Stabilization of fractional hybrid systems with applications in biomedical dynamics

Volume 41, Issue 3, pp 406--420 https://dx.doi.org/10.22436/jmcs.041.03.07

Publication Date: November 14, 2025 Submission Date: June 04, 2025 Revision Date: August 07, 2025 Accteptance Date: August 22, 2025

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Authors

I. Alraddadi - Department of Mathematics, Faculty of Science, Islamic University of Madinah, Saudi Arabia. M. P. Ineh - Department of Mathematics and Computer Science, Ritman University, Ikot Ekpene, Akwa Ibom State, Nigeria. D. K. Igobi - Department of Mathematics, University of Uyo, Uyo, Nigeria. U. Ishtiaq - Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore, Pakistan. I.-L Popa - Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, Alba Iulia 510009, Romania. - Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, Brasov 500091, Romania.

Abstract

This work introduces a novel analytical framework for analyzing the stability of Caputo fractional dynamic equations on time scales (CFDE\(\mathbb{T}\)) using a two-measure approach combined with comparison principle. By applying paired measures \((m_0, m)\) and vector Lyapunov functions, we derive sufficient conditions for both \((m_0, m)\)-stability and asymptotic stability. The method simplifies analysis by relating the system to a well-understood comparison system, reducing the task to verifying quasimonotonicity and avoiding the need for explicit solutions. The framework’s effectiveness is demonstrated through two biological models, an immune response system and a 3D hypothalamic-pituitary-adrenal (HPA) axis, highlighting its ability to handle nonlinearities, hybrid time scales, and varying system dimensions. This study bridges theoretical stability analysis with practical biomedical applications, advancing the understanding of fractional-order hybrid dynamics.

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Keywords

• Immune response model
• neuro-endocrine regulation model
• two measure stabilization
• fractional calculus
• time scales
• comparison principle

MSC

•  34A08
•  34A38
•  34D20
•  34N05
•  92C50

References

• [1] J. O. Achuobi, E. P. Akpan, R. George, A. E. Ofem, STABILITY analysis of Caputo fractional time-dependent systems with delay using vector Lyapunov functions, AIMS Math., 9 (2024), 28079–28099
  • View Article
  • MathSciNet
  • Google Scholar


• [2] R. P. Agarwal, S. Hristova, D. O’Regan, Practical stability of Caputo fractional differential equations by Lyapunov functions, Differ. Equ. Appl., 8 (2016), 53–68
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] A. Ahmadkhanlu, M. Jahanshahi, On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, Bull. Iranian Math. Soc., 38 (2012), 241–252
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] J. E. Ante, M. P. Ineh, J. O. Achuobi, U. P. Akai, J. U. Atsu, N. A. O. Offiong, A novel Lyapunov asymptotic eventual stability approach for nonlinear impulsive Caputo fractional differential equations, Appl. Math., 4 (2024), 1600–1617
  • View Article
  • Google Scholar


• [5] J. E. Ante, O. O. Itam, J. U. Atsu, S. O. Essang, E. E. Abraham, M. P. Ineh, On the novel auxiliary Lyapunov function and uniform asymptotic practical stability of nonlinear impulsive Caputo fractional differential equations via new modelled generalized Dini derivative, Afr. J. Math. Stat. Stud., 7 (2024), 11–33
  • View Article
  • Google Scholar


• [6] M. Bohner, P. Allan, Dynamic equations on time scales: an introduction with applications, Birkhäuser/Springer Basel AG, Boston (2001)
  • View Article
  • Google Scholar
  • MATH


• [7] S. E. Ekoro, A. E. Ofem, F. A. Adie, J. Oboyi, G. I. Ogban, M. P. Ineh, On a faster iterative method for solving nonlinear fractional integro-differential equations with impulsive and integral conditions, Palest. J. Math., 12 (2023), 477–484
  • View Article
  • Google Scholar


• [8] S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18–56
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] D. K. Igobi, M. P. Ineh, Results on existence and uniqueness of solutions of dynamic equations on time scale via generalized ordinary differential, Int. J. Appl. Math., 37 (2024), 1–20
  • View Article
  • Google Scholar


• [10] D. K. Igobi, E. Ndiyo, M. P. Ineh, Variational stability results of dynamic equations on time-scales using generalized ordinary differential equations, World J. Appl. Sci. Technol., 15 (2024), 245–254
  • View Article
  • Google Scholar


• [11] D. Igobi, W. Udogworen, Results on Existence and Uniqueness of solutions of Fractional Differential Equations of Caputo- Fabrizio type in the sense of Riemann-Liouville, IAENG Int. J. Appl. Math., 54 (2024), 1163–1171
  • View Article
  • Google Scholar


