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 a susceptible-infected-removed type model studied by us recently in mean field approximation. There it had been found that cooperativity can lead to first order transitions from spreading to extinction. However, due to the rapid mixing implied by the mean field assumption, first order transitions required nonzero 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.
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