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We study the spreading of two mutually cooperative diseases on differentnetwork topologies, and with two microscopic realizations, both of which arestochastic versions of an SIR type model studied by us recently in mean fieldapproximation. There it had been found that cooperativity can lead tofirst-order spreading/extinction transitions. However, due to the rapid mixingimplied by the mean field assumption, first order transitions required non-zeroinitial densities of sick individuals. For the stochastic model studied herethe results depend strongly on the underlying network. First order transitionsare found when there are few short but many long loops: (i) No first ordertransitions exist on trees and on 2-d lattices with local contacts (ii) They doexist on Erdos-Renyi (ER) networks, on d-dimensional lattices with d >= 4, andon 2-d lattices with sufficiently long-ranged contacts; (iii) On 3-d latticeswith local contacts the results depend on the microscopic details of theimplementation; (iv) While single infected seeds can always lead to infiniteepidemics on regular lattices, on ER networks one sometimes needs finiteinitial densities of infected nodes; (v) In all cases the first ordertransitions are actually "hybrid", i.e. they display also power law scalingusually associated with second order transitions. On regular lattices, ourmodel can also be interpreted as the growth of an interface due to cooperativeattachment of two species of particles. Critically pinned interfaces in thismodel seem to be in different universality classes than standard criticallypinned interfaces in models with forbidden overhangs. Finally, the detailedresults mentioned above hold only when both diseases propagate along the samenetwork of links. If they use different links, results can be rather differentin detail, but are similar overall.

Phase Transitions in Cooperative Coinfections: Simulation Results for Networks and Lattices