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Functional Central Limit Theorem and SPDE for epidemic model with memory of the last infection and waning immunity

Arsene Brice Zotsa-Ngoufack

TL;DR

The paper proves a Functional Central Limit Theorem for a memory-enabled, age- and trait-structured stochastic epidemic model, showing that the fluctuation process around the deterministic FLLN limit converges to a Gaussian process governed by a nonlinear stochastic integral equation. The authors build a rigorous diffusion-limit description using weighted Sobolev spaces to establish tightness and identify the limit, and they derive an SPDE for the density component driven by Gaussian noise, along with a stochastic Volterra representation. This extends prior non-memory results by incorporating memory from previous infections and waning immunity, while connecting to existing SPDE and Volterra frameworks. The results yield a tractable diffusion approximation for fluctuations in the force of infection and susceptibility, with potential implications for inference and numerical simulation in non-Markovian epidemic models.

Abstract

We study the fluctuations of a stochastic epidemic model with memory of the last infections, varying infectivity, and waning immunity, as introduced in Guerin and Zotsa-Ngoufack:arXiv preprint arXiv:2505.00601. The dynamics of the epidemic model are described by a measure-valued process with respect to infection age and individual traits. The Functional Law of Large Numbers (FLLN) is formulated as an integral equation, which is solved by a deterministic measure. In this article, we establish the Functional Central Limit Theorem (FCLT), capturing the fluctuations of the stochastic model around its deterministic limit. The limit of the FCLT is given by a nonlinear stochastic integral equation which is solved by a random signed-measure. We further derive the weak solution in the form of a stochastic partial differential equation (SPDE) and propose an alternative representation of the FCLT, as fluctuations in the average total force of infection and average susceptibility.

Functional Central Limit Theorem and SPDE for epidemic model with memory of the last infection and waning immunity

TL;DR

The paper proves a Functional Central Limit Theorem for a memory-enabled, age- and trait-structured stochastic epidemic model, showing that the fluctuation process around the deterministic FLLN limit converges to a Gaussian process governed by a nonlinear stochastic integral equation. The authors build a rigorous diffusion-limit description using weighted Sobolev spaces to establish tightness and identify the limit, and they derive an SPDE for the density component driven by Gaussian noise, along with a stochastic Volterra representation. This extends prior non-memory results by incorporating memory from previous infections and waning immunity, while connecting to existing SPDE and Volterra frameworks. The results yield a tractable diffusion approximation for fluctuations in the force of infection and susceptibility, with potential implications for inference and numerical simulation in non-Markovian epidemic models.

Abstract

We study the fluctuations of a stochastic epidemic model with memory of the last infections, varying infectivity, and waning immunity, as introduced in Guerin and Zotsa-Ngoufack:arXiv preprint arXiv:2505.00601. The dynamics of the epidemic model are described by a measure-valued process with respect to infection age and individual traits. The Functional Law of Large Numbers (FLLN) is formulated as an integral equation, which is solved by a deterministic measure. In this article, we establish the Functional Central Limit Theorem (FCLT), capturing the fluctuations of the stochastic model around its deterministic limit. The limit of the FCLT is given by a nonlinear stochastic integral equation which is solved by a random signed-measure. We further derive the weak solution in the form of a stochastic partial differential equation (SPDE) and propose an alternative representation of the FCLT, as fluctuations in the average total force of infection and average susceptibility.

Paper Structure

This paper contains 19 sections, 17 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Model and Results
  4. Model description
  5. Some known results
  6. Main Results
  7. Weighted Sobolev spaces
  8. Main results
  9. SPDE
  10. Link with literature
  11. Proof of Central Limit Theorem
  12. Tightness
  13. Preliminary results
  14. Suitable Sobolev space to establish the tightness of $\hat{\mu}^N$
  15. Proof of tightness
  16. ...and 4 more sections

Key Result

Theorem 2.1

Under Assumption V-As-1 and Assumption V-As-2, as $N\to\infty,\,\mu ^N$ converges to $\mu$ in distribution to a measure $\mu \in\mathbb{D}(\mathbb{R}_+;\mathcal{P}(\mathbb{R}_+\times\Theta)),$ which is the unique solution of for any bounded function $f$ on $\mathbb{R}_+\times\mathbb{R}_+\times\Theta$, and of class $\mathcal{C}^1$ with respect to the first two variables, where the operator $R$ is

Theorems & Definitions (38)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Definition 2.8
  • Proposition 2.9
  • Lemma 3.1
  • ...and 28 more