Vector fields as a framework for modelling the mobility of commodities.

Commodities flow through trade networks across the world, with trajectories that can be effectively modelled using approaches similar to those used in human mobility studies. Yet, documenting these movements comprehensively is challenging due to data sparsity, cost, and privacy constraints. Origin-destination (OD) matrices provide a widely used framework for representing mobility, although they inherently omit locations not directly observed as either origins or destinations. This incompleteness creates gaps across different geographical scales, constraining our ability to characterise movement patterns in underrepresented areas. In this study, we introduce a vector-field-based method to address these persistent data challenges. By transforming OD data into continuous vector fields, we capture spatial flow patterns more comprehensively than traditional network approaches, while also enabling robust analysis of mobility directions. Our approach incorporates interpolation techniques that handle incomplete and sparse datasets effectively; when approximately 500 out of 853 areas are removed, 189 areas (36%) maintain degree deviations of less than 15 degrees, showing that the general direction of flow is preserved for over one-third of the impacted areas and enabling continuous spatial analysis. We apply this framework to cattle trade data from Minas Gerais, Brazil. Cattle movements are particularly significant as they directly impact disease transmission, including foot-and-mouth disease. Accurately modelling these flows supports effective disease surveillance and preparedness, with benefits for both animal health and economic stability. Our analysis reveals distinct spatial clusters of trade behaviour, temporal patterns in flow directions, and seasonally varying critical points likely associated with known periodicities in cattle trade driven by breeding cycles, slaughter schedules, and fluctuations in global demand. While previous vector-field studies focused on human mobility, our framework addresses the distinct challenges of commodity flows, where aggregated OD data, sparse observations, and lack of data are the norm. It enables inference in unobserved areas which is a critical capability for modelling scenarios such as disease spread. This approach enhances our capacity to infer flow patterns from incomplete datasets and advances understanding of large-scale commodity trade dynamics.