Mechanistic epidemic calibration can be sensitive to short-lived reporting anomalies that give a few observations high leverage under least squares (LSQ). We develop direct forward-solver least absolute deviations (LAD) calibration for time-varying susceptible-exposed-infectious-removed models fitted to reporting-interval incidence, with finite-horizon regularity and large-sample results. A phase-aware Monte Carlo study compares LAD and LSQ across three transmission drivers, four epidemic phases, five reporting mechanisms, and four forecast horizons. All 120,000 primary fits converged. LAD was favored in 56.7% of mean-absolute-error comparisons and 57.9% of weighted-interval-score comparisons overall; under isolated spikes, backlog release, or temporary underreporting, these proportions rose to 73.6% and 72.2%, with strong gains near the epidemic peak. Rolling-origin validation on four synthetic RAPIDD Ebola scenarios also identifies settings where LSQ is preferred, demonstrating the framework's ability to distinguish robustness gains from sustained-trend behavior. Multiple uncertainty and identifiability diagnostics, complete code and data, and an interactive application provide a reproducible computational workflow.

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