Mathematical modeling of tuberculosis using Caputo fractional derivative: a comparative analysis with real data.

In this study, we investigate an epidemiological model for tuberculosis in China using the Caputo fractional-order derivative. To ensure dimensional consistency, we appropriately adjust model parameters to maintain uniform units. The mathematical properties of the model, including the non-negativity, boundedness, existence, and uniqueness of its solutions, are thoroughly examined and established. Sensitivity analysis, based on the basic reproduction number, is performed to evaluate the impact of critical parameters on disease dynamics, and additional insights are provided through 3D mesh and contour plots, which illustrate how key parameters influence tuberculosis transmission. We estimate the model parameters, including the fractional-order derivative, and determine that the optimal fractional order, which best fits the real data, is approximately 0.93. Numerical simulations are performed using the Adams-Bashforth-Moulton method. By utilizing the root mean square error (RMSE) metric, the fractional-order model demonstrates an efficiency improvement of approximately 28.5% compared to its integer-order counterpart, highlighting the superior accuracy of fractional-order models in describing tuberculosis transmission dynamics. These findings underscore the significance of fractional-order models in epidemiological analysis. They provide a more refined approach for modeling infectious diseases and aiding in public health decision-making.