We consider the spread of an infectious disease on a heterogeneous metapopulation defined by any (correlated or uncorrelated) network. The infection evolves under transmission, recovery and migration mechanisms. We study some spectral properties of a connectivity matrix arising from the continuous-time equations of the model. In particular we show that the classical sufficient condition of instability for the disease-free equilibrium, well known for the particular case of uncorrelated networks, works also for the general case. We give also an alternative condition that yields a more accurate estimation of the epidemic threshold for correlated (either assortative or dissortative) networks

Supported by Ministry of Economy and Competitiveness of the Spanish Government grant numbers MTM2011-27739-C04-03 (first author) and DPI2011-25822 (second author). Also by Generalitat de Catalunya projects 2009-SGR-345 (first author) and 2009-SGR-1228 (second author).