Abstract / Summary
Abstract Many data-driven biological systems must be reconstructed from observations, forecast under uncertainty, and steered toward desired outcomes, yet these tasks are often assigned different model representations. In computational epidemiology, this fragmentation obscures parameter meaning and makes model outputs harder to compare and communicate. We introduce 'control-polygon epidemiology', a unified Bernstein-Bezier framework in which an effective reproduction-number trajectory is encoded by a small set of interpretable control points. Historical polygons are estimated by constrained least squares or Bayesian negative-binomial renewal inference; forecasts attach new, smoothly joined Bernstein segments instead of extrapolating the historical polynomial; and inverse control maps a desired epidemic trajectory back to feasible intervention intensity, implementation speed, uncertainty, and cost. The representation yields global convex-hull threshold certificates, finite-difference momentum and curvature, coefficient-level uncertainty propagation, spectral calibration to mechanistic models, and structural lower bounds: C transversal threshold crossings require at least C + 1 control points, while E isolated interior extrema require at least E + 2. Synthetic experiments span renewal, SIR, SEIR, SIRS, vector-borne, age-structured, vaccination, isolation/hospitalization, and overdispersed observation models. In Philippine COVID- 19 rolling-origin tests, a history-only peak-aware continuation improved on the original joined-Bernstein forecast in 87.5% of origin–horizon comparisons and on a constant-Rt baseline in 75%, while stress tests correctly exposed the impossibility of forecasting genuinely unobserved structural shocks from case history alone. The framework provides a low-dimensional bridge between statistical learning, mechanistic transmission models, probabilistic forecasting, and interactive decision science.