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Mathematical Modeling of Biofilm Eradication Using Optimal Control

Rehan Akber, Adnan Khan

TL;DR

This work develops a 1-D moving-boundary biofilm model with susceptible, persister, and resistant bacteria, incorporating conjugation-mediated horizontal gene transfer and antibiotic dynamics. It analyzes steady states and compares continuous, periodic, and optimally controlled dosing strategies, showing periodic dosing can reduce total antibiotic use. Using an extended Pontryagin Maximum Principle for a mixed parabolic–hyperbolic PDE system, it derives a tapered, cost-efficient dosing protocol that achieves eradication. Numerical simulations corroborate the theory and demonstrate robust, dose-modulation–driven improvements across parameter regimes.

Abstract

We propose and analyze a model for antibiotic resistance transfer in a bacterial biofilm and examine antibiotic dosing strategies that are effective in bacterial elimination. In particular, we consider a 1-D model of a biofilm with susceptible, persistor and resistant bacteria. Resistance can be transferred to the susceptible bacteria via horizontal gene transfer (HGT), specifically via conjugation. We analyze some basic properties of the model, determine the conditions for existence of disinfection and coexistence states, including boundary equilibria and their stability. Numerical simulations are performed to explore different modeling scenarios and support our theoretical findings. Different antibiotic dosing strategies are then studied, starting with a continuous dosing; here we note that high doses of antibiotic are needed for bacterial elimination. We then consider periodic dosing, and again observe that insufficient levels of antibiotic per dose may lead to treatment failure. Finally, using an extended version of Pontryagin's maximum principle we determine efficient antibiotic dosing protocols, which ensure bacterial elimination while keeping the total dosing low; we note that this involves a tapered dosing which reinforces results presented in other clinical and modeling studies. We study the optimal dosing for different parameter values and note that the optimal dosing schedule is qualitatively robust.

Mathematical Modeling of Biofilm Eradication Using Optimal Control

TL;DR

This work develops a 1-D moving-boundary biofilm model with susceptible, persister, and resistant bacteria, incorporating conjugation-mediated horizontal gene transfer and antibiotic dynamics. It analyzes steady states and compares continuous, periodic, and optimally controlled dosing strategies, showing periodic dosing can reduce total antibiotic use. Using an extended Pontryagin Maximum Principle for a mixed parabolic–hyperbolic PDE system, it derives a tapered, cost-efficient dosing protocol that achieves eradication. Numerical simulations corroborate the theory and demonstrate robust, dose-modulation–driven improvements across parameter regimes.

Abstract

We propose and analyze a model for antibiotic resistance transfer in a bacterial biofilm and examine antibiotic dosing strategies that are effective in bacterial elimination. In particular, we consider a 1-D model of a biofilm with susceptible, persistor and resistant bacteria. Resistance can be transferred to the susceptible bacteria via horizontal gene transfer (HGT), specifically via conjugation. We analyze some basic properties of the model, determine the conditions for existence of disinfection and coexistence states, including boundary equilibria and their stability. Numerical simulations are performed to explore different modeling scenarios and support our theoretical findings. Different antibiotic dosing strategies are then studied, starting with a continuous dosing; here we note that high doses of antibiotic are needed for bacterial elimination. We then consider periodic dosing, and again observe that insufficient levels of antibiotic per dose may lead to treatment failure. Finally, using an extended version of Pontryagin's maximum principle we determine efficient antibiotic dosing protocols, which ensure bacterial elimination while keeping the total dosing low; we note that this involves a tapered dosing which reinforces results presented in other clinical and modeling studies. We study the optimal dosing for different parameter values and note that the optimal dosing schedule is qualitatively robust.

Paper Structure

This paper contains 14 sections, 9 theorems, 59 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical Modeling of Biofilm
  3. Steady State Solutions
  4. Autonomous System
  5. Numerical Results
  6. Continuous Dosing
  7. Periodic Dosing
  8. Optimal Control
  9. PMP for Mixed Parabolic–Hyperbolic Optimal Control Problem
  10. Numerical Scheme
  11. Results of Treatment through Optimal Dosing
  12. Dose Tapering
  13. Effects of kdr and beta on Optimal Control Strategy
  14. Conclusion

Key Result

Lemma 1

Define $m_1 := 1 - k_d A_0, \; m_2 := 1 - k_d^r A_0, \; m_1^\ast := \frac{g_p\,k_l}{\,g_p + k_r\,}$ and let $\lambda_1,\lambda_2,\lambda_3$ are the eigen values of the autonomous system eq:Bs_s2- eq:Br_s2 obtained by linearizing it at the trivial equilibrium $(b_s,b_p,b_r)=(0,0,0)$, then

Figures (10)

  • Figure 1: Completely Non-Trivial Steady State (Total antibiotic used: 100 units)
  • Figure 2: Partially Non-Trivial Steady State (Total antibiotic used: 250 units)
  • Figure 3: Trivial Steady State (Total antibiotic used: 1000 units)
  • Figure 4: Completely Non-Trivial Steady State (Total antibiotic used: 80 units)
  • Figure 5: Partially Non-Trivial Steady State (Total antibiotic used: 128 units)
  • ...and 5 more figures

Theorems & Definitions (10)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5