Dynamical behaviours and stability analysis of a generalized fractional model with a real case study
Introduction: Mathematical modelling is a rapidly expanding field that offers new and interesting opportunities for both mathematicians and biologists. Concerning COVID-19, this powerful tool may help humans to prevent the spread of this disease, which has affected the livelihood of all people badly...
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| Format: | Article |
| Language: | English |
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Elsevier
2023-06-01
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| Series: | Journal of Advanced Research |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2090123222001916 |
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| author | D. Baleanu S. Arshad A. Jajarmi W. Shokat F. Akhavan Ghassabzade M. Wali |
| author_facet | D. Baleanu S. Arshad A. Jajarmi W. Shokat F. Akhavan Ghassabzade M. Wali |
| author_sort | D. Baleanu |
| collection | DOAJ |
| description | Introduction: Mathematical modelling is a rapidly expanding field that offers new and interesting opportunities for both mathematicians and biologists. Concerning COVID-19, this powerful tool may help humans to prevent the spread of this disease, which has affected the livelihood of all people badly. Objectives: The main objective of this research is to explore an efficient mathematical model for the investigation of COVID-19 dynamics in a generalized fractional framework. Methods: The new model in this paper is formulated in the Caputo sense, employs a nonlinear time-varying transmission rate, and consists of ten population classes including susceptible, infected, diagnosed, ailing, recognized, infected real, threatened, diagnosed recovered, healed, and extinct people. The existence of a unique solution is explored for the new model, and the associated dynamical behaviours are discussed in terms of equilibrium points, invariant region, local and global stability, and basic reproduction number. To implement the proposed model numerically, an efficient approximation scheme is employed by the combination of Laplace transform and a successive substitution approach; besides, the corresponding convergence analysis is also investigated. Results: Numerical simulations are reported for various fractional orders, and simulation results are compared with a real case of COVID-19 pandemic in Italy. By using these comparisons between the simulated and measured data, we find the best value of the fractional order with minimum absolute and relative errors. Also, the impact of different parameters on the spread of viral infection is analyzed and studied. Conclusion: According to the comparative results with real data, we justify the use of fractional concepts in the mathematical modelling, for the new non-integer formalism simulates the reality more precisely than the classical framework. |
| first_indexed | 2024-03-13T08:07:57Z |
| format | Article |
| id | doaj.art-fb773f1f49034673b143eb1a6117c06d |
| institution | Directory Open Access Journal |
| issn | 2090-1232 |
| language | English |
| last_indexed | 2024-03-13T08:07:57Z |
| publishDate | 2023-06-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Journal of Advanced Research |
| spelling | doaj.art-fb773f1f49034673b143eb1a6117c06d2023-06-01T04:35:30ZengElsevierJournal of Advanced Research2090-12322023-06-0148157173Dynamical behaviours and stability analysis of a generalized fractional model with a real case studyD. Baleanu0S. Arshad1A. Jajarmi2W. Shokat3F. Akhavan Ghassabzade4M. Wali5Department of Mathematics, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey; Institute of Space Sciences, P.O.Box, MG-23, R 76900, Magurele-Bucharest, Romania; Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, TaiwanCOMSATS University Islamabad, Lahore Campus, Lahore 54000, PakistanDepartment of Electrical Engineering, University of Bojnord, P.O. Box, 94531-1339, Bojnord, Iran; Corresponding author.COMSATS University Islamabad, Lahore Campus, Lahore 54000, PakistanDepartment of Mathematics, Faculty of Sciences, University of Gonabad, Gonabad, IranCOMSATS University Islamabad, Lahore Campus, Lahore 54000, PakistanIntroduction: Mathematical modelling is a rapidly expanding field that offers new and interesting opportunities for both mathematicians and biologists. Concerning COVID-19, this powerful tool may help humans to prevent the spread of this disease, which has affected the livelihood of all people badly. Objectives: The main objective of this research is to explore an efficient mathematical model for the investigation of COVID-19 dynamics in a generalized fractional framework. Methods: The new model in this paper is formulated in the Caputo sense, employs a nonlinear time-varying transmission rate, and consists of ten population classes including susceptible, infected, diagnosed, ailing, recognized, infected real, threatened, diagnosed recovered, healed, and extinct people. The existence of a unique solution is explored for the new model, and the associated dynamical behaviours are discussed in terms of equilibrium points, invariant region, local and global stability, and basic reproduction number. To implement the proposed model numerically, an efficient approximation scheme is employed by the combination of Laplace transform and a successive substitution approach; besides, the corresponding convergence analysis is also investigated. Results: Numerical simulations are reported for various fractional orders, and simulation results are compared with a real case of COVID-19 pandemic in Italy. By using these comparisons between the simulated and measured data, we find the best value of the fractional order with minimum absolute and relative errors. Also, the impact of different parameters on the spread of viral infection is analyzed and studied. Conclusion: According to the comparative results with real data, we justify the use of fractional concepts in the mathematical modelling, for the new non-integer formalism simulates the reality more precisely than the classical framework.http://www.sciencedirect.com/science/article/pii/S2090123222001916Fractional modelCOVID-19 pandemicExistence and uniqueness resultsStability analysisNumerical method |
| spellingShingle | D. Baleanu S. Arshad A. Jajarmi W. Shokat F. Akhavan Ghassabzade M. Wali Dynamical behaviours and stability analysis of a generalized fractional model with a real case study Journal of Advanced Research Fractional model COVID-19 pandemic Existence and uniqueness results Stability analysis Numerical method |
| title | Dynamical behaviours and stability analysis of a generalized fractional model with a real case study |
| title_full | Dynamical behaviours and stability analysis of a generalized fractional model with a real case study |
| title_fullStr | Dynamical behaviours and stability analysis of a generalized fractional model with a real case study |
| title_full_unstemmed | Dynamical behaviours and stability analysis of a generalized fractional model with a real case study |
| title_short | Dynamical behaviours and stability analysis of a generalized fractional model with a real case study |
| title_sort | dynamical behaviours and stability analysis of a generalized fractional model with a real case study |
| topic | Fractional model COVID-19 pandemic Existence and uniqueness results Stability analysis Numerical method |
| url | http://www.sciencedirect.com/science/article/pii/S2090123222001916 |
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