A disease outbreak is like a wildfire: the fire runs through dry grass, and its speed depends on wind and humidity. But an infection has its own 'wind'—how often people infect each other. When this rate changes along with the fraction of already infected, the epidemic wave can behave differently.
Sometimes the leading edge of infection is determined by the swiftest sparks: those who got infected first and ran ahead. Such a wave is called 'pulled': its speed is dictated by lone individuals at the front. In other cases, the main mass of infected pushes from behind, and the front becomes 'pushed'—it is driven by the whole system. The authors calculated on a computer and confirmed with theory where the boundary between these regimes lies.
The main surprise is the discovery of a zone where both types of motion are possible. It is as if a fire could simultaneously smolder and blaze, depending on how it was ignited. This duality is important for forecasts: knowing the initial conditions, one can predict how rapidly the epidemic will spread and which scenario will unfold.
The speed of a 'pushed' wave is always higher than that of a 'pulled' one, but it requires a stronger initial push to start, like a rocket that needs a powerful launch to reach orbit.
In Stephen King's novel 'The Stand', a super-flu spreads at a terrifying speed, but no one there wondered whether it is 'pulled' by patient zero or 'pushed' by the crowd.