Discrete Fractional-Order Modeling of Recurrent Childhood Diseases Using the Caputo Difference Operator

Discrete Fractional-Order Modeling of Recurrent Childhood Diseases Using the Caputo Difference Operator

This paper presents a new SIRS model for recurrent childhood diseases under the Caputo fractional difference operator. The existence theory is established using Brouwer’s fixed-point theorem and the Banach contraction principle, providing a comprehensive mathematical foundation for the model. Ulam s...

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Bibliographic Details
Main Authors: Yasir A. Madani, Zeeshan Ali, Mohammed Rabih, Amer Alsulami, Nidal H. E. Eljaneid, Khaled Aldwoah, Blgys Muflh
Format: Article
Language:English
Published: MDPI AG 2025-01-01
Series:Fractal and Fractional
Subjects:
childhood disease modeling
Caputo fractional difference operator
existence theory
stability analysis
sensitivity analysis
numerical simulations
Online Access:https://www.mdpi.com/2504-3110/9/1/55
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Summary:This paper presents a new SIRS model for recurrent childhood diseases under the Caputo fractional difference operator. The existence theory is established using Brouwer’s fixed-point theorem and the Banach contraction principle, providing a comprehensive mathematical foundation for the model. Ulam stability is demonstrated using nonlinear functional analysis. Sensitivity analysis is conducted based on the variation of each parameter, and the basic reproduction number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi mathvariant="script">R</mi><mn>0</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula> is introduced to assess local stability at two equilibrium points. The stability analysis indicates that the disease-free equilibrium point is stable when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">R</mi><mn>0</mn></msub><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, while the endemic equilibrium point is stable when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">R</mi><mn>0</mn></msub><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> and otherwise unstable. Numerical simulations demonstrate the model’s effectiveness in capturing realistic scenarios, particularly the recurrent patterns observed in some childhood diseases.
ISSN:2504-3110

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