  In spite of the study of epidemic dynamics on single-layer networks has
received considerable attention, the epidemic dynamics on multiplex networks is
still limited and is facing many challenges. In this work, we consider the
susceptible-infected-susceptible-type (SIS) epidemic model on multiplex
networks and investigate the effect of overlap among layers on the spreading
dynamics. To do so, we assume that the prerequisite of one $S$-node to be
infected is that there is at least one infectious neighbor in each layer. A
remarkable result is that the overlap can alter the nature of the phase
transition for the onset of epidemic outbreak. Specifically speaking, the
system undergoes a usual continuous phase transition when two layers are
completely overlapped. Otherwise, a discontinuous phase transition is observed,
accompanied by the occurrence of a bistable region in which a disease-free
phase and an endemic phase are coexisting. As the degree of the overlap
decreases, the bistable region is enlarged. The results are validated by both
simulation and mean-field theory.
