Evaluating the Adequacy of Gravity Models as a Description of Human Mobility for Epidemic Modelling

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Abstract
Gravity models have a long history of use in describing and forecasting the movements of people as well as goods and services, making them a natural basis for disease transmission rates over distance. In agent-based micro-simulations, gravity models can be directly used to represent movement of individuals and hence disease. In this paper, we consider a range of gravity models as fits to movement data from the UK and the US. We examine the ability of synthetic networks generated from fitted models to match those from the data in terms of epidemic behaviour; in particular, times to first infection. For both datasets, best fits are obtained with a two-piece ‘matched’ power law distance distribution. Epidemics on synthetic UK networks match well those on data networks across all but the smallest nodes for a range of aggregation levels. We derive an expression for time to infection between nodes in terms of epidemiological and network parameters which illuminates the influence of network clustering in spread across networks and suggests an approximate relationship between the log-likelihood deviance of model fit and the match times to infection between synthetic and data networks. On synthetic US networks, the match in epidemic behaviour is initially poor and sensitive to the initially infected node. Analysis of times to infection indicates a failure of models to capture infrequent long-range contact between large nodes. An assortative model based on node population size captures this heterogeneity, considerably improving the epidemiological match between synthetic and data networks. An accurate representation of disease transmission between spatially-distinct regions is an essential part of modelling epidemic behaviour on a national or international scale. Gravity models, which describe movement fluxes between regions in terms of their populations and distance from each other, have a history of successful use in the geography and economics and are increasingly used in epidemiology. We look at the ability of a range of gravity models to fit human movement data from the UK and the US. In particular, we compare the behaviour of a simple flu-like epidemic model on synthetic networks generated by fitted gravity models and on the original network present in the data, using time to first infection. For UK data, epidemic behaviour on synthetic networks matches that on the original data quite closely. For US movement data, synthetic networks perform much worse. We develop an analytic expression for infection time between two regions which indicates that our gravity models fail to capture long range connections between large populations. A model with assortative mixing based on population size greatly improves the match between synthetic and data networks.