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The spatial spread of an epidemic is investigated in the case of a bistable dynamics, where the effective transmission rate depends on the fraction of infected and the state of no epidemic is linearly stable. The front propagation phenomenon is investigated both numerically and theoretically, by an analysis in a four-dimensional phase plane. A good agreement between numerical and theoretical results has been found both for the front profiles and for the speed of invasion. We discovered a novel phenomenon of front stoppage: In some regime of parameters, the front solution ceases to exist, and the propagating pulse of infection decays despite the initial outbreak.