Accurate forecasting of COVID-19 dynamics is essential for timely intervention. We compare the classical SIR model of Kermack and McKendrick with an operator-modified SIR formulation based on the Makinde differential operator. Using weekly Nigerian data (April–May 2025) on COVID-19, we estimate classical transmission and recovery rates. We also derive curvature-corrected operator rates using second-order incidence changes, yielding smoother estimates. Each model is discretized to predict new cases and evaluated by root-mean-square error (RMSE), Akaike Information Criterion (AIC), and peak-magnitude comparison. The classical SIR exactly reproduces in-sample data, whereas the modified model yields a smoother predicted peak, indicating improved generalizability. These results affirm the utility of memory-augmented epidemic models in forecasting and intervention-sensitive policy design under data uncertainty.