Stochastic Dynamics of Sensitive and Resistant Bacteria Under Capacity-Limited Antibiotic Treatment

Antimicrobial resistance is a major challenge in the effective treatment of bacterial infections because antibiotic exposure can selectively suppress sensitive bacteria while allowing resistant populations to persist and expand. In this study, we develop a finite-state stochastic population model for the joint dynamics of antibiotic-sensitive and antibiotic-resistant bacteria under capacity-limited treatment. The model is formulated as a bivariate continuous-time Markov chain with density-dependent logistic growth, sensitive-to-resistant mutation, differential antibiotic susceptibility, finite carrying capacity, and bounded treatment-removal capacity. Because the extinction state is absorbing, the ordinary stationary distribution of the full finite-state process is concentrated at extinction, and long-lived non-extinct behavior is therefore analyzed through transient and quasi-stationary perspectives. Exact first-moment equations are derived, showing that mutation transfers individuals from the sensitive population to the resistant population without creating additional bacterial individuals. Local stability near extinction is characterized by the growth rates rs=λs-μ-ν and rr=λr-ϵμ. The positive equilibria of the deterministic fluid approximation are then derived under the different treatment-capacity regimes, while capacity-induced critical transitions are characterized through the equilibrium equations and the singularity condition detJ=0, where J is the Jacobian of the deterministic drift. Near a nondegenerate fold, the model exhibits square-root equilibrium scaling and critical slowing down. Gillespie Monte Carlo simulations complement the analytical results by quantifying finite-horizon survival probability, extinction time, conditional bacterial burden, resistant fraction, and cumulative bacterial exposure. The resulting framework provides a mathematically consistent description of how bacterial growth, mutation, differential susceptibility, and limited treatment capacity jointly influence extinction and long-lived persistence.