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Formulation a new SIR model with non-local mobility

Ciana Applegate, Jiaxu Li, Dan Han

TL;DR

This work addresses how nonlocal mobility alters the classical SIR dynamics by formulating a SIR model with a nonlocal mobility operator $\mathcal{L}$ on $\mathbb{Z}^d$ and studying first- and second-moment dynamics. The authors derive closed-form first-moment equations $\partial_t m_1^{S,I,R}$ with $\kappa \mathcal{L}$ and define a mobility-adjusted reproduction number $R_0^m=\frac{\kappa\hat{a}(k)+\beta}{\gamma}$, together with second-moment analysis and an intermittency diagnostic. They solve the moment equations in homogeneous and inhomogeneous spaces using Fourier methods, obtain explicit solutions and asymptotics, and show that mobility can either suppress or enhance spread and induce clustering. The work thus provides a flexible, analytically tractable framework for epidemic propagation that accounts for realistic mobility patterns and offers insights for targeted interventions.

Abstract

In this manuscript, we develop a mobility-based Susceptible-Infectious-Recovered (SIR) model to elucidate the dynamics of pandemic propagation. While traditional SIR models within the field of epidemiology aptly characterize transitions among susceptible, infected, and recovered states, they typically neglect the inherent spatial mobility of particles. To address this limitation, we introduce a novel dynamical SIR model that incorporates nonlocal spatial motion for three distinct particle types, thereby bridging the gap between epidemiological theory and real-world mobility patterns. This paper primarily focuses on analyzing the long-term behavior of this dynamic system, with specific emphasis on the computation of first and second moments. We propose a new reproduction number $R_0^m$ and compare it with the classical reproduction number $R_0$ in the traditional SIR model. Furthermore, we rigorously examine the phenomenon of intermittency within the context of this enhanced SIR model. The results contribute to a more comprehensive understanding of pandemic spread dynamics, considering both the interplay between disease transmission and population mobility and the impact of spatial motion on the system's behavior over time.

Formulation a new SIR model with non-local mobility

TL;DR

This work addresses how nonlocal mobility alters the classical SIR dynamics by formulating a SIR model with a nonlocal mobility operator on and studying first- and second-moment dynamics. The authors derive closed-form first-moment equations with and define a mobility-adjusted reproduction number , together with second-moment analysis and an intermittency diagnostic. They solve the moment equations in homogeneous and inhomogeneous spaces using Fourier methods, obtain explicit solutions and asymptotics, and show that mobility can either suppress or enhance spread and induce clustering. The work thus provides a flexible, analytically tractable framework for epidemic propagation that accounts for realistic mobility patterns and offers insights for targeted interventions.

Abstract

In this manuscript, we develop a mobility-based Susceptible-Infectious-Recovered (SIR) model to elucidate the dynamics of pandemic propagation. While traditional SIR models within the field of epidemiology aptly characterize transitions among susceptible, infected, and recovered states, they typically neglect the inherent spatial mobility of particles. To address this limitation, we introduce a novel dynamical SIR model that incorporates nonlocal spatial motion for three distinct particle types, thereby bridging the gap between epidemiological theory and real-world mobility patterns. This paper primarily focuses on analyzing the long-term behavior of this dynamic system, with specific emphasis on the computation of first and second moments. We propose a new reproduction number and compare it with the classical reproduction number in the traditional SIR model. Furthermore, we rigorously examine the phenomenon of intermittency within the context of this enhanced SIR model. The results contribute to a more comprehensive understanding of pandemic spread dynamics, considering both the interplay between disease transmission and population mobility and the impact of spatial motion on the system's behavior over time.
Paper Structure (13 sections, 8 theorems, 55 equations, 1 figure, 5 tables)

This paper contains 13 sections, 8 theorems, 55 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Formulation of SIR model with mobility
  3. The First Moments
  4. Deriviation of the Differential Equations
  5. Solving for the First Moments
  6. Analyzing the First Moments
  7. The Second Moments
  8. Deriving the Differential Equations
  9. Solving for the Second Moments
  10. Analyzing the Second Moments
  11. Intermittency Analysis
  12. Discussion
  13. Compliance with Ethical Standards

Key Result

Theorem 1

The differential equations for the first moment of the susceptible, infected, and recovered groups are where $m_{1}^{D}(t,x) = E[D(t,x)]$ for $D= S, I,$ or $R$ and the discrete Laplace operator is the generator of the underlying random walk of the individuals and

Figures (1)

  • Figure 1: This comparison highlights the critical differences in epidemic spread dynamics under local versus nonlocal movement patterns. Over 50,000 iterations, local movement patterns driven by Brownian motion or simple random walk tend to confine the spread within smaller areas, often resulting in centralized outbreaks, whereas our nonlocal modeling approach facilitates a pattern in wide and more dispersed distributions of infections.

Theorems & Definitions (17)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 1: Fourier Transform
  • Definition 2: Inverse Fourier Transform
  • Definition 3: Intensity of Mobility Effect
  • Lemma 1: Fourier Transform of function $\mathcal{L}f(k)$
  • Lemma 2
  • proof
  • ...and 7 more