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Functional Central limit theorems for epidemic models with varying infectivity and waning immunity

Arsene-Brice Zotsa-Ngoufack

TL;DR

This work establishes a functional central limit theorem for a non-Markovian epidemic model with random, time-varying infectivity and gradual loss of immunity, building on a prior FLLN result. The fluctuations around the deterministic limit are governed by a two-dimensional Gaussian process and a coupled system of stochastic nonlinear integral equations, capturing the joint dynamics of the average force of infection and average susceptibility. Central to the analysis are new moment bounds for stochastic integrals with Poisson random measures, a quarantine-based approximation to bound descendant counts, and careful tightness arguments that exploit Regime decompositions and the proposed integral mappings. The results extend non-Markovian and interacting epidemic limit theories and provide a rigorous description of the stochastic fluctuations around endemic/exit dynamics with waning immunity, with potential implications for uncertainty quantification in similar infectious-disease models.

Abstract

We study an individual-based stochastic epidemic model in which infected individuals become susceptible again following each infection (generalized SIS model). Specifically, after each infection, the infectivity is a random function of the time elapsed since the infection, and each recovered individual loses immunity gradually (equivalently, becomes gradually susceptible) after some time according to a random susceptibility function. The epidemic dynamics is described by the average infectivity and susceptibility processes in the population together with the numbers of infected and susceptible/uninfected individuals. In \cite{forien-Zotsa2022stochastic}, a functional law of large numbers (FLLN) is proved as the population size goes to infinity, and asymptotic endemic behaviors are also studied. In this paper, we prove a functional central limit theorem (FCLT) for the stochastic fluctuations of the epidemic dynamics around the FLLN limit. The FCLT limit for the aggregate infectivity and susceptibility processes is given by a system of stochastic non-linear integral equation driven by a two-dimensional Gaussian process.

Functional Central limit theorems for epidemic models with varying infectivity and waning immunity

TL;DR

This work establishes a functional central limit theorem for a non-Markovian epidemic model with random, time-varying infectivity and gradual loss of immunity, building on a prior FLLN result. The fluctuations around the deterministic limit are governed by a two-dimensional Gaussian process and a coupled system of stochastic nonlinear integral equations, capturing the joint dynamics of the average force of infection and average susceptibility. Central to the analysis are new moment bounds for stochastic integrals with Poisson random measures, a quarantine-based approximation to bound descendant counts, and careful tightness arguments that exploit Regime decompositions and the proposed integral mappings. The results extend non-Markovian and interacting epidemic limit theories and provide a rigorous description of the stochastic fluctuations around endemic/exit dynamics with waning immunity, with potential implications for uncertainty quantification in similar infectious-disease models.

Abstract

We study an individual-based stochastic epidemic model in which infected individuals become susceptible again following each infection (generalized SIS model). Specifically, after each infection, the infectivity is a random function of the time elapsed since the infection, and each recovered individual loses immunity gradually (equivalently, becomes gradually susceptible) after some time according to a random susceptibility function. The epidemic dynamics is described by the average infectivity and susceptibility processes in the population together with the numbers of infected and susceptible/uninfected individuals. In \cite{forien-Zotsa2022stochastic}, a functional law of large numbers (FLLN) is proved as the population size goes to infinity, and asymptotic endemic behaviors are also studied. In this paper, we prove a functional central limit theorem (FCLT) for the stochastic fluctuations of the epidemic dynamics around the FLLN limit. The FCLT limit for the aggregate infectivity and susceptibility processes is given by a system of stochastic non-linear integral equation driven by a two-dimensional Gaussian process.
Paper Structure (33 sections, 46 theorems, 317 equations, 1 figure)

This paper contains 33 sections, 46 theorems, 317 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and Results
  3. Model description
  4. Already known results
  5. Main Results
  6. Assumptions
  7. Statement of the main theorem
  8. Relaxing Assumption \ref{['TCL-AS-lambda-0']}
  9. Proof of Theorem \ref{['TCL']}
  10. Decomposition of the fluctuations $(\hat{\mathfrak S}^N_1,\hat{\mathfrak F}^N_1)$
  11. Two continuous integral mappings
  12. Proof of Theorem \ref{['TCL']}
  13. Alternative expressions of $(\hat{\mathfrak S}^N_2,\hat{\mathfrak F}^N_2)$ and their limits
  14. Some preliminary results
  15. Tightness criterion
  16. ...and 18 more sections

Key Result

Theorem 2.2

Under Assumption TCL-AS-lambda-0, where $(\overline{\mathfrak{S}},\overline{\mathfrak{F}})$ satisfies the following system of equations, Given the solution $(\overline{\mathfrak{S}},\overline{\mathfrak{F}})$, where $(\overline{U},\overline{I})$ is given by

Figures (1)

  • Figure 1: Illustration of a typical realization of the random infectivity and susceptibility functions of an individual from the time of infection to the time of recovery, and then to the time of losing immunity and becoming fully susceptible (or in general, partially susceptible).

Theorems & Definitions (68)

  • Theorem 2.2
  • Remark 2.3
  • Definition 2.7
  • Remark 2.8
  • Lemma 2.9
  • proof
  • Theorem 2.10
  • Corollary 2.11
  • Lemma 3.1
  • Proposition 3.2
  • ...and 58 more