Rank and relative performance of import risk estimation models.

The different import probability models are compared via their rank (A) and relative performance (B), with the highest values representing the best approach. The rank and relative performance are shown for each (black dots) of the six comparison measures (corr, logcorr, RMSE, logRMSE, cpc, τKendall) the box illustrates the interquartile range, the horizontal line the median and the red triangle the mean. The colors of the boxes illustrate the different distance measures in use. The outlier measure of the import risk models (I.R.) is the logRMSE, where the gravity models with effective distance are performing best. See Material and methods for definitions of comparison measures and Figs E, F in S1 Text for absolute and detailed relative performance.</p

Source countries’ prediction quality and WAN outflow.

The correlation between the logarithm of the import risk and the reference import probability improves with the outflow of the respective source country (top). Examples of source countries with particularly low (ER, Eritrea) and high (GB, Great Britain) outflow and log_corr are shown with their import risk and reference import risk to target countries (middle row). Countries with exceptionally low log_corr measures compared to source countries with a comparable outflow are either historically linked to specific regions as Australia (AU) and Israel (IL) to European countries (lower right panel) or politically as Macao (MO) as a special administrative region of China.</p

Estimates of import probability by the gravity model with exponentially (1st column) and power law (2nd column) decaying distance function, the radiation model (3rd. column) and by the import risk model (4th column).

The first three models (1st-3rd column) use as distance the geodesic (1st row), geodesic path (2nd row) and the effective (3rd row) distance. The import risk model is computed from the WAN with the geodesic distance (D) or the effective distance (L) as a weight for the exit probabilities or without weighting (H), i.e. in the last two cases (H, D) only WAN information is used. The orange line depicts the median and the gray line is y = x and illustrates perfect mapping.</p

The relation between WAN, OD-probability, SPT and effective distance.

A: The world air transportation network (WAN) represents the direct flight connections and maximal seat capacities between airports in 2014, here shown for flights starting from five selected countries. It is based on flight-schedule-data. The lines are bundled and do not represent the specific flight route, but illustrate the links to airports abroad. B: The reference import probability from Canada to all countries, based on the OD matrix (Origin-Destination) of the Global Transnational Mobility Data set [40, 47] in 2014. It combines origin and final-destination trips between countries from the SABRE and the World Tourism Organization (UNWTO). The lines illustrate the connection to the common source country. C: Based on the effective distance deff = d0 − ln(p) a shortest path tree (SPT) is constructed with the largest Canadian airport as source (YYZ: Toronto Pearson International Airport). The link color and thickness shows the hop distance, i.e. number of connecting flights. D: exponential decay of the reference import probability (as in B but for all countries as source) with the effective distance deff (derived from the SPT (C) of the WAN (A)). Each dot represents a country-country link, the lines are medians including either all source countries or only from a specific continent. Maps are created with geopandas [48].</p

Correlation analysis: Disease arrival time vs. the effective model distance.

Each model’s import probability is converted to an effective distance dM(i|n0) = −ln(p(i|n0)) with n0 as the outbreak country of the respective disease. The correlation results C(tA, dM) with the arrival time tA(i) of the disease in the target country i are grouped by model (A) and by the disease (B). As comparison distances, the correlation of the geodesic, geodesic path (on the effective shortest path tree) and the effective distance with tA are shown. Each dot represents a correlation result of the 21 considered outbreaks (H1N1 in 2009, COVID-19 in 2020 and the spread of 18 of its variants in the years 2020–2022).</p

S1 Text -

Disease propagation between countries strongly depends on their effective distance, a measure derived from the world air transportation network (WAN). It reduces the complex spreading patterns of a pandemic to a wave-like propagation from the outbreak country, establishing a linear relationship to the arrival time of the unmitigated spread of a disease. However, in the early stages of an outbreak, what concerns decision-makers in countries is understanding the relative risk of active cases arriving in their country—essentially, the likelihood that an active case boarding an airplane at the outbreak location will reach them. While there are data-fitted models available to estimate these risks, accurate mechanistic, parameter-free models are still lacking. Therefore, we introduce the ‘import risk’ model in this study, which defines import probabilities using the effective-distance framework. The model assumes that airline passengers are distributed along the shortest path tree that starts at the outbreak’s origin. In combination with a random walk, we account for all possible paths, thus inferring predominant connecting flights. Our model outperforms other mobility models, such as the radiation and gravity model with varying distance types, and it improves further if additional geographic information is included. The import risk model’s precision increases for countries with stronger connections within the WAN, and it reveals a geographic distance dependence that implies a pull- rather than a push-dynamic in the distribution process.</div

Fig 5 -

Classification of the 10 countries with the highest import probability by the gravity model with exponentially (1st column) and power law decaying (2nd column) distance function, the radiation model (3rd. column) and by the import risk model (4th column). A true or false positive (T. Pos. or F. Pos.) means that the country is or is not among the 10 countries with the highest reference import probability . A false negative (F. Neg.) means that it belongs to the reference set but was not detected by the respective model. The pie chart illustrates the sensitivity of the models.</p

Import risk scheme.

Starting from the transition network (left) the shortest path tree is computed based on the effective distance (center bottom). Based on the shortest path tree, the exit probabilities q∘ = q(∘|⋆) are computed. In the formula, the geometric symbols represent the estimated population of the respective node, which can also be distance-weighted (depending on the exact model). A random walk-process with exit probability is defined (top): at each step, the walker either exits the node with prob. q∘ = q(∘|⋆), or continues walking with prob. (1 − q∘). The import risk p∞(∘|⋆) (right) is the probability of a walker to exit at node ∘ given it started at node ⋆ under consideration of all possible paths.</p

Import risk aggregated on regional level “to target” vs. “from source” and its geodesic distance dependence.

The geodesic distance between regions predicts the import risk p∞ to a single target from all sources (A, B) better than from a single source to all targets (D, E) as can be seen by the p-values (C) of the power law fit p∞(d) = c⋅d−α that is illustrated for each selected examples by a grey line (A, B, D, E). The fitted exponent α of the import risk to a single target decreases with the respective regional WAN flow out of the target region (F), i.e. the more connected a region, the weaker the import risk decays with distance. The dashed horizontal lines show the average import risk of a single target (A, B) or a single source (D, E). The color of the dots corresponds to the depicted world regions (right). Maps are created with geopandas [48].</p