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Compared with traditional models, semi-random epidemic network models may be more reasonable to describe the real dynamics of many epidemics. In this paper, we first investigate the optimal control problem (OCP) of semi-random epidemic networks. By using the Pontryagin’s minimum principle, we obtain the optimal control strategy aimed to minimize the total epidemic incidence and control cost. We then define a centrality index which can measure average control strength of the optimal control. Based on this index, the OCP is converted into a static OCP (SOCP), whose solution is utilized to design a nonidentical constant control (NCC). NCC is suboptimal as it is optimal on a subset of the whole control set, and is determined by only the network’s clustering coefficient and initial condition. We finally propose an effective targeted constant quarantine control by using this centrality index. The results uncover the relationship between the optimal control and the network’s topological structure, provide a convenient method to determine suboptimal control, and present a strategy for targeted constant control. This paper can help to design effective control strategies for more general epidemic networks in the real world. [ABSTRACT FROM AUTHOR]