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The dynamics of human mobility have been known to play a critical role in the spread of infectious diseases like COVID-19. In this paper, we present a simple compact way to model the transmission of infectious disease through transportation networks using widely available aggregate mobility data in the form of a zone-level origin-destination (OD) travel flow matrix. A key feature of our model is that it not only captures the propagation of infection via direct connections between zones (first-order effects) as in most existing studies but also transmission effects that are due to subsequent interactions in the remainder of the system (higher-order effects). We demonstrate the importance of capturing higher-order effects in a simulation study. We then apply our model to study the first wave of COVID-19 infections in (i) Italy, and, (ii) the New York Tri-State area. We use daily data on mobility between Italian provinces (province-level OD data) and between Tri-State Area counties (county-level OD data), and daily reported caseloads at the same geographical levels. Our empirical results indicate substantial predictive power, particularly during the early stages of the outbreak. Our model forecasts at least 85% of the spatial variation in observed weekly COVID-19 cases. Most importantly, our model delivers crucial metrics to identify target areas for intervention. [ABSTRACT FROM AUTHOR]