THE SI AND SIR EPIDEMICS ON GENERAL NETWORKS
Abstract: Intuitively one expects that for any plausible parametric epidemic model, there will be
some region in parameter-space where the epidemic affects (with high probability) only a
small proportion of a large population, another region where it affects (with high probability)
a non-negligible proportion, with a lower-dimensional “critical” interface. This dichotomy is
certainly true in well-studied specific models, but we know of no very general results. A
recent result stated for a bond percolation model can be restated as giving weak conditions
under which the dichotomy holds for an SI epidemic model on arbitrary finite
networks. This result suggests a conjecture for more complex and more realistic
SIR epidemic models, and the purpose of this article is to record the conjecture.
2010 AMS Mathematics Subject Classification: Primary: 60K35; Secondary:
92D60.
Keywords and phrases: SI epidemic, SIR epidemic.