Epidemic dynamics as a Wasserstein gradient flow on viral load distributions

Classical epidemic models assume that infected individuals are homogeneous, reducing disease progression to a single aggregate variable. This hides the role of within‑host variability, in particular differences in viral load that directly influence both infectivity and recovery. We propose a geometric perspective in which the epidemic is described not as a system of compartments, but as an evolving distribution of individual infection states, allowing heterogeneity at the within‑host level to be explicitly represented. Within this framework, transmission and recovery emerge from interacting biological mechanisms rather than fixed rates. The model distinguishes between infectivity and susceptibility, incorporates variability in immune response, and can recover classical compartmental models as limiting cases. We calibrate the model against longitudinal SARS‑CoV‑2 viral‑load data from a prospective cohort, showing that it reproduces the full distribution of recovery durations, and provides uncertainty bounds on infectiousness curves. As a consistency check (using the same calibration data, not an independent validation), the stationary Ornstein-Uhlenbeck approximation yields a persistent-shedding fraction of 32.8%, matching the 32.4% observed in the cohort. The calibrated framework illustrates how distinct clinical burdens can arise under the same reproduction number, a feature not captured by simple mean-field models. This provides a unified description that bridges individual‑level disease dynamics with population‑level epidemic behavior, while offering a richer interpretation of epidemic thresholds and intervention effects.