Sampling for global epidemic models and the topology of an international airport network

Abstract

Mathematical models that describe the global spread of infectious diseases such as influenza, severe acute respiratory syndrome (SARS), and tuberculosis (TB) often consider a sample of international airports as a network supporting disease spread. However, there is no consensus on how many cities should be selected or on how to select those cities. Using airport flight data that commercial airlines reported to the Official Airline Guide (OAG) in 2000, we have examined the network characteristics of network samples obtained under different selection rules. In addition, we have examined different size samples based on largest flight volume and largest metropolitan populations. We have shown that although the bias in network characteristics increases with the reduction of the sample size, a relatively small number of areas that includes the largest airports, the largest cities, the most-connected cities, and the most central cities is enough to describe the dynamics of the global spread of influenza. The analysis suggests that a relatively small number of cities (around 200 or 300 out of almost 3000) can capture enough network information to adequately describe the global spread of a disease such as influenza. Weak traffic flows between small airports can contribute to noise and mask other means of spread such as the ground transportation.

Figure 1. Map of the cities selected in (a) the population-based sample, n = 52 and (b) the volume-based sample, n = 52.

Figure 2. Log-log relation between the city and flight volume size in the data.

Ordinary regression is represented with a solid black line, while an orthogonal regression is a dashed red. Orthogonal regression minimizes the orthogonal distance from the regression line as opposed to minimizing vertical distance in ordinary regression. The spread of the residuals is comparable to the range of the data indicating high variation of the flight volume for cities of the similar sizes. Orthogonal regression provides a useful reference line because it treats flight volume and city size as equal variables and could be viewed as the principle component capturing the essence of the relationship between the two variables. At the same time ordinary regression considers flight volume as a function of the city size. The lower slope in the regression line compared to the orthogonal regression indicates that there are many large cities that have a small flight volume as well as small cities with high flight volume.

Figure 3 a–d.. The relationships between network characteristics and sizes of network samples.

Samples are selected based on flight volume (solid black line) and population (broken red line). Subplots correspond to the following network characteristics: (a) betweenness, (b) degree, (c) clustering, and (d) geodesic.

Figure 4. Proportion of cities sharing the top ranks between 2 or more characteristics.

For each pair of network characteristics we consider 2 lists of cities ordered by each of the characteristics. The list size is shown on the horizontal axis. The proportion of cities shared by both lists is shown on the vertical axis. The minimum value of the shared proportion, 0, occurs when the lists have no city in common. The maximum value, 1, occurs when the two lists are identical. This is achieved only in the whole sample. For the 52 city lists, only 11 cities are among top 52 in all four characteristics.

Figure 5. Inflation coefficient indicating the union of two ranking lists is larger than the size of a single list.

For each pair of network characteristics, we consider two lists ordered by each of the characteristics. The single list size is shown on the horizontal axis. The inflation coefficient is shown on the vertical axis. The minimum value of the inflation coefficient, 1, occurs when the two lists are identical. This is achieved only in the whole sample. The maximum value of the inflation coefficient for the union of two categories, 2, occurs when the lists do not share any cities. For all four categories the maximum value of the inflation coefficient is 4. The size of the union of 52-city lists in all four categories is 2.2 times larger than the size of a single list, which translates into 114 cities.