Title: Spatial Disease Propagation With Hubs
Domain: Math

Question: How can disease propagation through common hubs be rigorously modeled using random bipartite geometric (RBG) graphs, and what are the key theoretical insights regarding: | | the degree statistics of agents and hubs under general connection functions, | | the conditions under which a critical hub density exists or vanishes depending on the support of the connection function, and | | the impact of dispersion in the connection function on percolation thresholds and large-scale outbreak risk?

Background: # Background Hints
Airborne infectious diseases spread via physical contact or proximity, and common hubs (e.g., supermarkets, airports) are key to their transmission. Understanding how agent density, hub density, and connectivity patterns jointly affect spread is vital for prevention.

Traditional homogeneous mixture models assume equal contact chances between individuals, with the basic reproduction number as a core indicator, but they fail to capture heterogeneous, location-dependent human interactions. Spatial models (e.g., discrete square lattice Bernoulli percolation, continuum Poisson Boolean models) address this by focusing on proximity-driven interactions and percolation (disease spreading to large components). Poisson random connection models generalize these, with dispersive functions aiding percolation, yet most spatial models overlook hubs.

Some abstract bipartite graph models include hubs but lack geometric dependence. The AB continuum percolation model uses two Poisson point processes but has limited connectivity rules, creating a need for a more general model integrating hubs, geometric dependence, and flexible connection functions.

Definitions: Remember the key definitions: | | * **Random Bipartite Geometric (RBG) Graph**: $G(\Phi,\Psi,f)$, where $\Phi$ are agents (density $\lambda$), $\Psi$ are hubs (density $\mu$), and edges exist with probability $f(\|x-y\|)$. | * **Connection Function $f$**: a non-increasing function controlling visit probability, possibly bounded or unbounded in support. | * **Degree Statistics**: Mean degree of an agent is | | $$ | \mathbb{E}[\deg(\text{agent})] = \mu c_d d \int_0^\infty f(r) r^{d-1} dr, | $$ | | with symmetric form for hubs. | * **Critical Hub Density $\mu_c$**: the minimal hub density required for the RBG graph to percolate. | * **Dispersed Connection Function**: $f_p(r) = p f(\sqrt{d}\,pr)$, which spreads connections farther while preserving mean degree.

Methodology: * **Palm calculus & Campbell’s theorem** for deriving mean degrees under point processes. | * **Mass transport principle** to connect agent and hub degree expectations. | * **Coupling arguments** (comparing RBG graphs to Boolean models or Galton–Watson branching processes) to establish percolation thresholds. | * **Probability generating functional (PGFL)** of PPPs to handle edge indicators and variance computations. | * **Comparison between site percolation and bond percolation** to analyze dispersion effects.

