  The study of how diseases spread has greatly benefited from advances in
network modeling. Recently, a class of networks known as multilayer graphs has
been shown to describe more accurately many real systems, making it possible to
address more complex scenarios in epidemiology such as the interaction between
different pathogens or multiple strains of the same disease. In this work, we
study in depth a class of networks that have gone unnoticed up to now, despite
of its relevance for spreading dynamics. Specifically, we focus on directed
multilayer networks, characterized by the existence of directed links, either
within the layers or across layers. Using the generating function approach and
numerical simulations of a stochastic susceptible-infected-susceptible (SIS)
model, we calculate the epidemic threshold for these networks for different
degree distributions of the networks. Our results show that the main feature
that determines the value of the epidemic threshold is the directionality of
the links connecting different layers, regardless of the degree distribution
chosen. Our findings are of utmost interest given the ubiquitous presence of
directed multilayer networks and the widespread use of disease-like spreading
processes in a broad range of phenomena such as diffusion processes in social
and transportation systems.
