Dynamics of Tuberculosis transmission: A fractional-order modelling approach with modified Mittag-Leffler kernel and epidemiological data validation

Abstract

In this work, we analyze a fractional-order epidemiological model for tuberculosis ($\tb$) transmission in China, formulated using the modified Mittag–Leffler kernel ($\m$) fractional operator, and calibrated against the reported $\tb$ cases from 2005 to 2016. The mathematical properties of the system, including non-negativity, boundedness, existence, uniqueness, and stability of solutions, are rigorously established to ensure the well-posedness of the model. Model parameters, along with the fractional derivative order, are estimated through real $\tb$ case data, and the optimal fractional order is identified as $\kappa \approx 0.88$. Numerical simulations are carried out using the Adams–Bashforth predictor–corrector method. To evaluate predictive performance, the root mean square error (RMSE) is employed as a statistical measure of model–data agreement. The classical integer-order model produced an RMSE of $\varepsilon = 8.28 \times 10^{5}$ ($\approx 824,000$ cases), while the Caputo formulation reduced the error to $\varepsilon = 5.93 \times 10^{5}$ ($\approx 593,000$ cases), yielding a 28.4\% improvement. Furthermore, the $\mathcal{MMLK}$ fractional formulation provided an even closer fit to the data, lowering the error to $\varepsilon = 4.70 \times 10^{5}$ ($\approx 470,000$ cases), corresponding to a 43.2\% improvement relative to the classical model and a further 20.7\% gain over the Caputo case. These findings demonstrate that fractional-order modeling, particularly the $\mathcal{MMLK}$ approach, offers superior accuracy and flexibility in capturing $\tb$ dynamics compared to traditional integer-order frameworks.