  Models of contagion arise broadly both in the biological and social sciences,
with applications ranging from the transmission of infectious diseases to the
diffusion of innovations and the spread of cultural fads. In this Letter, we
introduce a general model of contagion which, by explicitly incorporating
memory of past exposures to, for example, an infectious agent, rumor, or new
product, includes the main features of existing contagion models and
interpolates between them. We obtain exact solutions for a simple version of
the model, finding that under general conditions only three classes of
collective dynamics exist, two of which correspond to familiar epidemic
threshold and critical mass dynamics, while the third is a distinct
intermediate case. We find that for a given length of memory, the class into
which a particular system falls is determined by two parameters, each of which
ought to be measurable empirically. Our model suggests novel measures for
assessing the susceptibility of a population to large contagion events, and
also a possible strategy for inhibiting or facilitating them.
