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Spatial Disease Propagation With Hubs

Ke Feng, Martin Haenggi

TL;DR

The paper develops a random bipartite geometric (RBG) graph model to study disease spread via hubs, with agents and hubs distributed by independent spatial point processes and edges formed with probability $f(\\|x-y\\|)$. It derives mean degree formulas for general point processes and Poisson PPPs, defines quantities counting two-edge paths and shared connections, and analyzes percolation thresholds. A key result is that the existence of a giant outbreak-capable component depends on the boundedness of the support of $f$, with dispersion of $f$ (long-range connections) lowering the percolation threshold and making disease spread more likely; bounds are obtained via coupling to unipartite models and Galton-Watson processes. The work also discusses extensions to dependent hub-agent placements, spatiotemporal dynamics, and SIS-type propagation, highlighting the practical implication that limiting long-distance travel can curb large-scale outbreaks. Overall, the RBG framework provides a tractable, geometry-aware lens on hub-driven disease propagation in continuous space.

Abstract

Physical contact or proximity is often a necessary condition for the spread of infectious diseases. Common destinations, typically referred to as hubs or points of interest, are arguably the most effective spots for the type of disease spread via airborne transmission. In this work, we model the locations of individuals (agents) and common destinations (hubs) by random spatial point processes in $\mathbb{R}^d$ and focus on disease propagation through agents visiting common hubs. The probability of an agent visiting a hub depends on their distance through a connection function $f$. The system is represented by a random bipartite geometric (RBG) graph. We study the degrees and percolation of the RBG graph for general connection functions. We show that the critical density of hubs for percolation is dictated by the support of the connection function $f$, which reveals the critical role of long-distance travel (or its restrictions) in disease spreading.

Spatial Disease Propagation With Hubs

TL;DR

The paper develops a random bipartite geometric (RBG) graph model to study disease spread via hubs, with agents and hubs distributed by independent spatial point processes and edges formed with probability . It derives mean degree formulas for general point processes and Poisson PPPs, defines quantities counting two-edge paths and shared connections, and analyzes percolation thresholds. A key result is that the existence of a giant outbreak-capable component depends on the boundedness of the support of , with dispersion of (long-range connections) lowering the percolation threshold and making disease spread more likely; bounds are obtained via coupling to unipartite models and Galton-Watson processes. The work also discusses extensions to dependent hub-agent placements, spatiotemporal dynamics, and SIS-type propagation, highlighting the practical implication that limiting long-distance travel can curb large-scale outbreaks. Overall, the RBG framework provides a tractable, geometry-aware lens on hub-driven disease propagation in continuous space.

Abstract

Physical contact or proximity is often a necessary condition for the spread of infectious diseases. Common destinations, typically referred to as hubs or points of interest, are arguably the most effective spots for the type of disease spread via airborne transmission. In this work, we model the locations of individuals (agents) and common destinations (hubs) by random spatial point processes in and focus on disease propagation through agents visiting common hubs. The probability of an agent visiting a hub depends on their distance through a connection function . The system is represented by a random bipartite geometric (RBG) graph. We study the degrees and percolation of the RBG graph for general connection functions. We show that the critical density of hubs for percolation is dictated by the support of the connection function , which reveals the critical role of long-distance travel (or its restrictions) in disease spreading.

Paper Structure

This paper contains 21 sections, 11 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related Work
  4. Contribution
  5. System Model
  6. The RBG Graph
  7. Dispersed RBG Graph
  8. Degrees of the RBG Graph
  9. General Point Processes
  10. Poisson Point Processes
  11. Examples
  12. Percolation
  13. Definitions
  14. Some Bounds on the Percolation Thresholds
  15. Dispersed RBG Graph
  16. ...and 6 more sections

Key Result

Theorem 1

For the PPP, the means of $M,~N$ are and Further, the variance of $N$ is given in (eq: E_N_o^2), and the variance of $M$ satisfies $\mathbb{V} M \geq \mathbb{E} M$.

Figures (2)

  • Figure 1: Illustrating one realization of the RBG graph on $\mathbb{R}^2$, where only its segment in a square window $[-0.5,0.5]^2$ is presented. Hubs are marked with circles, agents with dots, and edges with line segments. Agents without edges are in light gray. In both figures, the density of agents is 100 and the density of hubs is 10; the average number of edges of the typical hub is 5.
  • Figure 2: Mean and standard deviation of $N$ and $M$ for $f =\mathbbm{1}{(r\leq 0.2122)}$. We compare $(\lambda,\mu) = (5,50), (50,5)$ respectively. $\mathbb{E} N=5$. $\sqrt{\mathbb{V}{M}}$ is obtained via simulation whereas the rest via Corollary 2.

Theorems & Definitions (30)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Corollary 1
  • proof
  • Remark 4
  • Corollary 2
  • proof
  • ...and 20 more