  In this paper we use comparison theorems from classical ODE theory in order
to rigorously show that the N-Intertwined Mean-Field Approximation (NIMFA)
model provides an upper estimate on the exact stochastic process. The proof of
the results relies on the observation that the epidemic process is negatively
correlated (in the sense that the probability of an edge being in the
susceptible-infected state is smaller than the product of the probabilities of
the nodes being in the susceptible and infected states, respectively), which we
also prove rigorously. The results in the paper hold for arbitrary weighted and
directed networks. We cast the results in a more general framework where
alternative closures, other than that assuming the independence of nodes
connected by an edge, are possible and provide a succinct summary of the
stability analysis of the resulting more general mean-field models.
