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Epidemic disease spreading is conventionally often modelled and analyzed by means of rate and diffusion equations, following the paradigms of well-controlled chemical reactions and diffusive dynamics in a test tube. Yet, serious worries that this suggestive and appealing similarity might be a false friend were already voiced by the pioneers of mathematical epidemiology. A century later, we can draw on cross-fertilizations from network and game theory and the emerging field of eco-evolutionary dynamics to substantiate them. Epidemiological spreading is thereby revealed as a fundamentally heterogeneous and erratic process that shares certain properties with more unwieldy phenomena, such as earthquakes, hurricanes, traffic jams, and stock crashes. They are all characterised by high tail risks that materialize very rarely but fatally. That they arise from bursts of unlikely chains of localized random "superspreader" events, by which micro-scale fluctuations and uncertainties may get heftily magnified, makes their accurate prediction and control intrinsically and notoriously hard. That epidemic disease spreading is moreover closely intertwined with equally heterogeneous genetic drift and information feedback adds new challenges -- and chances.