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Collateral sensitivity, where resistance to one drug confers heightened sensitivity to another, offers a promising strategy for combating antimicrobial resistance, yet predicting resultant evolutionary dynamics remains a significant challenge. We propose here a mathematical model that integrates fitness trade-offs and adaptive landscapes to predict the evolution of collateral sensitivity pathways, providing insights into optimizing sequential drug therapies. Our approach embeds collateral information into a network of switched systems, allowing us to abstract the effects of sequential antibiotic exposure on antimicrobial resistance. We analyze the system stability at disease-free equilibrium and employ set-control theory to tailor therapeutic windows. Consequently, we propose a computational algorithm to identify effective sequential therapies to counter antibiotic resistance. By leveraging our theory with data on collateral sensivity interactions, we predict scenarios that may prevent bacterial escape for chronic Pseudomonas aeruginosa infections. 

Switched systems are important for modeling biomedical control problems, where the control action can be considered to be a switching signal to select the active mode (e.g., a drug therapy or intervention). Practical implementations impose constraints not only on the variable magnitudes, but also on each mode’s active and inactive times (AT and IT). To address this, a model predictive control strategy is proposed using an enlarged model with integer state variables to track past AT/IT for each mode. Two biomedical applications were selected to demonstrate the controller’s effectiveness through simulations. The results highlight that our approach is suitable for biomedical applications with intricate temporal requirements. 

The evolution of antibiotic resistance in bacteria is a significant public health risk influenced by several factors. Switched systems can abstract the evolutionary aspects driven by antibiotic use in a given population. However, mathematical models are not perfect, and uncertain dynamics remain. Based on a set theory approach, our main result is the development of an algorithm to demonstrate the stabilizability of a robust invariant set for the uncertain switched system. The algorithm also provides a characterization of invariant regions for switched systems under perturbations. Our findings provide insights into how to incorporate uncertainties in switched systems. This paves the way for selecting antibiotics to tackle drug-resistant infections.