Caputo fractional-order differential equations for tuberculosis transmission and relapse dynamics with empirical validation

Tuberculosis (TB) mortality has declined by only 23% since 2015, far below the WHO End TB Strategy target. Classical integer-order models, rooted in Markovian assumptions, do not adequately capture the memory-driven nature of TB transmission, in which progression depends on cumulative infection history. We formulate Caputo fractional-order differential equations for TB transmission in a SELIT framework with two-stage latency, direct relapse from the treated to the infectious class, and waning immunity, where the fractional order α quantifies epidemiological memory. We establish positivity and boundedness of solutions, derive the basic reproduction number \(\:{R}₀\) via the next-generation matrix, and prove global asymptotic stability of the equilibrium using Lyapunov and Volterra-type functionals. Sensitivity and bifurcation analyses show that relapse rates above a critical threshold induce backward bifurcation with bistability for \(\:{R}₀<\) 1, so that \(\:{R}₀<\:\) 1 is necessary but not sufficient for elimination. The model is calibrated and validated with Ethiopian surveillance data (2005–2020), yielding α = 0.914 (95% CI: 0.882–0.941) and \(\:{R}₀\) = 1.67. The fractional model reduces validation error by 54.7% relative to its integer-order analogue (ΔAIC = 22.4), and thereby provides empirical evidence of memory in TB transmission and a rigorous basis for elimination planning.