The influence of the network characteristics on the virus spread is analyzed in a new-the N-intertwined Markov chain-model, whose only approximation lies in the application of mean field theory. The mean field approximation is quantified in detail. The N-intertwined model has been compared with the exact 2N-state Markov model and with previously proposed "homogeneous"or "local"models. The sharp epidemic threshold τc, which is a consequence of mean field theory, is rigorously shown to be equal to τc= (λmax (A)), where λmax (A) is the largest eigenvalue-the spectral radius-of the adjacency matrix A.A. A continued fraction expansion of the steady-state infection probability at node j is presented as well as several upper bounds. © 2008 IEEE.