  We consider the emergent behavior of viral spread when agents in a large
population interact with each other over a contact network. When the number of
agents is large and the contact network is a complete graph, it is well known
that the population behavior -- that is, the fraction of susceptible, infected
and recovered agents -- converges to the solution of an ordinary differential
equation (ODE) known as the classical SIR model as the population size
approaches infinity. In contrast, we study interactions over contact networks
with generic topologies and derive conditions under which the population
behavior concentrates around either the classic SIR model or other
deterministic models. Specifically, we show that when most vertex degrees in
the contact network are sufficiently large, the population behavior
concentrates around an ODE known as the network SIR model. We then study the
short and intermediate-term evolution of the network SIR model and show that if
the contact network has an expander-type property or the initial set of
infections is well-mixed in the population, the network SIR model reduces to
the classical SIR model. To complement these results, we illustrate through
simulations that the two models can yield drastically different predictions,
hence use of the classical SIR model can be misleading in certain cases.
