Fractional-Order Event-Triggered Control of HIV on Heterogeneous Networks: Robust Contraction and Spectral Targeting

We analyze a campaign-renewed Caputo model of HIV transmission on heterogeneous contact networks, with risk-triggered screening and antiretroviral-treatment linkage. The model has a spectral next-generation threshold and an explicit positive matrix Mittag–Leffler jump–flow operator. Positivity, global extinction under a positive linear resolvent comparison, and conditional robust hybrid-contraction theorems are established without assuming that the nonlinear infected–treated subsystem is cooperative. A common weighted norm and watchdog-bounded intervention intervals are required; these assertions do not apply unmodified to a fixed-history Caputo system. An independent reconstruction of the complete 80-dimensional cycle matrix gives spectral radius 1.0688 for the stated nominal baseline (α=0.85, one-year cycle), correcting an earlier noncontractive setting that had been described as stabilizing. A separate six-mode intervention family admits a common weighted-norm bound below 0.9981. Newly executed nonlinear 20-year experiments, including mandatory watchdog events, report actual campaign counts, ART transfers and burden under single-factor and nominal-budget-matched comparisons; a periodic strategy can outperform risk-triggered intervention at the same total nominal pulse budget. Three seeded 100-graph synthetic ensembles clarify structural variability but do not certify arbitrary switching. Beijing surveillance counts from 2005 to 2020 are independently redrawn with one missing AIDS observation marked explicitly; these data do not empirically identify the fractional order, contact network or controller.