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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| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2021/9512371 |
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| 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 |
| collection | DOAJ |
| 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 |
| id | doaj-art-99f41a30ef3344cd9131fe1295bc8bf7 |
| institution | OA Journals |
| issn | 1607-887X |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-99f41a30ef3344cd9131fe1295bc8bf72025-08-20T02:20:16ZengWileyDiscrete Dynamics in Nature and Society1607-887X2021-01-01202110.1155/2021/9512371Mathematical Analysis of the TB Model with Treatment via Caputo-Type Fractional DerivativeXiao-Hong Zhang0Aatif Ali1Muhammad Altaf Khan2Mohammad Y. Alshahrani3Taseer Muhammad4Saeed Islam5College of ScienceDepartment of MathematicsInstitute for Ground Water StudiesDepartment of Clinical Laboratory SciencesDepartment of MathematicsDepartment of MathematicsIn 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.http://dx.doi.org/10.1155/2021/9512371 |
| spellingShingle | 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 Discrete Dynamics in Nature and Society |
| title | 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_short | 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 |
| url | http://dx.doi.org/10.1155/2021/9512371 |
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