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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Main Authors: Tian-Chuan Sun, Mahmoud H. DarAssi, Wafa F. Alfwzan, Muhammad Altaf Khan, Abdulaziz Saad Alqahtani, Saeed S. Alshahrani, Taseer Muhammad
Format: Article
Language:English
Published: MDPI AG 2023-03-01
Series:Fractal and Fractional
Subjects:
Online Access:https://www.mdpi.com/2504-3110/7/3/234
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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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