Bacteriophages, or phages (viruses of bacteria), play significant roles in shaping the diversity of bacterial communities within the human gut. A phage-infected bacterial cell can either immediately undergo lysis (virulent/lytic infection) or enter a stable state within the host as a prophage (lysogeny) until a trigger event, called prophage induction, initiates the lysis process. We develop an approach based on a model structured in terms of time since bacterial infection. We derive important threshold parameters for the asymptotic dynamics of the system and demonstrate that the model’s qualitative behavior can range from the extinction of all bacterial strains to the persistence of a single strain (either lysogen or non-lysogen bacteria) or the coexistence of all strains at a positive steady state. We highlight the existence of critical induction rate values that lead to the coexistence of all states through periodic oscillations. We also conduct a global sensitivity analysis for an effective bacterial clearance. In scenarios where antibiotics are not sufficiently effective, we identify four key phage parameter traits: (i) the phage induction probability, describing the capacity of prophages to be induced, (ii) the probability of absorption, describing the phages’ ability to invade susceptible bacteria, (iii) the reproduction number of susceptible bacteria in the absence of antibiotics, and (iv) the latent period, describing the time since absorption. The obtained results emphasize the effective therapeutic potential of selected phages.