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.