Can conformalized physics-informed weak regression estimate epidemic parameters?

Derivatives were the weak point of physics-informed regression

Compartment models describe an epidemic as a system of ordinary differential equations, and the coefficients for transmission, recovery and transition are the quantities epidemiologists want. In many of these models the right-hand side is linear in a coefficient vector whose entries are physical parameters or products of them, so the coefficients can be recovered by ordinary least squares and the physical parameters by an explicit map back. Physics-informed regression (PIR) exploits exactly this: it estimates the time derivatives of the observed states from the data, evaluates the model library at the data, and solves one least-squares problem.

The catch is the derivative estimate. Finite differences amplify measurement noise, and their truncation error depends on the sampling interval. The paper shows the consequence directly: on the COVID-19 benchmark without noise, second-order finite differences leave 0.76–0.95% error, which is the truncation error at a 5.05-day grid, while sixth-order differences reach 0.002–0.003%. On the measles benchmark, whose outbreak ends within about 0.3 years, no finite-difference estimator at the 0.2-year spacing does better than 33%.

PIWR removes that dependence. Instead of differentiating the data, it integrates the model over time, so the fit uses the area under the trajectory rather than its slope. It needs no derivatives of the data, estimates the initial state jointly with the parameters, and is high-order accurate with cubic-spline quadrature. The practical meaning is that coarse or noisy sampling hurts less: on measles at the same spacing PIWR gives 9.4% error, and halving the interval twice brings it to 10⁻³%.

Accuracy gains hold where sampling resolves the dynamics

The main comparison uses the eleven benchmark models of disease-informed neural networks (DINNs), regenerated with the original initial conditions, time grids and parameter values, at zero noise and at 1% noise. Across the 22 model/noise settings, PIWR is the most accurate estimator in 15 and PIR with sixth-order finite differences in the remaining seven. Where the sampling resolves the dynamics, PIWR or PIR-FD6 improves on the reported DINN errors by one to four orders of magnitude for COVID-19, Ebola, Dengue, Pneumonia, Polio and Tuberculosis, and is comparable for HIV and Zika.

Two benchmarks stay out of reach at their DINN sampling. Anthrax has outbreaks of a few days sampled every 40 days, and all estimators have median errors of 7–85%. For Zika, the row scaling chosen on tuning draws is suboptimal for the literature case, giving 6.3% against 0.19% for PIR-FD6 without noise. Smallpox and measles are also less accurate than the reported DINN values without noise (6.9% against 5.1%, and 9.4% against 3.4%).

The paper is explicit that the DINN comparison is not like-for-like. Shaier et al. restricted every parameter to a search range typically ±1–10% around the literature value, which bounds their errors by construction, and reported estimates for some parameters that do not occur in the equations or cannot be separated from others. PIR and PIWR search a ±50% box instead, and the least-squares structure exposes non-identifiability as collinear library columns rather than hiding it.

Conformal intervals come from the model's own simulated epidemics

A point estimate is only half the answer, because measurement noise, errors-in-variables bias and model mismatch all propagate into the coefficients, and the map from coefficients to parameters can amplify them further through ratios. The paper quantifies this with split conformal prediction, a distribution-free method that turns a calibration population into intervals with a finite-sample coverage guarantee. The unusual step is where the calibration population comes from: the physics-based model itself generates simulated epidemics, drawn from a ±50% box around the literature values, sampled on the same time grid, given the same measurement noise, and passed through the same estimator.

The resulting intervals are simultaneous for all parameters, not marginal one at a time. On the DINN benchmarks the 90% intervals achieved 87–92% empirical coverage in all 88 settings, measured on 2,000 independent test problems. A residual-adaptive factor, PIWR-RACC, rescales the intervals to each data set using a quantity computed without ground truth; it narrows them where the estimate is well resolved and widens them where it is not, at no cost in coverage.

The guarantee has a stated scope. It is marginal over the calibration population and assumes the model is correct, so a misspecified model or a test case far from the prior box is not covered. Coverage at one fixed parameter vector over noise realizations is a conditional quantity the method does not control: for PIR-FD6 that conditional coverage was 96–100% for all models but one, and for PIWR 86–100%.

On 14 real data points, the coefficient constraint decides the answer

The real-data test is the 1978 boarding-school influenza outbreak, with only 14 daily counts of boys in bed. Here the SIR model is written so that only the infected compartment is observed, and the three coefficients satisfy one constraint that a free fit ignores. With so few samples, fitting two parameters rather than three free coefficients matters: the finite-difference and free-coefficient PIWR estimates are calibrated at 95–96% coverage but useless, because their intervals include negative rates, and their median test error is 55% for PIR-FD2 and 33% for PIWR.

