  We investigate the spread of diseases, computer viruses or information on
complex networks and also immunization strategies to prevent or control the
spread. When an entire population cannot be immunized and the effect of
immunization is not perfect, we need the targeted immunization with
immunization rate. Under such a circumstance we calculate epidemic thresholds
for the SIR and SIS epidemic models. It is shown that, in scale-free networks,
the targeted immunization is effective only if the immunization rate is equal
to one. We analyze here epidemic spreading on directed complex networks, but
similar results are also valid for undirected ones.
