  We study the spreading of two mutually cooperative diseases on different
network topologies, and with two microscopic realizations, both of which are
stochastic versions of an SIR type model studied by us recently in mean field
approximation. There it had been found that cooperativity can lead to
first-order spreading/extinction transitions. However, due to the rapid mixing
implied by the mean field assumption, first order transitions required non-zero
initial densities of sick individuals. For the stochastic model studied here
the results depend strongly on the underlying network. First order transitions
are found when there are few short but many long loops: (i) No first order
transitions exist on trees and on 2-d lattices with local contacts (ii) They do
exist on Erdos-Renyi (ER) networks, on d-dimensional lattices with d >= 4, and
on 2-d lattices with sufficiently long-ranged contacts; (iii) On 3-d lattices
with local contacts the results depend on the microscopic details of the
implementation; (iv) While single infected seeds can always lead to infinite
epidemics on regular lattices, on ER networks one sometimes needs finite
initial densities of infected nodes; (v) In all cases the first order
transitions are actually "hybrid", i.e. they display also power law scaling
usually associated with second order transitions. On regular lattices, our
model can also be interpreted as the growth of an interface due to cooperative
attachment of two species of particles. Critically pinned interfaces in this
model seem to be in different universality classes than standard critically
pinned interfaces in models with forbidden overhangs. Finally, the detailed
results mentioned above hold only when both diseases propagate along the same
network of links. If they use different links, results can be rather different
in detail, but are similar overall.
