Healthcare-associated infections (HAIs) are a major problem in hospital infection control. Although HAIs can be suppressed using contact precautions, such precautions are expensive, and we can only apply them to a small fraction of patients (i.e., a limited budget). In this work, we focus on two clinical problems arising from the limited budget: (a) choosing the best patients to be placed under precaution given a limited budget to minimize the spread (the Isolation problem), and (b) choosing the best patients to release when limited budget requires some of the patients be cleared from precaution (the Clearance problem). One of the critical challenges in solving them is that HAIs have multiple transmission pathways such that locations can also accumulate 'load' and spread the disease. Hence, standard propagation models like Independent Cascade (IC)/Susceptible-Infected-Susceptible (SIS) cannot capture such mechanisms. To account for this challenge, using non-linear system theory, we develop a novel spectral characterization of a recently proposed 2-Mode-SIS model on people/location networks to capture HAIs spread dynamics. We formulate the two clinical problems with this spectral characterization and explain previously observed counter-intuitive behaviors like the `dilution effect'. We also develop effective and efficient algorithms to solve the formulated problems. Our experiments show that our methods outperform several natural structural and clinical approaches on real-world hospital testbeds and pick meaningful solutions.