  We demonstrate that the susceptible-infected-susceptible (SIS) model on
complex networks can have an inactive Griffiths phase characterized by a slow
relaxation dynamics. It contrasts with the mean field theoretical prediction
that the SIS model on complex networks is active at any nonzero infection rate.
The dynamic fluctuation of infected nodes, ignored in the mean field approach,
is responsible for the inactive phase. It is proposed that the question whether
the epidemic threshold of the SIS model on complex networks is zero or not can
be resolved by the percolation threshold in a model where nodes are occupied in
the degree-descending order. Our arguments are supported by the numerical
studies on scale-free network models.
