Travel distance, frequency of return and the spread of disease

Abstract / Central Thesis

• Paper examines whether the invariant pattern in urban mobility – the product of travel distance and visit frequency $v:=r\cdot f$ ("exploration velocity") – also governs epidemic spread.
• Uses two large‐scale mobility traces (New York City, USA & Dakar, Senegal) + agent–based SI, SIR and SEIR simulations calibrated for COVID-19.
• Key findings:
  • Epidemic speed $\tau$ and spatial dispersion both collapse onto a single curve when plotted against $v=r\cdot f$, irrespective of $r$ and $f$ individually.
  • Empirical form: $\tau(r\cdot f)=(r\cdot f)^a\,b$ with city/model–specific $a,b$.
  • Relationship persists across disease models, different $R_0$, and synthetic trajectories generated with a modified Preferential Exploration–Preferential Return (PEPR) mobility model.
  • Policy implication: limiting distance alone is insufficient; bounding $r\cdot f$ can slow and localize outbreaks more efficiently.

Background & Motivation

• Mobility ➜ vector for pathogen transmission; policy responses to COVID-19 focused largely on geographic radius limits (neighborhood, city, state, country closures).
• Recent mobility discovery (Schläpfer et al. 2021): for any urban location, the mean total inbound distance per visitor is constant; mathematically $N(r,f)\propto 1/(rf)^2$ – a universal inverse-square law.
• Hypothesis: if the same invariant governs infections, then both travel distance $r$ and return frequency $f$ must jointly determine epidemic dynamics.

Datasets

New York City (NYC)

• Source: GPS pings from X-Mode (Feb 2020).
• Raw: $\approx 4.8\times10^5$ anonymised users; analysis subsample: $N=10{,}000$ appearing daily.
• Duration: 28 days.

Dakar, Senegal

• Source: Call Detail Records (D4D, SET2, Jan 2013).
• After cleaning: 173 000 users, 173 cell towers; simulation sub-sample: 10 000.
• Duration: 14 days.

Data Pre-processing

NYC (high-resolution GPS)

1. DBSCAN clustering (eps $=0.0004^\circ\approx56\,\text{m}$, minPts $=5$) → distinct places.
2. Home = most visited cluster.
3. Discard visits < $\tau_{\text{min}}=15\,\text{min}$.
4. Impose distance cap $r$ via haversine; drop trips beyond.
5. Impose frequency cap $f$ by randomly keeping $f$ distinct visits / location (home exempt).

Dakar (coarse CDR)

1. Each cell tower = location.
2. Home = tower with longest cumulative residence.
3. Drop visits < $\tau_{\text{min}}=10\,\text{min}$.
4. Distance cap $r$: haversine between towers; drop beyond.
5. Frequency cap $f$ exactly as NYC (home exempt).

Simulation Framework

• Agent-based models with $N=10{,}000$ agents following processed trajectories.
• Time step: 900 s (NYC) or 600 s (Dakar).
• Disease states:
  • SI, SIR, SEIR; baseline calibration to early COVID-19 (Chen 2020).
• Epidemiological parameters (daily):
  $\beta=\frac{R_0}{\text{infectious period}}=\frac{3.58}{5.8}=0.617$, $\sigma=1/5.2$, $\gamma=1/5.8$.
• Per-timestep conversion ((s)=steps per day):
  $\beta^<em>=\beta/s$, $\sigma^</em>=1-\sqrt[s]{1-\sigma}$,
  $\gamma^*=1-\sqrt[s]{1-\gamma}$.
• Infection occurs if two agents are within:
  • NYC: 190 m radius
  • Dakar: same tower
    Probability: $P[S\to E]=\beta^*\,I<em>{\text{local}}/N</em>{\text{local}}$.
• Initial prevalence: 5 % (NYC); 5 % or 1 % (Dakar SI/SIR).
• Restrictions explored on $r\in[?]$, $f\in{1,2,3,4,6,10}$ to modulate $v=r\,f$.

Key Empirical Results

Epidemic Speed $\tau$

• For fixed $f$, $\tau$ decreases monotonically with radius $r$.
• For fixed $r$, $\tau$ decreases with frequency $f$.
• Rescale abscissa $r\to r\,f$ ⇒ curves collapse onto one logarithmic‐like line (Fig 1c–d).
• Best-fit (SEIR):
  • NYC: $a=-0.08$, $b=2292.58$, $R^2=0.981$;
  • Dakar: $a=-0.01$, $b=1853.57$, $R^2=0.902$.
• Same collapse holds for SI & SIR (Fig 5) and for other $R<em>0$ values (Delta $R</em>0\approx8.2$, H1N1 $R_0=1.46$; Fig 6).

Spatial Dispersion (Marcon–Puech $M(k)$)

• $M(k,r,f)$ high ⇒ clustering; low ⇒ homogeneous spread.
• For given radius $k$ (e.g.
  $k=700\,\text{m}$) $M$ falls as both $r$ and $f$ rise (Fig 2a).
• Rescaling $r\to r\,f$ collapses onto single curve (Fig 2b). Robust for multiple $k$ (Fig 7).
• Visual maps (Fig 2c–d): equal $r$, varied $f$ → broader reach; equal $r\,f$, varied components → near-identical patterns.

