Fig 1.
Epidemic spread in a two-city system with identical populations.
A: Schematic of the transport model with symmetric inter-city flows. B: Mean offset in the peak-infection day of the destination city relative to the origin city as a function of inter-city traffic (fraction of the population per day; logarithmic scale) across multiple levels of infectiousness (fraction of wild-type SARS-CoV-2 infectiousness). C: Epidemic curves for the origin city (blue) and the destination city (orange) for varying traffic (columns) and infectiousness (rows). Each panel summarizes 150 simulations with outliers removed. House and virus icons used in panel A are sourced from SVG Repo and Openclipart and are available under the CC0 1.0 Public Domain Dedication.
Fig 2.
Epidemic spread between two unequal-size cities.
A–B: Schematics of the two-city transport model with the outbreak seeded in a city of 10,000 agents (A) or 190,000 agents (B). C–D: Heatmaps of cumulative infections, peak daily infections, and peak-day timing (columns) for the origin city (top rows) and the neighboring city (bottom rows), as functions of (C) flow from the neighbor when the origin has 10,000 agents, and (D) flow from the origin when it has 190,000 agents. E: First-order sensitivity indices versus the city-size ratio for the flow from the origin (top panels), the flow from the neighbor (middle panels), and the resulting insensitivity index (bottom panels), shown for cumulative infections, peak daily infections, and peak-day timing (columns) in the origin city (blue) and the neighbor (orange). Infectiousness is set to the wild-type SARS-CoV-2 level. House, apartment, and virus icons used in panels A and B are sourced from SVG Repo and Openclipart and are available under the CC0 1.0 Public Domain Dedication.
Fig 3.
Hub-and-satellite transport model.
A: Schematic of the hub-satellite system represented as a graph, with outbreaks initiated in the hub (top) or in a satellite (bottom). Vertex labels indicate city population sizes (number of agents), and edge labels indicate the daily commuting fraction of the population. B: Epidemic curves for outbreaks starting in the hub (left) or in a satellite (right). Shaded areas denote the standard deviation across 150 simulations. C–D: Cumulative infections (C) and peak infections (D) in the hub-satellite system as functions of the day interventions in commuting flows are introduced, for outbreaks starting in the hub (left) or in a satellite (right). Curves of different colors correspond to different transport-flow interventions. The gray curve indicates the mean across 150 baseline simulations without interventions; shaded areas show standard deviations. Stars mark statistically significant differences (Student’s t-test, p < 0.05 with multiple-testing correction). Infectiousness is set to the wild-type SARS-CoV-2 level. House, apartment, and virus icons used in panel A are sourced from SVG Repo and Openclipart and are available under the CC0 1.0 Public Domain Dedication.
Table 1.
The magnitude of the reduction in target metrics when introducing transport restrictions on the 10th day of the epidemic.
Table 2.
The magnitude of the reduction in target metrics when introducing transport restrictions on the 70th day of the epidemic.
Fig 4.
Relative shifts among satellite epidemic curves in the hub-satellite system.
A–B: Satellite incidence curves (A) and phase portraits (B) for outbreaks seeded in the hub (top rows) or in a satellite (bottom rows), shown across traffic-flow multipliers (panel columns) with infectiousness set to the wild-type SARS-CoV-2 level. Phase portraits highlight temporal offsets that are less apparent in raw incidence curves.
Fig 5.
Detecting the outbreak’s origin in the hub-satellite model.
A: Schematic of the DTW-based method for identifying the origin city. B-C: Heatmaps of origin-detection accuracy versus epidemic day for systems with traffic flows equal to (B) or 100-fold smaller than (C) those of Moscow and the Moscow region. Testing is simulated by sampling detected infections from a binomial distribution.