A variable-projection variant, PIWR-VP, imposes the constraint and reduces the median test error to 7.6%, with intervals 5–12 times narrower. Its 95% simultaneous intervals are β ∈ [1.15, 2.47], γ ∈ [0.39, 0.52] and R₀ ∈ [2.97, 4.96] with the adaptive factor, at 94.4% mean coverage across five calibration/test splits. These intervals contain the Stan posterior means of β and R₀, and the γ interval overlaps the Bayesian credible interval, though the estimate γ = 0.46 lies below the posterior mean of 0.54.

The paper attributes that gap to weighting rather than to the weak form: an output-error least-squares fit of the simulated SIR trajectory to the counts also gives γ = 0.44, so the difference comes from the negative-binomial likelihood and priors in Stan versus least squares here. The credible intervals are narrower, most of all for β, but they are conditional on the model, prior and likelihood and are marginal per parameter, whereas the conformal intervals are simultaneous with coverage guaranteed over the calibration population.

A fixed parameter helps only if its uncertainty enters calibration

Parameters known from other sources can be moved out of the estimated vector and fixed at a nominal value, and the paper tests what that buys on the seven-compartment S3I3R model. With the three recovery rates known exactly, the intervals of φ₁ and φ₂ shrink from ±34% and ±33% to ±1.6%, a factor of twenty, and β's interval from ±1.8% to ±0.7%. Prior knowledge narrows the intervals of the parameters it decouples.

The precision of that knowledge decides whether fixing is worthwhile. Known only to ±5% or ±20%, the recovery rates still narrow φ₁ and φ₂ (to ±9% and ±19%), but they widen β and θ, which the data determine well when all rates are estimated (±1.8% and ±2.4%), to ±4.1% and ±17% at ±5% and to ±8% and ±33% at ±20%. Fixing a parameter known less precisely than the data can determine it is worse than estimating it.

The calibration population must also reflect how well the prior is actually known. Calibrated as exactly known, the intervals covered 90.9% of test epidemics with nominal recovery rates but only 2.4% when those rates deviated by up to ±20%: narrow intervals around biased estimates. Calibration at ±5% gave 59% on those test epidemics (74% with PIWR-RACC), and at ±20% gave 89.9%, the target. When the rates were in fact nominal, the wider calibrations were conservative at 100%.

Time-varying transmission stays a linear problem

The framework is not limited to constant parameters. Expanding a time-varying transmission rate β(t) in B-splines keeps the model linear in the coefficients, and a smoothness penalty turns the estimate into one penalized least-squares solve. On the benchmark of Millevoi et al., PIR and PIWR are more accurate than physics-informed neural networks in three of four cases and at least comparable in the noisiest, with simultaneous bands for β(t) and Rₜ.

The noisiest case shows the intervals behaving as intended. With 40% errors on every count over 120 days, the bands are wide and uninformative, with half-widths of 1.7 (PIR-FD6) and 4.6 (PIWR) in units of Rₜ, correctly reflecting that the data do not determine Rₜ more precisely. The paper contrasts this with the PINN bands of Millevoi et al., which are one standard deviation over training runs and measure optimization variability on one data set rather than uncertainty due to the data.

Cost stays small throughout. A single PIR fit takes 0.14–0.5 ms and a PIWR fit 0.5–1.6 ms, and the complete conformal pipeline for a model, including tuning, calibration and testing for four estimators and both noise levels, takes 19–108 s, dominated by simulating roughly 3·10⁴ calibration and test epidemics. The reported DINN training times range from 20 minutes to 91 hours on different hardware, which the paper calls indicative but not a matter of hardware.

Prefer PIWR when sampling is coarse or noise is large

The paper's own priority ordering is clear from the benchmark split. PIWR wins in 15 of 22 model/noise settings and PIR-FD6 in the other seven, and the pattern is not random: PIWR's advantage is largest where finite differences amplify the noise of small compartments, as in Pneumonia (11% against 13–57%), Ebola (25% against 37–40%), Measles (28% against 84–88%) and HIV (22% against 35–51%). On fine grids the sixth-order finite-difference estimator can overtake it, but only on grids eight times finer than the DINN grid for COVID-19.

Two conditions bound the recommendation. The sampled states must resolve the dynamics, which Anthrax fails at a 40-day spacing, and the library terms carrying a parameter must not be swamped by noise in the largest compartment, which limits some transmission rates in Ebola. Errors-in-variables bias also remains: both methods evaluate the library at noisy samples, and the weak form averages out the random part but not the bias, which is small at 1% noise but dominates for sparse, noisy counts. The conformal calibration includes that bias, so coverage is retained at the price of wider intervals.

The practical threshold the paper supports is about the data, not the model class. When the sampling resolves the epidemic and the coefficient constraint can be imposed, PIWR with variable projection delivers calibrated simultaneous intervals in milliseconds, as on the boarding-school data where the free fit's intervals included negative rates. When counts are sparse and noisy, a likelihood-based fit is still more efficient than least squares, and the paper says comparing against likelihood-based or Bayesian inference remains valuable as validation rather than being made obsolete.

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