There are two standard "Ising" models for the spreading of epidemics,
the "SIS" (susceptible-infected-susceptible) epidemic and the "SIR"
(susceptible-infected-removed) model. Both are related to percolation
- the first to directed, the second to undirected - and thus both show
continuous ("second-order") phase transitions when the conditions for
spreading are critical. Recently, much interest has been roused by
models which show first order transitions. After giving a short
overview I will concentrate on first order transitions in models with
cooperativity. This cooperativity can be of two very different types.
On the one hand, several agents on neighboring nodes in a network (or
sites on a lattice) can cooperate in infecting a new node (like three
friends who convince you more easily than any single one of them to
adopt a political opinion). On the other hand, also two pathogens
(like HIV and TB) can cooperate in the sense that one of them lowers
the resistance to the other. In the first case the transition from a
continuous to a discontinuous phase transition is a standard
tricritical point (with higher n-point interactions in a field
theoretic formalism), while in the second one one has a
multi-component order parameter. In several instances, the resulting
first order transitions are actually "hybrid", i.e. they involve also
features of second-order transitions like scaling. In one case, one even finds
that two order parameter definitions which coincide for ordinary
percolation display different transition orders.