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.
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