THE SI AND SIR EPIDEMICS ON GENERAL NETWORKS
Intuitively one expects that for any plausible parametric epide mic model, there wil l be some region in parameter-space where the epidemicaffects (with high probability) only a small proportion of a largepopulation, another region where it affects (with high probability) a nonnegligible proportion, with a lower-dimensional “critical” interface. This dichotomy is certainly true in well-studied specific models, but we know o fno very general results. A recent result stated for a bond percolation modelcan be restated as giving weak conditions under which the dichotomy holdsfor an SI epidemic model on arbitrary finite networks. This result suggestsa conjecture for more complex and more realistic SIR epidemic models, and the purpose of this article is to record the conjecture.