Open Access

ARTICLE

Modeling the Dynamics of Tuberculosis with Vaccination, Treatment, and Environmental Impact: Fractional Order Modeling

by Muhammad Altaf Khan^1,*, Mahmoud H. DarAssi², Irfan Ahmad³, Noha Mohammad Seyam⁴, Ebraheem Alzahrani⁵

1 Faculty of Natural and Agricultural Sciences, University of the Free State, Bleomfontein, 9300, South Africa
2 Department of Basic Sciences, Princess Sumaya University for Technology, Amman, 11941, Jordan
3 Department of Clinical Laboratory Sciences, College of Applied Medical Science, King Khalid University, Abha, 62529, Saudi Arabia
4 Mathematical Sciences Department, College of Applied Sciences, Umm Al-Qura University, Makkah, 24381, Saudi Arabia
5 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

* Corresponding Author: Muhammad Altaf Khan. Email:

(This article belongs to the Special Issue: Analytical and Numerical Solution of the Fractional Differential Equation)

Computer Modeling in Engineering & Sciences 2024, 141(2), 1365-1394. https://doi.org/10.32604/cmes.2024.053681

Received 07 May 2024; Accepted 18 July 2024; Issue published 27 September 2024

Abstract

A mathematical model is designed to investigate Tuberculosis (TB) disease under the vaccination, treatment, and environmental impact with real cases. First, we introduce the model formulation in non-integer order derivative and then, extend the model into fractional order derivative. The fractional system’s existence, uniqueness, and other relevant properties are shown. Then, we study the stability analysis of the equilibrium points. The disease-free equilibrium (DFE) is locally asymptotically stable (LAS) when . Further, we show the global asymptotical stability (GAS) of the endemic equilibrium (EE) for and for . The existence of bifurcation analysis in the model is investigated, and it is shown the system possesses the forward bifurcation phenomenon. Sensitivity analysis has been performed to determine the sensitive parameters that impact . We consider the real TB statistics from Khyber Pakhtunkhwa in Pakistan and parameterized the model. The computed basic reproduction number obtained using the real cases is . Various numerical results regarding disease elimination of the sensitive parameters are shown graphically.

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Keywords

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