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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2025-01-01
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| Series: | Fractal and Fractional |
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| author | Yasir A. Madani Zeeshan Ali Mohammed Rabih Amer Alsulami Nidal H. E. Eljaneid Khaled Aldwoah Blgys Muflh |
| author_facet | Yasir A. Madani Zeeshan Ali Mohammed Rabih Amer Alsulami Nidal H. E. Eljaneid Khaled Aldwoah Blgys Muflh |
| author_sort | Yasir A. Madani |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-99a794ee60534507be7d841662d4055c |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-99a794ee60534507be7d841662d4055c2025-01-24T13:33:31ZengMDPI AGFractal and Fractional2504-31102025-01-01915510.3390/fractalfract9010055Discrete Fractional-Order Modeling of Recurrent Childhood Diseases Using the Caputo Difference OperatorYasir A. Madani0Zeeshan Ali1Mohammed Rabih2Amer Alsulami3Nidal H. E. Eljaneid4Khaled Aldwoah5Blgys Muflh6Department of Mathematics, College of Science, University of Ha’il, Ha’il 55473, Saudi ArabiaSchool of Science, Monash University, Selangor 47500, MalaysiaDepartment of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi ArabiaDepartment of Mathematics, Turabah University College, Taif University, Taif 21944, Saudi ArabiaDepartment of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Islamic University of Madinah, Medina 42351, Saudi ArabiaDepartment of Mathematics, College of Science and Humanities, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi ArabiaThis 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.https://www.mdpi.com/2504-3110/9/1/55childhood disease modelingCaputo fractional difference operatorexistence theorystability analysissensitivity analysisnumerical simulations |
| spellingShingle | Yasir A. Madani Zeeshan Ali Mohammed Rabih Amer Alsulami Nidal H. E. Eljaneid Khaled Aldwoah Blgys Muflh Discrete Fractional-Order Modeling of Recurrent Childhood Diseases Using the Caputo Difference Operator Fractal and Fractional childhood disease modeling Caputo fractional difference operator existence theory stability analysis sensitivity analysis numerical simulations |
| title | Discrete Fractional-Order Modeling of Recurrent Childhood Diseases Using the Caputo Difference Operator |
| title_full | Discrete Fractional-Order Modeling of Recurrent Childhood Diseases Using the Caputo Difference Operator |
| title_fullStr | Discrete Fractional-Order Modeling of Recurrent Childhood Diseases Using the Caputo Difference Operator |
| title_full_unstemmed | Discrete Fractional-Order Modeling of Recurrent Childhood Diseases Using the Caputo Difference Operator |
| title_short | Discrete Fractional-Order Modeling of Recurrent Childhood Diseases Using the Caputo Difference Operator |
| title_sort | discrete fractional order modeling of recurrent childhood diseases using the caputo difference operator |
| topic | childhood disease modeling Caputo fractional difference operator existence theory stability analysis sensitivity analysis numerical simulations |
| url | https://www.mdpi.com/2504-3110/9/1/55 |
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