The influence of the network characteristics on the virus spread is analyzed in a new-the -intertwined Markov chain-model, whose only approximation lies in the application of mean field theory. The mean field approximation is quantified in detail. The -intertwined model has been compared with the exact 2 -state Markov model and with previously proposed "homogeneous" or "local" models. The sharp epidemic threshold , which is a consequence of mean field theory, is rigorously shown to be equal to = 1 ( max ( )), where max ( ) is the largest eigenvalue-the spectral radius-of the adjacency matrix . A continued fraction expansion of the steady-state infection probability at node is presented as well as several upper bounds.Index Terms-Epidemic threshold, Markov theory, mean field theory, spectral radius, virus spread.

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