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A Nonlinear Optimal Control Approach for Bacterial Infections Under Antibiotics Resistance

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Abstract

The overuse and misuse of antibiotics has become a major problem for public health. People become resistant to antibiotics and because of this the anticipated therapeutic effect is never reached. In-hospital infections are often aggravated and large amounts of money are spent for treating complications in the patients’ condition. In this paper a nonlinear optimal (H-infinity) control method is developed for the dynamic model of bacterial infections exhibiting resistance to antibiotics. First, differential flatness properties are proven for the associated state-space model. Next, the state-space description undergoes approximate linearization with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instance around a time-varying operating point which is defined by the present value of the system’s state vector and by the last sampled value of the control inputs vector. For the approximately linearized model of the system a stabilizing H-infinity feedback controller is designed. To compute the controller’s gains an algebraic Riccati equation has to be repetitively solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. The proposed method achieves stabilization and remedy for the bacterial infection under moderate use of antibiotics.

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References

  1. Ibarguen-Modragan E, Mosquera S, Ceron M, et al., Mathematical modelling on bacterial resistance to multiple antibiotics causes by spontaneous mutations, Biosystems, 2014, 117): 60–67.

    Article  Google Scholar 

  2. Paterson I K, Hayle A, Ochan G, et al., Optimizing antibiotic usage to treat becterial infections, Journal of Scientific Reports, 2016, 6): 7853–37862.

    Google Scholar 

  3. Badzioch M, Bajger P, and Forys U, Competition between populations: Preventing domination of resistant population with optimal control, Applied Mathematical Modelling, 2023, 114): 671–693.

    Article  MathSciNet  Google Scholar 

  4. Blanquart F, Evolutionary epidemiology models to predict the dynamics of antibiotic resistance, Journal of Evolutionary Applications, 2010, 12): 365–383.

    Article  Google Scholar 

  5. Yan X, Zhap Z, Hu Y, et al., Dynamic analysis of a bacterial resistance model with impulsive state feedback control, Mathematical Biosciences and Engineering, 2024, 29): 20422–20436.

    MathSciNet  Google Scholar 

  6. Handels A, Margolis E, and Levin B R, Exploring the role of immune response in preventing antibiotic resistance, Journal of Theoretical Biology, 2009, 256): 655–662.

    Article  MathSciNet  Google Scholar 

  7. De Leenheer P and Cogan N G, Failure of antibiotic treatment in microbial populations, Mathematical Biology, 2009, 59): 563–579.

    Article  MathSciNet  Google Scholar 

  8. Zhao Y, Jia J, Dong Y, et al., Optimal dosing strategy and sensitivity analysis of a within-host drug resistance model with continuous and impulsive drug treatment, Journal of Applied Mathematics and Computing, 2023, 69): 2277–2293.

    Article  MathSciNet  Google Scholar 

  9. Mena H, Pfurtscheller L M, and Romero-Leiton J R, Random perturbations in a mathematical model of bacterial resistance: Analysis and optimal control, Mathematical Biosciences and Engineering, 2020, 17(5): 4477–4499.

    Article  MathSciNet  Google Scholar 

  10. Massad E, Nasciemento-Burratini M, and Bezerra-Coutinho F A, An optimization model for antibiotic use, Applied Mathematics and Computation, 2008, 201): 161–167.

    Article  MathSciNet  Google Scholar 

  11. Ankomah P and Levin B R, Exploring the collaboration between antibiotics and the immune response in the treatment of accute self-limiting infections, Proceedings of the National Academy of Sciences (PNAS), 2014, 111(23): 9331–8338.

    Article  Google Scholar 

  12. Li G, Leung C Y, Wardi Y, et al., Optimizing the timing and composition of therapeutic phage coctails: A control theoretic approach, Bulletin on Mathematical Biology, 2020, 82(75): 1–29.

    Google Scholar 

  13. Katriel G, Optimizing antimicrobial treatment schedules: Some fundamental analysis results, Bulletin of Mathematical Biology, 2023, 86(1): 1–37.

