  This study extends the SIS epidemic model for single virus propagation over
an arbitrary graph to an SI1SI2S epidemic model of two exclusive, competitive
viruses over a two-layer network with generic structure, where network layers
represent the distinct transmission routes of the viruses. We find analytical
results determining extinction, mutual exclusion, and coexistence of the
viruses by introducing the concepts of survival threshold and winning
threshold. Furthermore, we show the possibility of coexistence in SIS-type
competitive spreading over multilayer networks. Not only do we rigorously prove
a region of coexistence, we quantitate it via interrelation of central nodes
across the network layers. Little to no overlapping of layers central nodes is
the key determinant of coexistence. Specifically, we show coexistence is
impossible if network layers are identical yet possible if the network layers
have distinct dominant eigenvectors and node degree vectors. For example, we
show both analytically and numerically that positive correlation of network
layers makes it difficult for a virus to survive while in a network with
negatively correlated layers survival is easier but total removal of the other
virus is more difficult. We believe our methodology has great potentials for
application to broader classes of multi-pathogen spreading over multi-layer and
interconnected networks.
