Title: A generalized model of social and biological contagion,
Abstract: We present a model of contagion that unifies and generalizes existing models
of the spread of social influences and micro-organismal 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 \textit{epidemic threshold}, \textit{vanishing critical mass}, and
\textit{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.