    Article  MathSciNet  Google Scholar 

  14. Pena-Miller R, Lahnemann D, Schulenburg H, et al., Selecting against antibiotic resistant pathogens: Optimal treatments in the presence of commensal bacteria, Bulletin of Mathematical Biology, 2012, 14): 908–934.

    Article  MathSciNet  Google Scholar 

  15. Zilanova E M and Bratus A S, Optimal strategies in antibiotic treatment of microbial populations, Journal of Applicable Analysis, 2016, 95(7): 1534–1547.

    Article  MathSciNet  Google Scholar 

  16. Hunang O, Hua X, and Ruan R, Optimal control of environmental cleaning and antibiotic prescription in an epidimiological model of methicillin-resistant Staphylococcus aureus infections in hospitals, Mathematical Biosciences, 2019, 311): 13–30.

    Article  MathSciNet  Google Scholar 

  17. Bowong S, Optimal control of the transmission dynamics of tuberculosis, Nonlinear Dynamics, 2010, 61): 729–748.

    Article  MathSciNet  Google Scholar 

  18. Choi S and Jung E, Optimal tuberculosis prevention and control strategy from a mathematical model based on real data, Bulletin of Mathematical Biology, 2014, 76): 1566–1589.

    Article  MathSciNet  Google Scholar 

  19. Jung E, Lenhart S, and Jung E, Optimal control of treatment in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems: Series B, 2002, 2(4): 473–482.

    Article  MathSciNet  Google Scholar 

  20. Bonnard B and Rouot J, Optimal control of the controlled Lotka-Volterra equations with applications — The permanent case, SIAM Journal of Applied Dynamical Systems, 2023, 22(4): 2671–2691.

    Article  MathSciNet  Google Scholar 

  21. Ibarguen-Mongragan E, Esteva L, and Ceron-Gomez M, An optimal control problem applied to plasmid-mediated antibiotic resistance, Journal of Applied Mathematics and Computing, 2022, 68): 1635–1667.

    Article  MathSciNet  Google Scholar 

  22. Gao D P and Huang N J, Optimal control analysis of a tuberculosis model, Applied Mathematical Modelling, 2001, 58): 57–64.

    MathSciNet  Google Scholar 

  23. Das S, Das P, and Das P, Dynamics and control of multi-drug resistant bacterial infection in hospitals with multiple delays, Communications in Nonlinear Sciences and Numerical Simulation, 2020, 89): 105279.

    Article  Google Scholar 

  24. Leszczynski M, Ledzewicz U, and Schattler H, Optimal control for a mathematical model for chemotherapy with pharmacometrics, Mathematical Modelling of Natural Phenomena, 2020, 15): 1–23.

    Article  MathSciNet  Google Scholar 

  25. Issam S, Bouchaib K, Lobzai A, et al., Mathematical modeling and optimal control strategy for a discrete-time cholear model, Communications in Mathematical Biological Neurosciences, 2023, 135): 1–16.

    Google Scholar 

  26. Gwerryina R I, Kura M Y, and Ashezuo J T, Optimal control model for the infectiology of staphylococcus aurus with dual transmission pathways, Results in Control and Optimiztion, 2024, 14): 100364.

    Article  Google Scholar 

  27. Lunz C and Bonnans J F, Modelling and optimal control of a two-species bioproducing microbial consortium, SIAM Journal of Applied Mathematics, 2023, 83(1): 144–171.

    Article  Google Scholar 

  28. Dey A, Bokka V, and Sen S, Dependence of bacterial growth rate on dynamic temperature changes, IET Systems Biology, 2020, 14(2): 68–74.

    Article  Google Scholar 

  29. Bonnard B, Rouat J, and Silva C J, Geometric optimal control of the generalized Lotka-Volterra model of the intestinal microbioma, Optimal Control Applications and Methods, 2024, 45): 544–574.

