Discontinuous phase transitions via cooperative contagion

We study the spreading of two diseases that interact cooperatively (the presence of one helps the other one to spread) on different network topologies, and with two microscopic realizations, both of which are stochastic versions of an SIR type studied by us recently in mean field approximation. We had shown that cooperativity can lead to discontinuous transitions (DT). However, due to the rapid mixing implied by the mean field assumption, DTs were seen only when there were finite (non-zero) densities of sick individuals in the initial state.In this paper we find that the results for the stochastic model depend strongly on the underlying network. In particular, DTs are found when there are few short but many long loops: (i) No DTs exist on trees, due to the absence of loops; (ii) On 2-d lattices with local contacts there are no DTs either, but because of too many short loops; (iii) We do find DTs on Erdos-Renyi (ER) networks, on d-dimensional lattices with $d\geq 4$,and on 2-d lattices with sufficiently long-ranged contacts; (iv) On 3-d lattices with local contacts the results depend on the microscopic details of the implementation. All found discontinuous transitions are of ``hybrid" type, i.e. they display also scaling features usually associated with continuous transitions.