Phage-antibiotic therapy under density dependent bacterial defenses

Phage therapy is an alternative treatment method for bacterial infections. It has shown particular promise in reducing bacterial load while preventing antibiotic resistance. Here, we develop a mathematical model of a bacterial infection within a host to study phage therapy. It incorporates interactions between phages, bacteria, the immune system, and antibiotics. Additionally, the model includes bacterial social dynamics that provide protection from treatments and the innate immune response. We analytically and numerically identify all of the equilibria of the model and derive insights regarding the overall effectiveness of phage therapy. Without phage therapy, the model exhibits bistability: bacteria populations above a threshold grow and become entrenched, while those below it can be effectively suppressed by the immune system. We find that that phages destabilize the former equilibrium, and thus in combination with the immune system are able to suppress the bacteria. We conducted bifurcation analyses, which show that the equilibrium with a suppressed population of bacteria can become unstable. In this scenario, the system undergoes oscillations. However, these oscillations – which can be exacerbated by social dynamics – lead to minuscule bacterial populations, and thus, in practice, phage therapy is widely effective across the parameter space. We also demonstrate how suppression can be further improved by the addition of periodic dosing of antibiotics in a combination therapy.

💡 Research Summary

The authors develop a deterministic ODE model to explore the dynamics of a bacterial infection under the combined influence of bacteriophages, the host innate immune response, and antibiotics. The state variables are bacterial density (B), immune effector cells (I), phage particles (P), and antibiotic concentration (A). Bacterial growth follows logistic kinetics (maximum rate ρ, carrying capacity κ). Bacteria are removed by immune cells (kill rate ϵ), by phages (adsorption rate ϕ), and by antibiotics (Hill‑type killing α·A/(τ·A+ξ)). Crucially, the phage adsorption term is modulated by a quorum‑sensing factor 1/(1+B/γ), capturing the experimentally observed reduction of phage receptors at high bacterial densities. Immune effectors are produced at a basal rate σ and are up‑regulated by bacteria through a saturating activation term θ·B/(B+η); they die naturally at rate δ and are additionally killed by bacterial toxins at rate µ·B. Phages replicate with burst size β after successful infection and decay at rate ω. Antibiotics are administered periodically (dose D(t) every 24 h) and cleared at rate ν.

The analysis first omits antibiotics to focus on the intrinsic bacteria‑phage‑immune subsystem. Three classes of equilibria are identified: (i) the disease‑free state E₁=(0,σ/δ,0), which is unstable when ρ>ϵ·σ/δ, (ii) phage‑free bacterial‑immune equilibria E₂ that can be one, two, or three distinct points depending on parameter values, and (iii) a fully co‑existent equilibrium E₃ where bacteria, phage, and immune cells persist together. The system exhibits bistability in the absence of phage: low initial bacterial loads decay to E₁, whereas loads above a critical threshold converge to a high‑density E₂ (persistent infection). This reproduces earlier findings on infection outcomes being governed by initial conditions and immune strength.

Introducing phage creates a new stable equilibrium E₃ and can destabilize the high‑density E₂ through a transcritical or Hopf bifurcation. Numerical continuation shows that as the quorum‑sensing threshold γ decreases (stronger density‑dependent protection), the Hopf point moves, leading to sustained oscillations in B, I, and P. Despite oscillations, the minimum bacterial density during a cycle falls to 10⁻³–10⁻⁴ ×10⁶ cells mL⁻¹, effectively “eradicated” from a clinical perspective. The oscillatory regime is amplified when both the phage adsorption reduction (γ) and immune saturation (ζ) are strong, highlighting the role of social defenses such as biofilm formation.

The authors then incorporate periodic antibiotic dosing. A 100 mg L⁻¹ dose given at t = 100 h and repeated every 24 h, combined with a single phage administration at the same time, dramatically expands the parameter region where the infection is driven to the low‑density or disease‑free state. Antibiotics synergize with phage by lowering bacterial density, thereby weakening quorum‑sensing mediated protection and restoring phage efficacy. The combined therapy also suppresses the amplitude of oscillations, leading to faster convergence and reducing the risk of resurgence, even when resistant subpopulations are present.

Overall, the study provides a mechanistic framework that links density‑dependent bacterial social defenses to the success or failure of phage‑antibiotic combination therapy. Key insights include: (1) phage can break the bistable barrier that limits immune‑only clearance; (2) strong quorum‑sensing or biofilm protection can induce limit‑cycle oscillations, but these oscillations still drive bacterial numbers to clinically negligible levels; (3) periodic antibiotic dosing enhances stability and widens the therapeutic window; and (4) optimal timing and dosing of phage relative to antibiotics are critical for maximizing synergy. These results have practical implications for designing phage‑based treatments, suggesting that accounting for bacterial density and social behavior is essential for reliable infection control.