Fig 1.
The country of study and gravity and radiation model example results.
A) A population map of Kenya with district boundaries (grey) and major roads (black). For two districts, a schematic representation of B) a gravity model and C) a radiation model. The gravity model is based on the populations of the destination and origin as well as the distance between these locations. The radiation model is based on the destination and origin’s populations as well as the total population within a circle centered at the origin. This model is based on the premise that individuals living in a certain home location will consider the number of job opportunities (measured as the destination’s population proportional to the resident population) and will travel to the closest destination that would offer better benefits than the resident location.
Fig 2.
A summary of previously published papers incorporating human mobility models and infectious disease dynamics.
We reviewed nineteen papers that either simulated (simulated) disease dynamics or used epidemiological data (data). These papers covered a range of infectious diseases (see S1 Table). For each paper we identified the type of mobility model, disease model, location (high, low, or both high and low income country), and the type of movement quantified. Mobility models were classified as a gravity model, radiation model, a spatial transmission kernel, a network, or a risk surface. Disease models included a metapopulation model, metapopulation type model with stochastic fadeouts, time spent at locations with different risks of becoming infected, an individual based model (IBM), and a network. For each paper, the location of either the mobility and/or disease data determined the location of the paper with countries separated as high income or low income. Papers that focused on global disease spread were classified as both high and low income. Mobility was classified as either commuting, regional, or global movement patterns. If the paper did not explicitly state the type of mobility included in the paper the type of mobility was discerned from the spatial resolution of the data. Regional movement includes mobility between political admin units that are larger than a city, in general. If a paper included both global movements, such as airline flights, and localized commuting, then the paper was classified as global. We included papers describing various infectious diseases including cholera, malaria, dengue, measles, pertussis, rubella, influenza, and foot and mouth disease.
Fig 3.
The actual and predicted amounts of travel from two districts.
The A,D) actual amount of travel, predicted amounts of travel from the B,E) gravity model and C,F) radiation model are shown for two districts. For each map, a continuous density surface was constructed showing the relative spatial distribution of travel. The bar plot shows the amount of travel to all other districts from the district with values displayed in increasing order of distance. Nairobi A,B,C) is the most populated district in the country and includes the capital. In all three figures, the majority of travel (shown in dark blue) is to neighboring locations. The radiation model estimates more travel to the rest of the country than the data or gravity model. Garissa D,E,F) is a rural district bordering Somalia. The majority of actual travel occurs to Nairobi, which the radiation model did not capture. The gravity model was able to predict a large amount of travel to Nairobi, but greatly over predicts travel to the rest of the country.
Fig 4.
The results of fitting each spatial interaction model.
A) The predicted results from both gravity and radiation models. The gravity model (shown in blue) predicts larger amounts of the total volume of travel over the course of the data set (ratio of predicted values to data–mean: 12, 95% quantile interval: 0.34–43) than the data whereas the radiation model (shown in red) underpredicts the volume of travel (ratio of predicted values to data–mean: 0.5, 95% quantile interval: 0.0066–1.7). B) The ratio of predicted versus actual data from both models versus distance. For both models, the predictions over short distances were worse than over longer distances. C) We re-fit gravity models using Euclidean distance (red), travel times (blue) and road distance (green) between district centroids (circle) and population-weighted district (square) centroids. The reduction in deviance of these models is shown. In general, Euclidean distance based gravity models outperformed all other distance measures, except for travel between rural areas. For this type of travel, road distance outperformed Euclidean distance (Euclidean distance–reduction in deviance: 63%, road distance–reduction in deviance: 72%, see Supporting Information).
Fig 5.
The utility of each model to describe travel in various settings.
A) For travel between all pairs of locations, we compared the error in the actual versus predicted amount of travel. If this error was not within defined bounds (outside of the dotted black lines) (see Materials and Methods), we determined that both models do not adequately describe this travel (shown in black). For the remaining volumes and routes of travel, we determined if the gravity model (blue) or the radiation model (red) performed better to identify which model should be used in various settings. For example, the radiation model does much better than the gravity model predicting low volumes of travel and vice versa (shown in red versus blue). B) A schematic highlighting the situations when a radiation model is preferred over a gravity model and when caution should be taken using the predicted results of either model. Scenarios to use a gravity model over a radiation model include: travel to and from a major population centre, over short distances, and when predicting large volumes of travel. A radiation model should be used over a gravity model when describing travel between rural areas and low volumes of travel. Caution should be taken using either model if the travel is between locations of intermediate rural population and over short distances. C) We performed a logistic regression to determine when to use each model for various amounts of travel (a gravity factor). As the amount of travel increases, the number of times the gravity model outperforms the radiation model increases. For examples for the lowest amounts of travel 43% of the time a radiation model if preferable to a gravity model. In contrast for the largest amount of travel, all of these routes (100%) were better predicted with a gravity model than a radiation model.
Fig 6.
The impact of the duration of journeys on travel between districts.
A) For travel between pairs of locations, we compared the number of trips (log) versus the distance (km) for all journeys (grey) and the best fit lines for trips lasting up to between one and two weeks (red), between two weeks and one month (blue), between one and two months (green), between two and three months (orange), between three and four months (purple), and trips lasting four months or more (yellow). As the duration of journeys increased, the amount of travel between districts decreased. B) For all trips including any trip duration (red) and those lasting 4+ months (grey), the top 5% of of routes are shown based on the total amount of travel. For trips lasting short durations, there is a large amount of travel between nearby districts. For trips lasting long durations, the majority of these top routes are to/from major cities including Nairobi and Mombasa.
Table 1.
The gravity model parameters and fit for trips lasting various durations.