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

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 diseasefree equilibrium (DFE) D0 is locally asymptotically stable (LAS) when (v) pound < 1. Further, we show the global asymptotical stability (GAS) of the endemic equilibrium (EE) D & lowast; for (v)> pound 1 and D0 for (v) pound <= 1. 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 v pound. 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 (0) pound approximate to 3.6615. Various numerical results regarding disease elimination of the sensitive parameters are shown graphically.