Peak-Weighted Calibration of Implemented Integer-Order, Caputo, and Atangana–Baleanu–Caputo Schemes for Simultaneous COVID-19 and Influenza Transmission

COVID-19 and influenza may produce overlapping waves, yet an overall least-squares fit need not represent their high-activity weeks well. We compare implemented integer-order, Caputo, and Atangana–Baleanu–Caputo (ABC) discrete schemes for a common ten-compartment transition structure with seasonal transmission and external influenza input. The schemes are calibrated with ordinary mean-normalized weighted least squares (J1) and a peak-weighted objective (J2). We fit the first 66 of 83 weekly observation pairs and assess the remaining 17 weeks. ABC gives the smallest training peak root-mean-square error for both series under either objective. Within ABC, J2 reduces COVID-19 and influenza training peak errors by 9.50% and 42.98%, respectively, relative to its J1 fit. Integer order achieves the smallest ordinary loss and lower temporal validation error. The autonomous disease-free and conditional invasion thresholds characterize the underlying unforced vector field, not the fitted seasonal system. These findings support peak-weighted calibration for retrospective high-activity reconstruction in these data, but they do not establish superior future-peak prediction.