Mechanistic Explanation via Contact Network

• Treat movement as dynamic proximity network; disease speed linked to degree stats.
• Known SIR formula for characteristic time $\hat{\tau}$:
  $\hat{\tau}=\frac{\langle k \rangle}{\langle k^2 \rangle-(\gamma+\beta)\langle k \rangle}$.
• Simulations show:
  • $\langle k \rangle$ and $\langle k^2 \rangle$ scale logarithmically with $v=r\,f$ (Fig 3a–b).
  • Plugging into formula predicts observed $\tau$ (Fig 3c).
• Intuition: limiting $r$ or $f$ lowers mean contacts and compresses variance by pruning long-distance, high-frequency trips.

Synthetic Mobility Experiments (PEPR Model)

• Original PEPR (
  always travelling) reproduces mobility law but not $\tau$–$r\,f$ relation; reason: no home time ⇒ inflated contacts.
• Modification: introduce stay-home probability $P_{\text{travel}}$.
  • $P_{\text{travel}}=0.25$ (matches NYC home proportion) yields clear $\tau$–$r\,f$ collapse (Fig 4a–b, $R^2=0.875/0.924$).
  • $P_{\text{travel}}=1$ (original) fails (Fig 4c–d).
  • $P_{\text{travel}}=0.40$ similar success (Fig 8).
• Model details:
  • At each step, with prob. $P_{\text{travel}}$: move; else stay home.
  • Exploration vs return: new‐location prob.
    $P_{\text{new}}=\rho S^{-\gamma}$ with fitted $\rho=0.500$, $\gamma=0.267$.
  • New‐location distance $\Delta r$ drawn from heavy tail $P(\Delta r)\propto|\Delta r|^{-1-\alpha}$, $\alpha=0.55$.

Discussion & Implications

• Empirical universal curve traces back to inverse-square visitation law $N(r,f)\propto1/(rf)^2$; exact derivation of $\tau(rf)$ remains open.
• Policy insight:
  • Short, high-frequency trips can be as dangerous as long, rare trips.
  • Effective containment requires bounding $v=r\,f$, not distance alone.
  • Possibility to allow infrequent long trips (e.g.
    to hospitals) if daily commute frequency reduced – supports hybrid/remote work policies.
• Limitations & Future Work
  • Behavioural response to velocity caps unknown.
  • Classic compartment models (SI/SIR/SEIR) simplify biology, assume equal infectivity within radius $\varepsilon$, identical agents.
  • Data city-level only; need inter-city / international validation via metapopulation models.
  • Further theoretical work to derive $\tau(rf)$ from $N(r,f)\propto1/(rf)^2$.

Ethical & Practical Considerations

• Data anonymised; access requires permission from providers (X-Mode, ORANGE/SONATEL).
• Potential application in real-time mobility dashboards for public-health decision-makers.
• Balancing privacy, freedom of movement, and health benefits crucial.

Numerical & Statistical Highlights

• Simulation horizon: NYC 28 d, Dakar 14 d; each scenario averaged over 5 runs.
• Best-fit $\tau(rf)$ parameters:
  • SEIR NYC $a=-0.08$, $b=2292.6$ ((R^2=0.981)).
  • SEIR Dakar $a=-0.01$, $b=1853.6$ ((R^2=0.902)).
  • SI/SIR fits similar (Fig 5).
• Spatial clustering always significantly non-random: empirical $M(k)$ never overlaps 99 % Monte-Carlo bands.

Methodological Appendix (condensed)

• Distance computation: haversine great-circle.
• DBSCAN parameters: \eps=0.0004^\circ,\;\text{minPts}=5.
• Epidemic parameters tested:
  • Baseline COVID-19 (above);
  • Delta: $\beta=1.42$;
  • H1N1: $\beta=0.913,\;\gamma=1.6,\;\sigma=1$.
• Visualization units: $\tau$ measured in 10-min (Dakar) or 15-min (NYC) increments; spatial grid for maps sized to GIS clusters/towers.

Connections to Prior Literature

• Confirms earlier claims (Wilson 1995; Wesolowski 2015; etc.) that mobility drives epidemics, but adds the unifying scalar $r\,f$.
• Extends universal visitation law (Schläpfer 2021) to epidemiological consequences.
• Reinforces metapopulation insights (Watts 2005; Colizza 2007) on heterogeneity & multi-scale spread, by tying heterogeneity to a measurable scalar.

Concluding Takeaways

• Exploration velocity $v=r\,f$ is a single, city-level control knob for epidemic timing and geographic spread.
• Practical containment ≈ enforce $v\le v<em>{\text{crit}}$ ⇔ trade-off curve $r \propto 1/f$ at fixed acceptable $\tau</em>{\text{bound}}$.
• Supports nuanced, less disruptive mobility policies (e.g.
  allow long trips if infrequent) and bolsters arguments for part-time remote work.
• Opens interdisciplinary research path linking urban science, network epidemiology, and policy design.