Investigation of chaos control in a fractional-order model of lysogenic bacteria under phage therapy

Lal Khan; Kokab Khan; Hsien-Tsung Chang; Lal Khan; Kokab Khan; Hsien-Tsung Chang

doi:10.3934/math.20261233

AIMS Mathematics

2026, Volume 11, Issue 9: 31214-31249. doi: 10.3934/math.20261233

Previous Article Next Article

Research article

Investigation of chaos control in a fractional-order model of lysogenic bacteria under phage therapy

1.
Department of AI and Software, Gachon University, Seongnam-si 13120, Republic of Korea
2.
Department of Mathematics, Ghazi University, D G Khan 32200, Pakistan
3.
Department of Artificial Intelligence, Chang Gung University, Taoyuan 333, Taiwan, R.O.C
4.
Department of Computer Science and Information Engineering, Chang Gung University, Taoyuan 333, Taiwan, R.O.C
5.
Center for Artificial Intelligence in Medicine, Chang Gung Memorial Hospital at Linkou, Taoyuan 333, Taiwan, R.O.C

Received: 03 June 2026 Revised: 11 August 2026 Accepted: 11 September 2026 Published: 23 September 2026
MSC : 92Bxx, 92Fxx

This paper investigates a fractional-order model of lysogenic bacteria and its chaos control, with a focus on phage therapy. A mathematical model is formulated to describe bacterial dynamics, and chaos control techniques are applied to stabilize the system. The Caputo fractional operator is used to construct a fractional-order framework that incorporates memory effects and enables reliable numerical approximation. The proposed model is analyzed with respect to positivity, boundedness, and the existence and uniqueness of solutions. The Lipschitz condition and fixed-point theory are utilized to establish the uniqueness of solutions. The local and global stability of the disease-free and endemic equilibrium points are examined through Lyapunov stability theory. Chaos control conditions are derived to investigate the influence of the model's parameters on the emergence and suppression of chaotic behavior. Numerical simulations based on the Caputo fractional operator illustrate the dynamics of bacterial populations under phage therapy. Furthermore, the effects of the model's parameters and fractional-order values are analyzed using a two-step Lagrange polynomial numerical scheme, demonstrating the effectiveness of the proposed approach in controlling infected bacterial populations.
- lysogenic bacteria,
- phage therapy,
- stochastic model,
- chaos control,
- Caputo fractional derivative
Citation: Lal Khan, Kokab Khan, Hsien-Tsung Chang. Investigation of chaos control in a fractional-order model of lysogenic bacteria under phage therapy[J]. AIMS Mathematics, 2026, 11(9): 31214-31249. doi: 10.3934/math.20261233

Related Papers:

Abstract

This paper investigates a fractional-order model of lysogenic bacteria and its chaos control, with a focus on phage therapy. A mathematical model is formulated to describe bacterial dynamics, and chaos control techniques are applied to stabilize the system. The Caputo fractional operator is used to construct a fractional-order framework that incorporates memory effects and enables reliable numerical approximation. The proposed model is analyzed with respect to positivity, boundedness, and the existence and uniqueness of solutions. The Lipschitz condition and fixed-point theory are utilized to establish the uniqueness of solutions. The local and global stability of the disease-free and endemic equilibrium points are examined through Lyapunov stability theory. Chaos control conditions are derived to investigate the influence of the model's parameters on the emergence and suppression of chaotic behavior. Numerical simulations based on the Caputo fractional operator illustrate the dynamics of bacterial populations under phage therapy. Furthermore, the effects of the model's parameters and fractional-order values are analyzed using a two-step Lagrange polynomial numerical scheme, demonstrating the effectiveness of the proposed approach in controlling infected bacterial populations.

