  We propose a spreading model in multilayer networks and study the nature of
nonequilibrium phase transition in the model. The model integrates the
susceptible-infected-susceptible (or susceptible-infected-recovered) spreading
dynamics with a biased diffusion process among different layers. A parameter
$\alpha$ is introduced to control the bias of the diffusion process, such that
each individual prefers to move to one layer with more infected (or recovered)
neighbors for larger values of $\alpha$. Using stochastic simulations and
mean-field theory, we show that the type of phase transition from a
disease-free phase to an endemic phase depends on the value of $\alpha$. When
$\alpha$ is small enough, the system undergoes a usual continuous phase
transition as an effective spreading rate $\beta$ increases, as in single-layer
networks. Interestingly, when $\alpha$ exceeds a critical value the system
shows either a hybrid two-step phase transition or a one-step discontinuous
phase transition as $\beta$ increases. The former contains a continuous
transition between the disease-free phase and a low-prevalence endemic phase,
and a discontinuous transition between the low-prevalence endemic phase and a
high-prevalence endemic phase. For the latter, only a discontinuous transition
occurs from the disease-free phase directly to the high-prevalence endemic
phase. Moreover, we show that the discontinuous transition is always
accompanied by a spontaneous symmetry breaking in occupation probabilities of
individuals in each layer.