Answer: | Model spatial disease propagation through **random bipartite geometric (RBG) graphs**. Agents are distributed according to a stationary point process $\Phi$ of intensity $\lambda$, and hubs are distributed according to another point process $\Psi$ of intensity $\mu$. An agent $x \in \Phi$ connects to a hub $y \in \Psi$ with probability | | $$ | \mathbb{P}((x,y)\in E) = f(\|x-y\|), | $$ | | where $f:\mathbb{R}_+ \to [0,1]$ is a non-increasing **connection function**. | | --- | | ## 1. Degree Statistics | | For a typical agent at the origin, the expected degree is given by **Campbell’s theorem**: | | $$ | \mathbb{E}_o \Bigg[\sum_{y\in\Psi} I(o,y)\Bigg] = \mu c_d d \int_0^\infty f(r) r^{d-1} dr, | $$ | | where $c_d$ is the volume of the unit ball in $\mathbb{R}^d$. | Symmetrically, for a typical hub, the mean degree is: | | $$ | \mathbb{E}\big[\deg(h)\big] = \lambda c_d d \int_0^\infty f(r) r^{d-1} dr. | $$ | | For **independent Poisson point processes (PPPs)**, let | | $$ | M = \# \{\text{agents connected to origin}\}, \quad N = \# \{\text{two-edge paths from origin to other agents}\}. | $$ | | Then: | | $$ | \mathbb{E}[N] = \lambda \mu \left( \int_{\mathbb{R}^d} f(\|x\|) dx \right)^2, \tag{1} | $$ | | $$ | \mathbb{E}[M] = \lambda \int_{\mathbb{R}^d} \Big(1 - e^{-\mu \int_{\mathbb{R}^d} f(\|y\|) f(\|x-y\|) dy}\Big) dx. \tag{2} | $$ | | Both $M, N$ are **super-Poisson** random variables, with variances exceeding their means. | | --- | | ## 2. Critical Hub Density and Percolation | | We define the **percolation region**: | | $$ | D = \{ (\lambda,\mu): G(\Phi,\Psi,f)\ \text{percolates} \}. | $$ | | Percolation means the existence of an infinite connected component almost surely. | | ### Theorem 1 (Threshold Existence via Support of $f$) | | * If $f$ has **bounded support** on $[0,a]$, then the critical hub density satisfies | | $$ | \mu_c = \zeta(2a) > 0, | $$ | | where $\zeta(2a)$ is the percolation threshold of the Boolean model with radius $2a$. | | * If $f$ has **unbounded support**, then | | $$ | \mu_c = 0, | $$ | | i.e., *arbitrarily small hub density suffices for percolation if agent density is high enough*. | | This dichotomy shows that **long-distance interactions (unbounded $f$) fundamentally remove the epidemic threshold**. | | --- | | ## 3. Dispersion Effect | | For any $p \in (0,1]$, define the **dispersed connection function**: | | $$ | f_p(r) = p f(\sqrt{d}\,pr). | $$ | | This transformation spreads out connections without changing mean degree: | | $$ | \int_0^\infty f(r) r^{d-1} dr = \int_0^\infty f_p(r) r^{d-1} dr. | $$ | | ### Theorem 2 (Effect of Dispersion) | | For all $f$ and $p \in (0,1]$, | | $$ | \lambda_c^p(\mu) \leq \lambda_c(\mu), \quad \mu_c^p \leq \mu_c. | $$ | | Thus, **increasing dispersion lowers the percolation threshold** and enlarges the percolation region. Epidemiologically, this means allowing agents to travel further (even at lower probability) makes large-scale outbreaks more likely. | | --- | | ## 4. Practical Implications | | * The RBG graph model shows that **hub density thresholds** are not universal: they depend critically on whether human mobility is bounded. | * **Bounding travel distances** (finite support of $f$) guarantees the existence of a threshold below which epidemics cannot spread globally. | * **Allowing rare long-range travel** (infinite support of $f$) destroys this threshold: even sparse hubs can sustain percolation. | * **Increased dispersion** (e.g., flights or long commutes) significantly lowers the epidemic threshold, amplifying outbreak risk. | | --- | | ## Final Synthesis | | The RBG graph framework unifies stochastic geometry and epidemic theory by explicitly linking: | | 1. **Degree distributions** → heterogeneity of individual risk. | 2. **Support of the connection function** → existence vs. absence of epidemic thresholds. | 3. **Dispersion** → sensitivity of outbreak risk to long-range mobility. | | These rigorous results establish that curbing long-distance travel or limiting hub density is mathematically necessary to prevent large-scale disease outbreaks. | |

Checklist: 1. **Model setup**: The answer should correctly describe the RBG graph framework, including the roles of agents, hubs, and the connection function $f$. | 2. **Degree statistics**: The answer should derive or explain the mean degree formulas for agents and hubs using integrals of $f$. | 3. **Percolation threshold condition**: The answer should explain that the existence of a positive hub density threshold depends on whether $f$ has bounded support. | 4. **Critical cases**: The answer should explicitly state the dichotomy: bounded support ⇒ $\mu_c > 0$; unbounded support ⇒ $\mu_c = 0$. | 5. **Dispersion effect**: The answer should address how dispersing the connection function lowers the percolation threshold and enlarges the percolation region. | 6. **Methodological tools**: The answer should mention key analytical tools such as stochastic geometry, Palm calculus, Campbell’s theorem, coupling arguments, or branching processes. | 7. **Practical implications**: The answer should connect the theoretical results to real-world epidemic insights, such as the role of long-distance travel and hub density in determining outbreak risk.