Theoretical and Numerical Analysis of the <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">SIR</mi></semantics></math></inline-formula> Model and Its Symmetric Cases with Power Caputo Fractional Derivative
This paper introduces a novel fractional Susceptible-Infected-Recovered (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">SIR</mi></semantics></math><...
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| Format: | Article |
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2025-04-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/4/251 |
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| author | Mohamed S. Algolam Mohammed Almalahi Khaled Aldwoah Amira S. Awaad Muntasir Suhail Fahdah Ayed Alshammari Bakri Younis |
| author_facet | Mohamed S. Algolam Mohammed Almalahi Khaled Aldwoah Amira S. Awaad Muntasir Suhail Fahdah Ayed Alshammari Bakri Younis |
| author_sort | Mohamed S. Algolam |
| collection | DOAJ |
| description | This paper introduces a novel fractional Susceptible-Infected-Recovered (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">SIR</mi></semantics></math></inline-formula>) model that incorporates a power Caputo fractional derivative (PCFD) and a density-dependent recovery rate. This enhances the model’s ability to capture memory effects and represent realistic healthcare system dynamics in epidemic modeling. The model’s utility and flexibility are demonstrated through an application using parameters representative of the COVID-19 pandemic. Unlike existing fractional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">SIR</mi></semantics></math></inline-formula> models often limited in representing diverse memory effects adequately, the proposed PCFD framework encompasses and extends well-known cases, such as those using Caputo–Fabrizio and Atangana–Baleanu derivatives. We prove that our model yields bounded and positive solutions, ensuring biological plausibility. A rigorous analysis is conducted to determine the model’s local stability, including the derivation of the basic reproduction number (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">R</mi><mn>0</mn></msub></semantics></math></inline-formula>) and sensitivity analysis quantifying the impact of parameters on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">R</mi><mn>0</mn></msub></semantics></math></inline-formula>. The uniqueness and existence of solutions are guaranteed via a recursive sequence approach and the Banach fixed-point theorem. Numerical simulations, facilitated by a novel numerical scheme and applied to the COVID-19 parameter set, demonstrate that varying the fractional order significantly alters predicted epidemic peak timing and severity. Comparisons across different fractional approaches highlight the crucial role of memory effects and healthcare capacity in shaping epidemic trajectories. These findings underscore the potential of the generalized PCFD approach to provide more nuanced and potentially accurate predictions for disease outbreaks like COVID-19, thereby informing more effective public health interventions. |
| format | Article |
| id | doaj-art-6423045e343e43e1a5bffa08cd774e73 |
| institution | OA Journals |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-6423045e343e43e1a5bffa08cd774e732025-08-20T02:18:11ZengMDPI AGFractal and Fractional2504-31102025-04-019425110.3390/fractalfract9040251Theoretical and Numerical Analysis of the <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">SIR</mi></semantics></math></inline-formula> Model and Its Symmetric Cases with Power Caputo Fractional DerivativeMohamed S. Algolam0Mohammed Almalahi1Khaled Aldwoah2Amira S. Awaad3Muntasir Suhail4Fahdah Ayed Alshammari5Bakri Younis6Department of Mathematics, College of Science, University of Ha’il, Ha’il 55473, Saudi ArabiaDepartment of Mathematics, College of Computer and Information Technology, Al-Razi University, Sana’a 12544, YemenDepartment of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi ArabiaDepartment of Mathematics, College of Science and Humanities, Al-Kharj Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi ArabiaDepartment of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi ArabiaDepartment of Biology, College of Science, Northern Border University, Arar 73241, Saudi ArabiaDepartment of Mathematics, Faculty of Arts and Science, Elmagarda, King Khalid University, Abha 61421, Saudi ArabiaThis paper introduces a novel fractional Susceptible-Infected-Recovered (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">SIR</mi></semantics></math></inline-formula>) model that incorporates a power Caputo fractional derivative (PCFD) and a density-dependent recovery rate. This enhances the model’s ability to capture memory effects and represent realistic healthcare system dynamics in epidemic modeling. The model’s utility and flexibility are demonstrated through an application using parameters representative of the COVID-19 pandemic. Unlike existing fractional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">SIR</mi></semantics></math></inline-formula> models often limited in representing diverse memory effects adequately, the proposed PCFD framework encompasses and extends well-known cases, such as those using Caputo–Fabrizio and Atangana–Baleanu derivatives. We prove that our model yields bounded and positive solutions, ensuring biological plausibility. A rigorous analysis is conducted to determine the model’s local stability, including the derivation of the basic reproduction number (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">R</mi><mn>0</mn></msub></semantics></math></inline-formula>) and sensitivity analysis quantifying the impact of parameters on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">R</mi><mn>0</mn></msub></semantics></math></inline-formula>. The uniqueness and existence of solutions are guaranteed via a recursive sequence approach and the Banach fixed-point theorem. Numerical simulations, facilitated by a novel numerical scheme and applied to the COVID-19 parameter set, demonstrate that varying the fractional order significantly alters predicted epidemic peak timing and severity. Comparisons across different fractional approaches highlight the crucial role of memory effects and healthcare capacity in shaping epidemic trajectories. These findings underscore the potential of the generalized PCFD approach to provide more nuanced and potentially accurate predictions for disease outbreaks like COVID-19, thereby informing more effective public health interventions.https://www.mdpi.com/2504-3110/9/4/251??ℝ modelgeneralized power fractional derivativestabilitysimulationsnumerical analysis |
| spellingShingle | Mohamed S. Algolam Mohammed Almalahi Khaled Aldwoah Amira S. Awaad Muntasir Suhail Fahdah Ayed Alshammari Bakri Younis Theoretical and Numerical Analysis of the <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">SIR</mi></semantics></math></inline-formula> Model and Its Symmetric Cases with Power Caputo Fractional Derivative Fractal and Fractional ??ℝ model generalized power fractional derivative stability simulations numerical analysis |
| title | Theoretical and Numerical Analysis of the <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">SIR</mi></semantics></math></inline-formula> Model and Its Symmetric Cases with Power Caputo Fractional Derivative |
| title_full | Theoretical and Numerical Analysis of the <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">SIR</mi></semantics></math></inline-formula> Model and Its Symmetric Cases with Power Caputo Fractional Derivative |
| title_fullStr | Theoretical and Numerical Analysis of the <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">SIR</mi></semantics></math></inline-formula> Model and Its Symmetric Cases with Power Caputo Fractional Derivative |
| title_full_unstemmed | Theoretical and Numerical Analysis of the <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">SIR</mi></semantics></math></inline-formula> Model and Its Symmetric Cases with Power Caputo Fractional Derivative |
| title_short | Theoretical and Numerical Analysis of the <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">SIR</mi></semantics></math></inline-formula> Model and Its Symmetric Cases with Power Caputo Fractional Derivative |
| title_sort | theoretical and numerical analysis of the inline formula math display inline semantics mi mathvariant double struck sir mi semantics math inline formula model and its symmetric cases with power caputo fractional derivative |
| topic | ??ℝ model generalized power fractional derivative stability simulations numerical analysis |
| url | https://www.mdpi.com/2504-3110/9/4/251 |
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