  In this paper, we propose two novel immunization strategies, i.e., combined
immunization and duplex immunization, for SIS model in directed scale-free
networks, and obtain the epidemic thresholds for them with linear and nonlinear
infectivities. With the suggested two new strategies, the epidemic thresholds
after immunization are greatly increased. For duplex immunization, we
demonstrate that its performance is the best among all usual immunization
schemes with respect to degree distribution. And for combined immunization
scheme, we show that it is more effective than active immunization. Besides, we
give a comprehensive theoretical analysis on applying targeted immunization to
directed networks. For targeted immunization strategy, we prove that immunizing
nodes with large out-degrees are more effective than immunizing nodes with
large in-degrees, and nodes with both large out-degrees and large in-degrees
are more worthy to be immunized than nodes with only large out-degrees or large
in-degrees. Finally, some numerical analysis are performed to verify and
complement our theoretical results. This work is the first to divide the whole
population into different types and embed appropriate immunization scheme
according to the characteristics of the population, and it will benefit the
study of immunization and control of infectious diseases on complex networks.
