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Mittag-Leffler Stability Analysis for a Fractional Order Model of Co-Contaminated Soil and Water Pollution |
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PP: 549-567 |
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doi:10.18576/pfda/120305
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Author(s) |
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Priya P.,
M. Qasymeh,
Roselyn Besi P.,
Revathy M.,
Ali Akgu ̈l,
Mahmoud Abdel-Aty,
Tan Yu Jie,
Mohamed Hafez,
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Abstract |
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| Environmental pollution is a serious and long-lasting problem, and understanding how it spreads and how it can be controlled requires robust mathematical models. In this work, we study a five-compartment fractional-order model that describes how pollution
moves between different environmental components, namely effluent, soil, water, and farmland. To capture the memory and hereditary
effects that naturally occur in environmental processes, we use the Atangana–Baleanu–Caputo (ABC) fractional derivative, which
provides a more realistic description than classical integer-order models. We first analyze the qualitative behavior of the model. Using
linearization, Lyapunov function, and tools from fractional calculus, we establish the local and global asymptotic stability of two
important equilibrium points: the pollution-free equilibrium (E0) and the endemic (polluted) equilibrium (E∗). In particular, we apply
the concept of Mittag-Leffler stability, which is well-suited to fractional-order systems. A central result of the analysis is the derivation
of the basic reproduction number R0 = LaN , which acts as a threshold parameter for pollution dynamics. If R0 < 1, the pollution- ΦC(Φ2+γ1)
free equilibrium E0 is globally asymptotically stable, meaning that pollution will eventually disappear from the environment. On the other hand, if R0 > 1, the pollution-free state becomes unstable, and a unique endemic equilibrium E∗ exists and is stable, indicating that pollution will persist in the long term. To support the theoretical results, we develop an adaptive predictor–corrector numerical scheme designed specifically for the ABC fractional-order model. This method is both accurate and computationally efficient, making it suitable for long-time simulations. Numerical experiments, particularly for the case R0 = 4.55, clearly show that the solution converges to the endemic equilibrium, which confirms the theoretical stability results and demonstrates the reliability of the proposed numerical approach. Overall, this study provides a clear mathematical framework for understanding pollution spread and control in complex environmental systems. The analytical results, together with the numerical simulations, offer valuable insights that can help in designing effective long-term pollution remediation and management strategies.
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