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Abstract Human mobility is a key driver of infectious disease spread. Recent literature has uncovered a clear pattern underlying the complexity of human mobility in cities: $$r \cdot f$$ r · f , the product of distance traveled r and frequency of return f per user to a given location, is invariant across space. This paper asks whether the invariant $$r\cdot f$$ r · f also serves as a driver for epidemic spread, so that the risk associated with human movement can be modeled by a unifying variable $$r\cdot f$$ r · f . We use two large-scale datasets of individual human mobility to show that there is in fact a simple relation between r and f and both speed and spatial dispersion of disease spread. This discovery could assist in modeling spread of disease and inform travel policies in future epidemics—based not only on travel distance r but also on frequency of return f.

We investigate a population of swarmalators, mobile versions of phase oscillators that both sync in time and swarm through space. We focus on an XY-type model of identical swarmalators running on a one-dimensional ring and subject to thermal noise. We uncover four distinct collective states, some of which capture the behavior of real-world swarmalators such as vinegar eels and sperm. Among these, the most intriguing is the “mixed state,” which blends two of the other states. We present a comprehensive phase diagram from the Fourier mode analysis with a high accuracy, which is in excellent agreement with numerical simulation results. Our model serves as a tractable toy model for thermal systems that both self-synchronize and self-assemble interdependently.