The network structure provides critical information for law enforcement agencies to develop effective strategies to interdict illicit supply networks. However, the complete structure of covert networks is often unavailable, thus it is crucially important to develop approaches to infer a more complete structure of covert networks. In this paper, we work on real-world multiplex drug trafficking networks extracted from an investigation report. A statistical approach built on the EM algorithm (DegEM) as well as other methods based on structural similarity are applied to reconstruct the multiplex drug trafficking network given different fractions of observed nodes and links. It is found that DegEM approach achieves the best predictive performance in terms of several accuracy metrics. Meanwhile, structural similarity-based methods perform poorly in reconstructing the drug trafficking networks due to the sparsity of links between nodes in the network. The inferred multiplex networks can be leveraged to (i) inform the decision-making on monitoring covert networks as well as allocating limited resources for collecting additional information to improve the reconstruction accuracy and (ii) develop more effective interdiction strategies.