In this study, we demonstrate a discrepancy between modeled and observed coastal epidemics and we ask what drives this breakdown of the gravity model assumptions. Because our question address spatial transmission, detailed human movement data to parallel the epidemic time series would be ideal for comparison but these are not available, particularly for children, during the 1940s and 50s 
. Instead, we used local population heterogeneities to test the gravity model predictions of epidemic persistence and synchrony.
Our initial, basic gravity model may have incorrectly predicted coastal persistence for two possible reasons. First, the towns along the coast may have had contact rates that were different from those of inland towns. For example, we considered the possibility that coastal locations may have experienced relative isolation-by-distance and low contact rates for most of the year, alternating with high contact rates during the summer months, due to travelers 
. In inland towns, contact rates are highest at the beginning of each school term, when epidemics take place. Cyclic demographic flux could cause the model to underestimate seasonal movement and coastal contacts, resulting in overestimated fadeout rates caused by not considering summer cases and predicting only school term epidemics, sparked by core cities. In this situation, coastal towns would show measles outbreaks in the summer, unique from the rest of the island, where contact rates and epidemics rise and peak during the school term. The data do not show this.
If coastal towns showed epidemic seasonality that indicated summer outbreaks, this would imply that they were somewhat isolated from inland towns and were not influenced by inland epidemic cycles as a result of unique contact rates. However, our analysis shows that coastal towns measles epidemics followed the term time forcing of the large inland cities and that coastal towns were not at all isolated from inland towns. Therefore, the data show us that coastal epidemic cycles were likely driven by core cities, which were only found inland, indicating that coastal and inland towns did not have different seasonal contact rates (). It has further been shown that the coast was an attractive location for suburban residences year round, as well as for seasonal holidays 
, not a continuously isolated edge as the model predicted in both our simulated ‘artificial metapopulation model’ and our England and Wales simulations. The data clearly show that coastal towns did not have reduced contact rates with inland towns.
A second possible reason that the gravity model overestimated fadeouts along the coast is that coastal and inland per capita contact rates are relatively similar. The distance-weighted, size-dependent spatial coupling element of the basic gravity model will always predict lower overall contact rates for coastal than for inland towns. Because coastal towns are partially surrounded by water, they have fewer populations at close proximity (small Dij
in equation 3), which greatly impacts the flux of infection between towns. However, if each coastal town approximately averages the same number of contacts per capita as inland locations (as the public transportation data suggest in figure S3
), and the model is unable to map social space over geographic space by assuming the opposite, then the prediction of reduced contacts along the coast would create a false “edge effect” of increased fadeouts. It is both unrealistic and counterintuitive to assume reduced individual coastal contacts; living along the coast does not reduce the need, for example, for medical attention, commerce, or social companionship. If observed contact rates are reasonably similar between coastal and inland towns, the model will underestimate contacts, transmission, and persistence along the coast. In this case, the observed coastal epidemic seasonality would not differ from inland seasonality, as it does not in this system.
If host mobility resulted in high contact rates along the coast year-round, even for distant cities, this would result in multiple measles introductions during local epidemics troughs. While these introductions would not have sparked new measles epidemics because of low susceptible density resulting from regular biennial outbreaks, they would have sparked isolated cases and led to decreased fadeout rates in coastal towns. However, it is very difficult to determine actual contact rates; even though we were able to obtain passenger train use volume, we did not have bus or road use data. Further, even with all those data, we would still fail to quantify the actual movement of children. Thus, while our train use data give us a good idea of host mobility and train use by town size, it is still only a vague approximation of the contacts we are actually interested in.
In the gravity model, the spatial coupling coefficient (Θ, equation 3) represents the amount of human movement from one town to another; as Θ increases, contacts increase and spatial synchrony increases. Based on our model predictions, the spatial coupling parameter estimation fits inland towns well but underestimates the connectivity of coastal towns.
In , we compare the residuals of the fadeouts on population size between the observed data, initial gravity model predictions, adjusted gravity model predictions for high train use coastal towns, and adjusted gravity model predictions for all coastal towns. Although the high train use adjustment gravity model predicts a slight bias towards coastal fadeouts, it corrects for most of the bias in the initial, unadjusted gravity model predictions and more accurately reflects the observed data. When we increased the coastal spatial coupling coefficient to more accurately map social space over geographic space for the purpose of increasing coastal contact rates, the adjustment corrected for the model's bias of reduced coastal contacts and increased coastal fadeouts (, , and ).
Contact from core cities to coastal regions introduced isolated measles cases during the troughs between epidemics. These stochastic introductions did not lead to out-of-phase epidemics along the coast; instead they resulted in a low level of persistence 
. When this occurred, coastal towns did not fade out as the model predicted because of the model's inaccurate assumption that locations at the edge of a system have reduced contact rates, simply because of their position. The observed data do not support this assumption, implying that (at least childhood) behavior and movement in this landscape do not isolate geographic edges.
The adjustments we have shown here crudely illustrate the gravity model's potential to accommodate spatial heterogeneities and host behavior in stochastic metapopulations by identifying important geographic features, which can influence host mixing behavior and affect disease transmission 
. The spatial coupling coefficient for edges can be increased when host mobility results in reduced isolation-by-distance.
In realistic landscapes, habitats often include variation in accessibility, land quality and resource availability. Populations establish centers and edges with respect to these features. The methods presented here can be applied as a first step to understanding disease dynamics and host movement across heterogeneous landscape peripheries. Dissecting the applied implications of these results is an important area for future work, especially in developing countries 
. It is clear that more sophisticated methods need to be developed to address these specific issues with spatial models but these findings make a satisfactory first step in identifying the problem and exploring solutions.