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Abstract
We present a model of contagion that unifies and generalizes existing models of the
spread of social influences and microorganismal infections. Our model incorporates
individual memory of exposure to a contagious entity (e.g. a rumor or disease), variable
magnitudes of exposure (dose sizes), and heterogeneity in the susceptibility of individuals.
Through analysis and simulation, we examine in detail the case where individuals may
recover from an infection and then immediately become susceptible again (analogous
to the so-called SIS model). We identify three basic classes of contagion models which
we call epidemic threshold, vanishing critical mass, and critical mass classes, where
each class of models corresponds to different strategies for prevention or facilitation.
We find that the conditions for a particular contagion model to belong to one of the
these three classes depend only on memory length and the probabilities of being infected
by one and two exposures, respectively. These parameters are in principle measurable
for real contagious influences or entities, thus yielding empirical implications for
our model. We also study the case where individuals attain permanent immunity once
recovered, finding that epidemics inevitably die out but may be surprisingly persistent
when individuals possess memory.