  It is often useful to represent the infectious dynamics of mobile agents by
metapopulation models. In such a model, metapopulations form a static network,
and individuals migrate from one metapopulation to another. It is known that
heterogeneous degree distributions of metapopulation networks decrease the
epidemic threshold above which epidemic spreads can occur. We investigate the
combined effect of heterogeneous degree distributions and diffusion on
epidemics in metapopulation networks. We show that for arbitrary heterogeneous
networks, diffusion suppresses epidemics in the sense of an increase in the
epidemic threshold. On the other hand, some diffusion rates are needed to
elicit epidemic spreads on a global scale. As a result of these opposing
effects of diffusion, epidemic spreading near the epidemic threshold is the
most pronounced at an intermediate diffusion rate. The result that diffusion
can suppress epidemics contrasts with that for diffusive SIS dynamics and its
variants when individuals are fixed at nodes on static networks.
