Fractional Order Derivative Model and Analysis of Tuberculosis Transmission

Abstract

This paper constructs a fractional derivative model of TB in the Caputo sense. By using the Laplace definition in fractional derivative and fractional integral along with the stated Lipschitz condition, the proposed model's mathematical acceptability and biological significance are confirmed. The basic reproduction number is calculated by incorporating the spectral radius of the next-generation matrix. We compute steady states and demonstrate the local stability of the TB-free equilibrium. Sensitivity index is computed and the most contributor parameter to TB disease persistence and extinction is identified. Analytical and numerical simulation analysis results are consistent with one another. Furthermore, the TB fractional model solutions demonstrate that a higher order of fractional derivative corresponds to a lower population infection rate. Since the population's memory is correlated with the order of fractional derivative, a higher level of activation of memory toward tuberculosis infection means that the disease will spread and advance less. The interactive numerical scheme is derived and the fractional TB model is solved to support the analytical results.