  We present a continuous formulation of epidemic spreading on multilayer
networks using a tensorial representation, extending the models of monoplex
networks to this context. We derive analytical expressions for the epidemic
threshold of the SIS and SIR dynamics, as well as upper and lower bounds for
the disease prevalence in the steady state for the SIS scenario. Using the
quasi-stationary state method we numerically show the existence of disease
localization and the emergence of two or more susceptibility peaks, which are
characterized analytically and numerically through the inverse participation
ratio. Furthermore, when mapping the critical dynamics to an eigenvalue
problem, we observe a characteristic transition in the eigenvalue spectra of
the supra-contact tensor as a function of the ratio of two spreading rates: if
the rate at which the disease spreads within a layer is comparable to the
spreading rate across layers, the individual spectra of each layer merge with
the coupling between layers. Finally, we verified the barrier effect, i.e., for
three-layer configuration, when the layer with the largest eigenvalue is
located at the center of the line, it can effectively act as a barrier to the
disease. The formalism introduced here provides a unifying mathematical
approach to disease contagion in multiplex systems opening new possibilities
for the study of spreading processes.
