A fractional model based on caputo derivative for tuberculosis transmission using real data from Kenya.

Abstract

Tuberculosis (TB) remains one of the top infectious disease killers worldwide. It is caused by Mycobacterium tuberculosis and spread through the air, posing a serious threat to vulnerable populations, especially those with weakened immune systems. These challenges emphasize the need for more robust and realistic modeling approaches to inform policy and intervention. This study has been carried out using a fractional-order TB model in Kenya incorporating the Caputo derivative. Fundamental mathematical features such as positivity, boundedness, uniqueness and existence are rigorously investigated for the model. Model parameters, including the fractional order, are estimated, with an optimal value of fractional order around 0.85 yielding the best alignment with real data. A sensitivity analysis, grounded in the basic reproduction number, identifies the influence of critical parameters on disease spread. Visualization through 3D mesh and contour plots further reveals the significant impact of these parameters on TB transmission dynamics. Numerical simulations are conducted by employing the Adams-Bashforth Predictor Corrector scheme, allowing us to account for memory effects and more accurately reflect real-world dynamics. By performing data fitting, we observe that our formulated model produces results that closely match real-world data. Overall, the simulations also explored various intervention strategies, including improved treatment access, faster diagnosis, and recognition of nonlinear transmission dynamics. These results emphasize the importance of timely intervention and provide actionable insights for strengthening TB control policies.