A mathematical study of combined bacteriophage and antibiotic therapies

Abstract

The rise of antimicrobial resistance poses a major global health threat, requiring innovative treatment strategies beyond conventional antibiotics. Among the most promising alternatives is the combined use of bacteriophages (phages) and antibiotics—a synergy that can potentially suppress or reverse bacterial resistance. In this work, we develop and analyze several mathematical models to study the population dynamics of bacteria under antibiotic and phage pressure, focusing on \textit{in vitro} systems involving \textit{Klebsiella pneumoniae}, a clinically relevant pathogen. We begin by analyzing a reduced antibiotic–bacteria model, characterizing its equilibria, stability, and bifurcations. Through the use of the variational equations, we identify the system's sensitivity to parameters—particularly the antibiotic-induced mutation rate and bacterial death rate—that govern extinction times and transient behaviors. We then extend the model to incorporate phage dynamics, resulting in a coupled system that allows the study of phage-antibiotic synergies. We investigate scenarios such as the temporary near-disappearance and subsequent re-emergence of bacterial populations, and study the role of pseudo-equilibria, where the phage population may not be at equilibrium. Our findings suggest that system behavior is highly sensitive to specific parameters such as antibiotic concentration, mutation rate (from resistant to sensitive), growth rate, and mortality rate. This modeling framework lays the groundwork for future experimental calibration and optimization of phage–antibiotic strategies, and opens up new questions regarding the influence of additional variables, such as temperature and pH causing both phage and bacteria decays, and the existence of scaling laws near bifurcations.