Our analysis demonstrates the degree of spatial locality in the large-scale geographical spread of influenza in both England and Wales and the US in 1918. However, it is difficult to directly compare the kernel power estimate from this study with those from other studies owing to differences in the functional forms used. For instance, Viboud et al. [5] estimate two power parameters above and below a given distance threshold when modelling the spread of seasonal influenza in the US. Gravity models used to describe the spread of measles in the UK by Xia et al. [18] assumed a kernel power of 1, rather than fitting this parameter. The distance power estimates we found for England and Wales and the US are quite different from each other. It is not surprising that there is a disparity in the distance kernel in England and Wales and the US, as the spatial scale in the US is much larger than in England and Wales. In comparing the US and UK, it should be noted that the mean distance between cities is of course much larger in the US (see electronic supplementary material, table S1). In theory, this gives better resolution for estimating the kernel shape, as the range of inter-city separations is an order of magnitude larger than for the UK. However, this is counterbalanced by the smaller size of the US dataset, which reduces inferential power. The low kernel power parameter estimates we have found in both England and Wales and the US suggests that long-distance interactions were important in spreading influenza between distant cities in both countries. At the start of the major autumn wave in 1918, the armistice was more than two months away and it is likely that travel relating to the war effort, including troop movements, might have enhanced the frequency of long-distance movements. Density-dependent gravity models are frequently used to explain the connectivity of urban centres for human diseases [5, 18] . There is good evidence from the estimates of 1 in this analysis for England and Wales that the density-dependent model underestimates the total force of infection on remote cities. The 95 per cent credible interval for 1 includes 1, which indicates that the density-independent model formulation cannot be definitively excluded as an explanation for the data. There was limited statistical power to estimate the degree of density dependence in the US context, but use of the England and Wales best-fit model to describe the US data gave a very similar DIC to the best-fit US model. Hence, it is not clear if the difference in the estimated density dependence found between the US and the England and Wales is because of the large differences in the degree of population coverage between the US and England and Wales datasets. Our results using a subset of the England and Wales dataset suggest that the degree of density dependence is a difficult parameter to estimate when coverage is low. The low power on destination city size found in fitting the gravity model to the England and Wales dataset shows that connectivity of a city increases sub-linearly with population increase. When modelling spread of influenza in the US, Viboud et al. [5] found very comparable low values of the population exponents with the infectious city lower than the susceptible. Differing results come from the analysis of measles data in Great Britain with a power coefficient on infectious populations estimated at approximately 1.5 [18] . We found the best-fit model in England and Wales does not include the population size of the infector (origin) city, a result which needs further examination in future work. Differences in our population parameter estimates and those from studies on contemporary populations are likely to differ owing to changes in human mobility patterns since 1918. Our estimate for England and Wales that the infectivity of a city is sub-linearly related to mortality, suggests that the rate of death in a city is not as important to infectivity as the presence or absence of disease. Other studies have used constant infectivity terms for the analysis of human seasonal influenza [21] . However, our estimates do support some level of mortality dependence, suggesting that cities with a very high influenza burden, usually later in the epidemic wave, are more infectious than newly infected cities. In the US dataset, the best-fit model gave constant infectiousness, but again this may be due to a lack of power to estimate such parameters from the US dataset. It may also be caused by non-uniform infection pressure from cities not in the dataset, which could mask an infectivity relationship for the cities that are given.