Mathematical Modeling of COVID-19 with Vaccination Using Fractional Derivative: A Case Study
Vaccination against any infectious disease is considered to be an effective way of controlling it. This paper studies a fractional order model with vaccine efficacy and waning immunity. We present the model’s dynamics under vaccine efficacy, the impact of immunization, and the waning of the vaccine...
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2023-03-01
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author | Tian-Chuan Sun Mahmoud H. DarAssi Wafa F. Alfwzan Muhammad Altaf Khan Abdulaziz Saad Alqahtani Saeed S. Alshahrani Taseer Muhammad |
author_facet | Tian-Chuan Sun Mahmoud H. DarAssi Wafa F. Alfwzan Muhammad Altaf Khan Abdulaziz Saad Alqahtani Saeed S. Alshahrani Taseer Muhammad |
author_sort | Tian-Chuan Sun |
collection | DOAJ |
description | Vaccination against any infectious disease is considered to be an effective way of controlling it. This paper studies a fractional order model with vaccine efficacy and waning immunity. We present the model’s dynamics under vaccine efficacy, the impact of immunization, and the waning of the vaccine on coronavirus infection disease. We analyze the model under their equilibrium points. The model under the equilibrium points is discussed and proven that it is locally asymptotically stable if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">R</mi><mi>v</mi></msub></semantics></math></inline-formula> is lesser than unity. We present the backward bifurcation hypothesis of the model and show that there is a parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>β</mi><mn>2</mn></msub></semantics></math></inline-formula> that causes the backward bifurcation in the imperfect vaccine model. We show certain assumptions when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ψ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> for the imperfect vaccine case; the model is both stable globally asymptotically at the disease-free (<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 endemic cases (<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>). By using infected cases from the recent wave throughout Pakistan, we shall estimate the model parameters and calculate the numerical value of the basic reproductive number <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>1.2591</mn></mrow></semantics></math></inline-formula>. We present the comprehensive graphical results for the realistic parameter values and show many useful suggestions regarding the elimination of the infection from society. The vaccination efficacy that provides an important role in disease elimination is discussed in detail. |
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spelling | doaj.art-74546625dc194644b6f4a441d3fd3d5f2023-11-17T11:12:11ZengMDPI AGFractal and Fractional2504-31102023-03-017323410.3390/fractalfract7030234Mathematical Modeling of COVID-19 with Vaccination Using Fractional Derivative: A Case StudyTian-Chuan Sun0Mahmoud H. DarAssi1Wafa F. Alfwzan2Muhammad Altaf Khan3Abdulaziz Saad Alqahtani4Saeed S. Alshahrani5Taseer Muhammad6School of Marxism (Department of Public Teaching and Researching), Huzhou College, Huzhou 313000, ChinaDepartment of Basic Sciences, Princess Sumaya University for Technology, Amman 11941, JordanDepartment of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi ArabiaFaculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South AfricaDepartment of Internal Medicine, College of Medicine, Najran University, Najran 55461, Saudi ArabiaDepartment of Surgery, College of Medicine, Najran University, Najran 55461, Saudi ArabiaDepartment of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi ArabiaVaccination against any infectious disease is considered to be an effective way of controlling it. This paper studies a fractional order model with vaccine efficacy and waning immunity. We present the model’s dynamics under vaccine efficacy, the impact of immunization, and the waning of the vaccine on coronavirus infection disease. We analyze the model under their equilibrium points. The model under the equilibrium points is discussed and proven that it is locally asymptotically stable if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">R</mi><mi>v</mi></msub></semantics></math></inline-formula> is lesser than unity. We present the backward bifurcation hypothesis of the model and show that there is a parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>β</mi><mn>2</mn></msub></semantics></math></inline-formula> that causes the backward bifurcation in the imperfect vaccine model. We show certain assumptions when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ψ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> for the imperfect vaccine case; the model is both stable globally asymptotically at the disease-free (<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 endemic cases (<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>). By using infected cases from the recent wave throughout Pakistan, we shall estimate the model parameters and calculate the numerical value of the basic reproductive number <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>1.2591</mn></mrow></semantics></math></inline-formula>. We present the comprehensive graphical results for the realistic parameter values and show many useful suggestions regarding the elimination of the infection from society. The vaccination efficacy that provides an important role in disease elimination is discussed in detail.https://www.mdpi.com/2504-3110/7/3/234coronavirus mathematical modelinfected datastability analysisnumerical results |
spellingShingle | Tian-Chuan Sun Mahmoud H. DarAssi Wafa F. Alfwzan Muhammad Altaf Khan Abdulaziz Saad Alqahtani Saeed S. Alshahrani Taseer Muhammad Mathematical Modeling of COVID-19 with Vaccination Using Fractional Derivative: A Case Study Fractal and Fractional coronavirus mathematical model infected data stability analysis numerical results |
title | Mathematical Modeling of COVID-19 with Vaccination Using Fractional Derivative: A Case Study |
title_full | Mathematical Modeling of COVID-19 with Vaccination Using Fractional Derivative: A Case Study |
title_fullStr | Mathematical Modeling of COVID-19 with Vaccination Using Fractional Derivative: A Case Study |
title_full_unstemmed | Mathematical Modeling of COVID-19 with Vaccination Using Fractional Derivative: A Case Study |
title_short | Mathematical Modeling of COVID-19 with Vaccination Using Fractional Derivative: A Case Study |
title_sort | mathematical modeling of covid 19 with vaccination using fractional derivative a case study |
topic | coronavirus mathematical model infected data stability analysis numerical results |
url | https://www.mdpi.com/2504-3110/7/3/234 |
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