Mathematical Analysis of the TB Model with Treatment via Caputo-Type Fractional Derivative

In this study, we formulate a noninteger-order mathematical model via the Caputo operator for the transmission dynamics of the bacterial disease tuberculosis (TB) in Khyber Pakhtunkhwa (KP), Pakistan. The number of confirmed cases from 2002 to 2017 is considered as incidence data for the estimation...

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Autores principales:	Xiao-Hong Zhang, Aatif Ali, Muhammad Altaf Khan, Mohammad Y. Alshahrani, Taseer Muhammad, Saeed Islam
Formato:	article
Lenguaje:	EN
Publicado:	Hindawi Limited 2021
Materias:	Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/99f41a30ef3344cd9131fe1295bc8bf7
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spelling	oai:doaj.org-article:99f41a30ef3344cd9131fe1295bc8bf72021-11-29T00:56:10ZMathematical Analysis of the TB Model with Treatment via Caputo-Type Fractional Derivative1607-887X10.1155/2021/9512371https://doaj.org/article/99f41a30ef3344cd9131fe1295bc8bf72021-01-01T00:00:00Zhttp://dx.doi.org/10.1155/2021/9512371https://doaj.org/toc/1607-887XIn this study, we formulate a noninteger-order mathematical model via the Caputo operator for the transmission dynamics of the bacterial disease tuberculosis (TB) in Khyber Pakhtunkhwa (KP), Pakistan. The number of confirmed cases from 2002 to 2017 is considered as incidence data for the estimation of parameters or to parameterize the model and analysis. The positivity and boundedness of the model solution are derived. For the dynamics of the tuberculosis model, we find the equilibrium points and the basic reproduction number. The proposed model is locally and globally stable at disease-free equilibrium, if the reproduction number ℛ0<1. Furthermore, to examine the behavior of the various parameters and different values of fractional-order derivative graphically, the most common iterative scheme based on fundamental theorem and Lagrange interpolation polynomial is implemented. From the numerical result, it is observed that the contact rate and treatment rate have a great impact on curtailing the tuberculosis disease. Furthermore, proper treatment is a key factor in reducing the TB transmission and prevalence. Also, the results are more precise for lower fractional order. The results from various numerical plots show that the fractional model gives more insights into the disease dynamics and on how to curtail the disease spread.Xiao-Hong ZhangAatif AliMuhammad Altaf KhanMohammad Y. AlshahraniTaseer MuhammadSaeed IslamHindawi LimitedarticleMathematicsQA1-939ENDiscrete Dynamics in Nature and Society, Vol 2021 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Mathematics QA1-939
spellingShingle	Mathematics QA1-939 Xiao-Hong Zhang Aatif Ali Muhammad Altaf Khan Mohammad Y. Alshahrani Taseer Muhammad Saeed Islam Mathematical Analysis of the TB Model with Treatment via Caputo-Type Fractional Derivative
description	In this study, we formulate a noninteger-order mathematical model via the Caputo operator for the transmission dynamics of the bacterial disease tuberculosis (TB) in Khyber Pakhtunkhwa (KP), Pakistan. The number of confirmed cases from 2002 to 2017 is considered as incidence data for the estimation of parameters or to parameterize the model and analysis. The positivity and boundedness of the model solution are derived. For the dynamics of the tuberculosis model, we find the equilibrium points and the basic reproduction number. The proposed model is locally and globally stable at disease-free equilibrium, if the reproduction number ℛ0<1. Furthermore, to examine the behavior of the various parameters and different values of fractional-order derivative graphically, the most common iterative scheme based on fundamental theorem and Lagrange interpolation polynomial is implemented. From the numerical result, it is observed that the contact rate and treatment rate have a great impact on curtailing the tuberculosis disease. Furthermore, proper treatment is a key factor in reducing the TB transmission and prevalence. Also, the results are more precise for lower fractional order. The results from various numerical plots show that the fractional model gives more insights into the disease dynamics and on how to curtail the disease spread.
format	article
author	Xiao-Hong Zhang Aatif Ali Muhammad Altaf Khan Mohammad Y. Alshahrani Taseer Muhammad Saeed Islam
author_facet	Xiao-Hong Zhang Aatif Ali Muhammad Altaf Khan Mohammad Y. Alshahrani Taseer Muhammad Saeed Islam
author_sort	Xiao-Hong Zhang
title	Mathematical Analysis of the TB Model with Treatment via Caputo-Type Fractional Derivative
title_short	Mathematical Analysis of the TB Model with Treatment via Caputo-Type Fractional Derivative
title_full	Mathematical Analysis of the TB Model with Treatment via Caputo-Type Fractional Derivative
title_fullStr	Mathematical Analysis of the TB Model with Treatment via Caputo-Type Fractional Derivative
title_full_unstemmed	Mathematical Analysis of the TB Model with Treatment via Caputo-Type Fractional Derivative
title_sort	mathematical analysis of the tb model with treatment via caputo-type fractional derivative
publisher	Hindawi Limited
publishDate	2021
url	https://doaj.org/article/99f41a30ef3344cd9131fe1295bc8bf7
work_keys_str_mv	AT xiaohongzhang mathematicalanalysisofthetbmodelwithtreatmentviacaputotypefractionalderivative AT aatifali mathematicalanalysisofthetbmodelwithtreatmentviacaputotypefractionalderivative AT muhammadaltafkhan mathematicalanalysisofthetbmodelwithtreatmentviacaputotypefractionalderivative AT mohammadyalshahrani mathematicalanalysisofthetbmodelwithtreatmentviacaputotypefractionalderivative AT taseermuhammad mathematicalanalysisofthetbmodelwithtreatmentviacaputotypefractionalderivative AT saeedislam mathematicalanalysisofthetbmodelwithtreatmentviacaputotypefractionalderivative
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Mathematical Analysis of the TB Model with Treatment via Caputo-Type Fractional Derivative

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