• [12] M. P. Ineh, J. O. Achuobi, E. P. Akpan, J. E. Ante, CDq on the uniform stability of Caputo fractional differential equations using vector Lyapunov functions, J. Niger. Assoc. Math. Phys., 68 (2024), 51–64
  • View Article
  • Google Scholar


• [13] M. P. Ineh, E. P. Akpan, Lyapunov uniform asymptotic stability of Caputo fractional dynamic equations on time scale using a generalized derivative, Trans. Niger. Assoc. Math. Phys., 20 (2024), 117–132
  • View Article
  • Google Scholar


• [14] M. P. Ineh, E. P. Akpan, On Lyapunov stability of Caputo fractional dynamic equations on time scale using vector Lyapunov functions, Khayyam J. Math., 11 (2025), 116–143
  • View Article
  • Google Scholar


• [15] M. P. Ineh, E. P. Akpan, U. D. Akpan, A. Maharaj, O. K. Narain, On Total Stability Analysis of Caputo Fractional Dynamic Equations on Time Scale, Asia Pac. J. Math., 12 (2025), 1–14
  • View Article


• [16] M. P. Ineh, E. P. Akpan, H. A. Nabwey, A novel approach to Lyapunov stability of Caputo fractional dynamic equations on time scale using a new generalized derivative, AIMS Math., 9 (2024), 34406–34434
  • View Article
  • MathSciNet
  • Google Scholar


• [17] M. P. Ineh, U. Ishtiaq, J. E. Ante, M. Garayev, I.-L. Popa, A robust uniform practical stability approach for Caputo fractional hybrid systems, AIMS Math., 10 (2025), 7001–7021
  • View Article
  • MathSciNet
  • Google Scholar


• [18] M. P. Ineh, V. N. Nfor, M. I. Sampson, J. E. Ante, J. U. Atsu, O. O. Itam, A novel approach for vector Lyapunov functions and practical stability of Caputo fractional dynamic equations on time scale in terms of two measures, Khayyam J. Math., 11 (2025), 61–89
  • View Article
  • Google Scholar
  • MATH


• [19] M. E. Koksal, Stability analysis of fractional differential equations with unknown parameters, Nonlinear Anal. Model. Control, 24 (2019), 224–240
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] V. Kumar, M. Malik, Existence, stability and controllability results of fractional dynamic system on time scales with application to population dynamics, Int. J. Nonlinear Sci. Numer. Simul., 22 (2021), 741–766
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] N. Laledj, A. Salim, J. E. Lazreg, S. Abbas, B. Ahmad, M. Benchohra, On implicit fractional q-difference equations: analysis and stability, Math. Methods Appl. Sci., 45 (2022), 10775–10797
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] J. Oboyi, M. P. Ineh, A. Maharaj, J. O. Achuobi, O. K. Narain, Practical stability of Caputo fractional dynamic equations on time scale, Adv. Fixed Point Theory, 15 (2025), 19 pages
  • View Article
  • Google Scholar


• [23] R. E. Orim, M. P. Ineh, D. K. Igobi, A. Maharaj, O. K. Narain, A novel approach to Lyapunov uniform stability of Caputo fractional dynamic equations on time scale using a new generalized derivative, Asia Pac. J. Math., 12 (2025), 16 pages
  • View Article


• [24] R. E. Orim, A. B. Panle, M. P. Ineh, A. Maharaj, O. K. Narain, Strict uniform stability analysis in terms of two measures of Caputo fractional dynamic systems on time scale, Adv. Fixed Point Theory, 15 (2025), 1–17
  • View Article
  • Google Scholar


• [25] R. E. Orim, A. B. Panle, M. P. Ineh, A. Maharaj, O. K. Narain, Integral Stability of Impulsive Dynamic Systems on Time Scale, Asia Pac. J. Math., 12 (2025), 1–16
  • View Article
  • Google Scholar


• [26] M. O. Udo, M. P. Ineh, E. J. Inyang, P. Benneth, Solving nonlinear Volterra integral equations by an efficient method, Int. J. Stat. Appl. Math., 7 (2022), 136–141
  • View Article
  • Google Scholar


• [27] J. A. Ugboh, C. F. Igiri, M. P. Ineh, A. Maharaj, O. K. Narain, A novel approach to Lyapunov eventual stability of Caputo fractional dynamic equations on time scale, Asia Pac. J. Math., 12 (2025), 19 pages
  • View Article
  • Google Scholar