Fig 1.
The Administrative districts and population distribution in Namibia.
A) The locations of the 105 districts (administrative level 2) in Namibia with district centroids of each shown with white circles. B) The population density of each district in calculated as the number of people per square kilometer from the 2010 WorldPop Project estimates of the total number of people per 100m grid cell within each district (www.worldpop.org). District-level shapefiles were acquired from DIVA-GIS (www.diva-gis.org) and are reused here under the Creative Commons Attribution License.
Fig 2.
Travel network topology shifts from heterogeneous to homogeneous as trip duration increases.
Maps of Namibia with travel volumes between districts that fall within three broad intervals of trip duration: A) 1–3 days, B) 7–14 days, and C) 30–60 days. For comparison, connectivity is defined as relative edge weight from 0 to 100%, which is calculated by scaling trip volume along the edges in each sub-network by the overall maximum trip volume observed in the full travel network. District centroids (nodes of the network) are indicated by the white circles. Shapefiles were acquired from DIVA-GIS (www.diva-gis.org) and are reused here under the Creative Commons Attribution License.
Fig 3.
Distribution of node strength and node centrality for models with different trip duration intervals and changepoint analysis showing which trip durations constitute shifts in network topology.
The empirical distributions of A) node strength (weighted degree per node) and B) node closeness (total weighted distance from all nodes) are plotted for four duration-restricted subnetworks which include trips of 1) 0–1 days, 5) 5–7 days, 11) 2–3 months, and 20) 11–12 months. C) The joint distribution of ηstrength and ηclose which measures the amount of network heterogeneity and structure based on the distributions of node strength and node closeness respectively. Each point represents one of the duration-restricted subnetworks and is colored according the duration interval shown in the color key to the right. The multivariate changepoint algorithm identified two significant shifts in network topology based on the joint distribution of ηstrength and ηclose that are placed at 5 and 60 days trip duration. The three nominal classes delineated by these thresholds are indicated by a circle for the heterogeneous class (1–5 days), a triangle for the homogeneous class (>60 days), and a square for intermediate class (6–60 days).
Table 1.
Table of trip duration intervals used, with number observations and total trip counts.
Fig 4.
The drivers of trip volume across varying trip durations.
The effect of distance to destination A) and destination population size B) on trip volume plotted across the 20 duration-restricted subnetworks. Each colored line represents one of the 105 districts with the population density of the origin district is indicated by the color bar. Dashed vertical lines indicate the network topology thresholds identified by the changepoint analysis. Scatterplots showing the joint distribution of effect sizes for distance (x-axis) and destination population (y-axis) for C) the model with the shortest duration (Model 1, 0–1 days) and D) the model with the longest duration (Model 20, 11–12 months). Comparison of C and D show that the effect size of both covariates is reduced to near-zero effect for longer trip durations.
Fig 5.
Change in connectivity and gravity model parameters fitted to travel data with increasing duration intervals compared to full model.
A) The distribution of connectivity values for duration-restricted models (y-axis) in comparison to the full model that includes all data (x-axis). The smoothed lines indicate the change in connectivity for duration-restricted models with larger duration intervals showing a more evenly distributed pattern across all locations compared to the null model. The dashed red line indicates connectivity values that are equal to the full model. In B) and C), the change in fitted gravity model parameters (distance parameter γ and destination population parameter ω2 respectively) for increasing trip duration intervals. The color gradient indicates the duration interval of each model and the dashed red line shows the fitted parameter value for the full model, which includes all trip durations.
Fig 6.
Predictability of spatial spread for a range of R0 and generation time values and the change in spatial predictability over time for 7 pathogens.
A) A heatmap with contour lines showing the values of initial spatial predictability (ϕ) calculated for hypothetical combinations of R0 and generation time (days). Example pathogens are indicated by colored circles: influenza, SARS-CoV-2, measles, Ebola, pertussis, P. falciparum malaria, tuberculosis. The level of spatial predictability is shown in the color bar to the right with schematic representations for scenarios where patterns of spatial spread are perfectly predictable (ϕ = 1) or completely unpredictable (ϕ = 0). B) The change in these initial values of spatial predictability over successive generations for each of the 7 example pathogens. Colored lines represent the mean value of spatial predictability calculated over 1000 replicates of the stochastic TSIR model (see Methods). In both analyses, the capital district of Namibia, Windhoek East, was used as the introduction district.
Fig 7.
The change in spatial predictability for 6 pathogens plotted over time for outbreaks introduced to districts with a range of population densities.
The results from simulations of outbreaks for 7 example pathogens (influenza, SARS-CoV-2, measles, Ebola, pertussis, P. falciparum malaria, and Tuberculosis). For each pathogen, outbreaks were introduced into each of the 105 districts in Namibia and the change in spatial predictability for successive generations was calculated. The heatmaps and contour lines show values of spatial predictability (ϕ) as they change over successive generations (x-axis) and the population density of the introduction district (y-axis). The number of days since the introduction is indicated on the top-axis. Annotations in the lower right indicate the pathogen, basic reproduction number (R0), generation time in days (γ), and proportion of the population that is susceptible (s) used in spatial simulations. Transmission parameters used in simulations were drawn from the literature and shown in S1 Table.