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.
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