References

[1]	G. M. Bodner, D. E. Gardner, M. W. Briggs, Models and modeling, in: Chemists' Guide to Effective Teaching (eds. N. Pienta, M. Cooper, T. Greenbowe), Prentice-Hall, Upper Saddle River, NJ, 2005, 67–76.
[2]	J. Cremer, A. Melbinger, E. Frey, Evolutionary and population dynamics: A coupled approach, Phys. Rev. E, 84 (2011), 051921. https://doi.org/10.1103/PhysRevE.84.051921 doi: 10.1103/PhysRevE.84.051921
[3]	A. A. Alonso, I. Molina, C. Theodoropoulos, Modeling bacterial population growth from stochastic single-cell dynamics, Appl. Environ. Microbiol., 80 (2014), 5241–5253. https://doi.org/10.1128/AEM.01423-14 doi: 10.1128/AEM.01423-14
[4]	C. Fritsch, J. Harmand, F. Campillo, A modeling approach of the chemostat, Ecol. Model., 299 (2015), 1–13. https://doi.org/10.1016/j.ecolmodel.2014.11.021 doi: 10.1016/j.ecolmodel.2014.11.021
[5]	J. Adam, Putting the X in biology: A review of The Mathematics of Life, Notices Amer. Math. Soc., 58 (2011), 1572–1578.
[6]	M. Carletti, Numerical solution of stochastic differential problems in the biosciences, J. Comput. Appl. Math., 185 (2006), 422–440. https://doi.org/10.1016/j.cam.2005.03.020 doi: 10.1016/j.cam.2005.03.020
[7]	A. Agila, D. Baleanu, R. Eid, B. Irfanoglu, Applications of the extended fractional Euler-Lagrange equations model to freely oscillating dynamical systems, Rom. J. Phys., 61 (2016), 350–359.
[8]	J. Singh, D. Kumar, M. A. Qurashi, D. Baleanu, A new fractional model for giving up smoking dynamics, Adv. Differ. Equ., 2017 (2017), 88. https://doi.org/10.1186/s13662-017-1139-9 doi: 10.1186/s13662-017-1139-9
[9]	I. Area, H. Batarfi, J. Losada, J. J. Nieto, W. Shammakh, A. Torres, On a fractional order Ebola epidemic model, Adv. Differ. Equ., 2015 (2015), 278. https://doi.org/10.1186/s13662-015-0613-5 doi: 10.1186/s13662-015-0613-5
[10]	L. Meng, X. Zhang, L. Lin, S. Shi, H. Wang, Collective dynamics of fractional oscillators with fluctuating frequency in Erdős–Rényi random networks, Commun. Nonlinear Sci. Numer. Simul., 159 (2026), 109919. https://doi.org/10.1016/j.cnsns.2026.109919 doi: 10.1016/j.cnsns.2026.109919
[11]	L. Lin, H. Wang, Time delay effects on collective resonant behaviors in two coupled fractional oscillators with mass fluctuations, Sci. Rep., 15 (2025), 2335. https://doi.org/10.1038/s41598-025-86080-1 doi: 10.1038/s41598-025-86080-1
[12]	L. Meng, R. Zhang, L. Yu, H. Wang, Tempered anomalous dynamics of globally coupled harmonic oscillators in the fluctuating potential field: stability, synchronism, and collective behaviors, Eur. Phys. J. Plus, 139 (2024), 63. https://doi.org/10.1140/epjp/s13360-024-04865-1 doi: 10.1140/epjp/s13360-024-04865-1
[13]	A. Ahmad, M. Farman, M. O. Ahmad, N. Raza, M. Abdullah, Dynamical behavior of SIR epidemic model with non-integer time fractional derivatives: A mathematical analysis, Int. J. Adv. Appl. Sci., 5 (2018), 123–129. https://doi.org/10.21833/ijaas.2018.01.016 doi: 10.21833/ijaas.2018.01.016
[14]	M. Farman, M. U. Saleem, M. O. Ahmed, A. Ahmad, Stability analysis and control of the glucose insulin glucagon system in humans, Chinese J. Phys., 56 (2018), 1362–1369. https://doi.org/10.1016/j.cjph.2018.03.037 doi: 10.1016/j.cjph.2018.03.037
[15]	Y. M. Chu, M. F. Khan, S. Ullah, S. A. A. Shah, M. Farooq, M. bin Mamat, Mathematical assessment of a fractional-order vector–host disease model with the Caputo–Fabrizio derivative, Math. Methods Appl. Sci., 46 (2023), 232–247. https://doi.org/10.1002/mma.8507 doi: 10.1002/mma.8507
[16]	M. Vellappandi, P. Kumar, V. Govindaraj, W. Albalawi, An optimal control problem for mosaic disease via Caputo fractional derivative, Alex. Eng. J., 61 (2022), 8027–8037. https://doi.org/10.1016/j.aej.2022.01.055 doi: 10.1016/j.aej.2022.01.055
[17]	H. L. Li, L. Zhang, C. Hu, Y. L. Jiang, Z. Teng, Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, J. Appl. Math. Comput., 54 (2017), 435–449. https://doi.org/10.1007/s12190-016-1017-8 doi: 10.1007/s12190-016-1017-8
[18]	H. L. Li, J. Cao, C. Hu, H. Jiang, F. E. Alsaadi, Synchronization analysis of discrete-time fractional-order quaternion-valued uncertain neural networks, IEEE T. Neural Netw. Lear., 35 (2024), 14178–14189. https://doi.org/10.1109/TNNLS.2023.3274959 doi: 10.1109/TNNLS.2023.3274959
[19]	M. Patil, S. Bhalekar, A new fractional order chaotic dynamical system and its synchronization using optimal control, arXiv: 2007.03168, 2020. Available from: https://arXiv.org/abs/2007.03168
[20]	A. Bayat, E. Khazaei, S. Mahdavi, Economic-Based Synchronization and Control of New Fractional-Order Chaotic System Based on Lyapunov Theorem, MPRA Paper No. 95394, University Library of Munich, Germany, 2019. Available from: https://mpra.ub.uni-muenchen.de/95394/
[21]	C. Li, D. Qian, Y. Q. Chen, On Riemann–Liouville and Caputo derivatives, Discrete Dyn. Nat. Soc., 2011 (2011), 562494. https://doi.org/10.1155/2011/562494 doi: 10.1155/2011/562494

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)