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In this paper, we propose two novel immunization strategies, i.e., combinedimmunization and duplex immunization, for SIS model in directed scale-freenetworks, and obtain the epidemic thresholds for them with linear and nonlinearinfectivities. With the suggested two new strategies, the epidemic thresholdsafter immunization are greatly increased. For duplex immunization, wedemonstrate that its performance is the best among all usual immunizationschemes with respect to degree distribution. And for combined immunizationscheme, we show that it is more effective than active immunization. Besides, wegive a comprehensive theoretical analysis on applying targeted immunization todirected networks. For targeted immunization strategy, we prove that immunizingnodes with large out-degrees are more effective than immunizing nodes withlarge in-degrees, and nodes with both large out-degrees and large in-degreesare more worthy to be immunized than nodes with only large out-degrees or largein-degrees. Finally, some numerical analysis are performed to verify andcomplement our theoretical results. This work is the first to divide the wholepopulation into different types and embed appropriate immunization schemeaccording to the characteristics of the population, and it will benefit thestudy of immunization and control of infectious diseases on complex networks.

Two novel immunization strategies for epidemic control in directed scale-free networks with nonlinear infectivity