Group testing, where individuals are tested initially in pools, is widely used to screen a large number of individuals for rare diseases. Triggered by the recent development of assays that detect multiple infections at once, screening programs now involve testing individuals in pools for multiple infections simultaneously. Tebbs, McMahan, and Bilder (2013, Biometrics) recently evaluated the performance of a two-stage hierarchical algorithm used to screen for chlamydia and gonorrhea as part of the Infertility Prevention Project in the United States. In this article, we generalize this work to accommodate a larger number of stages. To derive the operating characteristics of higher-stage hierarchical algorithms with more than one infection, we view the pool decoding process as a time-inhomogeneous, finite-state Markov chain. Taking this conceptualization enables us to derive closed-form expressions for the expected number of tests and classification accuracy rates in terms of transition probability matrices. When applied to chlamydia and gonorrhea testing data from four states (Region X of the United States Department of Health and Human Services), higher-stage hierarchical algorithms provide, on average, an estimated 11% reduction in the number of tests when compared to two-stage algorithms. For applications with rarer infections, we show theoretically that this percentage reduction can be much larger.