TY  - EJOU
AU  - Khan, Muhammad Altaf 
AU  - DarAssi, Mahmoud H. 
AU  - Ahmad, Irfan 
AU  - Seyam, Noha Mohammad 
AU  - Alzahrani, Ebraheem 

TI  - Modeling the Dynamics of Tuberculosis with Vaccination, Treatment, and Environmental Impact: Fractional Order Modeling
T2  - Computer Modeling in Engineering \& Sciences

PY  - 2024
VL  - 141
IS  - 2
SN  - 1526-1506

AB  - 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.
KW  - Tuberculosis; real data; stability analysis; parameter estimations; discussion

DO  - 10.32604/cmes.2024.053681