around age-structured epidemic limits in more complex settings.

Abstract

We study the stochastic SIR epidemic model with infection-age dependent infectivity for which a measure-valued process is used to describe the ages of infection for each individual. We establish a functional law of large numbers (FLLN) and a functional central limit theorem (FCLT) for the properly scaled measure-valued processes together with the other epidemic processes to describe the evolution dynamics. In the FLLN, assuming that the hazard rate function of the infection periods is bounded and the ages at time 0 of the infections of the initially infected individuals are bounded, we obtain a PDE limit for the LLN-scaled measure-valued process, for which we characterize its solution explicitly. The PDE is linear with a boundary condition given by the unique solution to a set of Volterra-type nonlinear integral equations. In the FCLT, we obtain an SPDE for the CLT-scaled measure-valued process, driven by two independent white noises coming from the infection and recovery processes. The SPDE is also linear and coupled with the solution to a system of stochastic Volterra-type linear integral equations driven by three independent Gaussian noises, one from the random infection functions in addition to the two white noises mentioned above. The solution to the SPDE can be also explicitly characterized, given this auxiliary process. The uniqueness of the SPDE solution is established under stronger assumptions (density and its derivative being locally bounded) on the distribution function of an infectious duration.

Paper Structure

This paper contains 13 sections, 19 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Notation
  4. Model and Results
  5. Model
  6. A linear first-order PDE
  7. FLLN
  8. FCLT
  9. Proof of Theorem \ref{['thm-FLLN']}
  10. Proof of Theorem \ref{['thm-FCLT-mu']}
  11. On the convergence of $(\widehat{S}^N, \widehat{\mathfrak{F}}^N)$
  12. On the convergence of $(\widehat{I}^N, \widehat{R}^N)$
  13. Declarations

Key Result

Theorem 2.1

Assume that the initial law $\bar{\mu}_0$ satisfies Assumption AS-g and that the hazard rate function $h$ is locally bounded. Then, as $N\to\infty$, in probability, where $(\bar{S}, \overline{\mathfrak{F}}) \in C\times D$ is the unique solution to the following set of integral equations, with If $\bar{\lambda}(\cdot) \in C$, then $\overline{\mathfrak{F}}(t)$ and $\overline{\Upsilon}(t)$ are con

Theorems & Definitions (48)

  • Theorem 2.1
  • Remark 2.1
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.2
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.1
  • ...and 38 more