    Article  MathSciNet  Google Scholar 

  30. Achanya S, Mondal B, Upadhyaya R K, et al., Exploring noise-induced dynamics and optimal control strategy of iSIR cholera transmission model, Nonlinear Dynamics, 2024, 112): 3951–3975.

    Article  Google Scholar 

  31. Basseville M and Nikiforov I, Detection of Abrupt Changes: Theory and Applications, Prentice-Hall, Upper Sudbury, New Jersey, 1993.

    Google Scholar 

  32. Rigatos G, Nonlinear Control and Filtering Using Differential Flatnesss Theory Approaches: Applications to Electromechanical Systems, Springer Cham, Switzerland, 2015.

    Book  Google Scholar 

  33. Rigatos G, Advanced Models of Neural Networks: Nonlinear Dynamics and Stochasticity in Biological Neurons, Springer, Berlin, 2013.

    Google Scholar 

  34. Rigatos G, Abbaszadeh M, and Siano P, Control and Estimation of Dynamical Nonlinear and Partial Differential Equation Systems: Theory and Applications, IET Publications, New York, 2021.

    Google Scholar 

  35. Rigatos G and Karapanou E, Advances in Applied Nonlinear Optimal Control, Cambridge Scholars Publishing, Cambridge, 2020.

    Google Scholar 

  36. Rigatos G and Tzafesta S, Extended Kalman filtering for fuzzy modelling and multi-sensor fusion, Mathematical and Computer Modelling of Dynamical Systems, 2007, 13): 251–266.

    Article  MathSciNet  Google Scholar 

  37. Rigatos G and Zhang Q, Fuzzy model validation using the local statistical approach, Fuzzy Sets and Systems, 2009, 60(7): 882–904.

    Article  MathSciNet  Google Scholar 

  38. Rigatos G, Busawon K, and Abbaszadeh M, Nonlinear optimal control of the acute inflammatory response, Biomedical Signal Processing and Control, 2020, 55): 101631.

    Article  Google Scholar 

  39. Rigatos G, Nonlinear optimal control for the multi-variable tumor-growth dynamics, Journal of Computer Methods in Biomechanics and Biomedical Engineering, 2023, 1–29, DOI: https://doi.org/10.1080/10255842.2023.2297660.

  40. Toussaint G, Basar T, and Bullo F, H∞ optimal tracking control techniques for nonlinear underactuated systems, Proc. IEEE CDC 2000, 39th IEEE Conference on Decision and Control, Sydney Australia, 2000.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Unit of Industrial Automation, Industrial Systems Institute, 26504, Rion Patras, Greece

    Gerasimos Rigatos

  2. Dept. of ECS Engineering, Rensselaer Polytechnic Institute, Troy, NY, 12065, USA

    Masoud Abbaszadeh

  3. Dept. of Management and Innovation Systems, University of Salerno, 84084, Fisciano, Italy

    Pierluigi Siano

  4. Dept. of Electrical Eng., King Saud University, Riyadh, 11421, Saudi Arabia

    Mohammed Al-Numay

  5. Laboratoire d’ Automatique, ENIT Tunisie, 1002, Tunis, Tunisia

    Farouk Zouari

Authors
  1. Gerasimos Rigatos
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  2. Masoud Abbaszadeh
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  3. Pierluigi Siano
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  4. Mohammed Al-Numay
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  5. Farouk Zouari
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Corresponding author

Correspondence to Gerasimos Rigatos.

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The authors declare no conflict of interest.

Additional information

This research was supported by the Unit of Industrial Automation, Industrial Systems Institute under Grant No. Ref. 301022, Greece and RSP2024R150 of King Saud University, Riyadh, Saudi Arabia.

This paper was recommended for publication by Editor SUN Jian.

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Rigatos, G., Abbaszadeh, M., Siano, P. et al. A Nonlinear Optimal Control Approach for Bacterial Infections Under Antibiotics Resistance. J Syst Sci Complex 37, 2293–2317 (2024). https://doi.org/10.1007/s11424-024-3566-5

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  • DOI: https://doi.org/10.1007/s11424-024-3566-5

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