A Fractional SEIAQR Model under Caputo and Atangana--Baleanu Operators: Dynamics, Sensitivity, and Estimation for the 2024 Algeria Diphtheria Outbreak

Abstract

Algeria recorded a marked diphtheria resurgence in 2024, with 1124 notified cases concentrated in the southern border wilayas. We analyse a six-compartment susceptible--exposed--symptomatic--asymptomatic--quarantined--recovered (SEIAQR) model under two non-local operators, the singular power-law Caputo kernel and the non-singular Mittag--Leffler Atangana--Baleanu--Caputo kernel, in order to quantify how epidemic memory shapes such an outbreak. For both operators we establish positive invariance, ultimate boundedness in a compact region bounded away from extinction, and well-posedness by the Banach fixed-point theorem. The basic reproduction number is obtained in closed form and shown to be independent of the memory order; the disease-free equilibrium is locally and globally asymptotically stable for $\Rzero\le1$, and the endemic equilibrium is globally stable for $\Rzero>1$ under an explicit Goh--Volterra condition. Closed-form normalized sensitivity indices identify the contact rate and the symptomatic recovery rate as the dominant levers. Calibrating a time-varying importation pulse against the 2024 INSP monthly series lowers the RMSE by about 30% and yields an endogenous $\Rzero\approx0.93<1$: local transmission is sub-critical and the outbreak importation-sustained. A fractional optimal-control problem with three interventions cuts the infected burden by about 75%.