Conformalized Physics-Informed Weak Regression for Parameter Estimation in Epidemic Models

Allan Peter Engsig-Karup

stat.MLOct 13, 2026

Conformalized Physics-Informed Weak Regression for Parameter Estimation in Epidemic Models

Authors: Allan Peter Engsig-Karup

Organizations: Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kongens Lyngby, Denmark

Abstract

Compartment models of infectious diseases are often linear in coefficients that are physical parameters or products or ratios of them. We exploit this with closed-form least squares and equip the estimates with distribution-free uncertainty. Physics-informed regression (PIR) fits the parameters to finite-difference derivatives of the data; we introduce physics-informed weak regression (PIWR), which integrates the model in time instead, needs no derivatives of the data, estimates the initial state jointly and is high-order accurate. To quantify uncertainty we wrap both estimators in split conformal prediction, with a calibration population generated by the physics-based model itself, giving simultaneous intervals for all parameters with a finite-sample coverage guarantee. A residual-adaptive factor, computed without ground truth, scales the intervals to each data set (PIWR-RACC). On the eleven benchmark models of disease-informed neural networks (DINNs), PIWR or PIR-FD6 is one to four orders of magnitude more accurate than the reported DINN results wherever the sampling resolves the dynamics, in about a millisecond per fit instead of minutes to hours of training, and the 90% intervals achieve 87-92% coverage in all 88 settings. On the S3I3R model of the original PIR study, PIWR lowers the reported PIR errors by about three orders of magnitude. On real data from the 1978 boarding-school influenza outbreak, a variable-projection variant gives calibrated 95% intervals that overlap those of a Bayesian analysis. On a time-varying transmission benchmark, a B-spline expansion of beta(t) keeps PIR and PIWR linear; they are more accurate than physics-informed neural networks in three of four cases and comparable in the noisiest, with simultaneous bands for beta(t). The complete conformal calibration takes less than two minutes for every benchmark.

Figures & tables

Model	$n$	$k$	$p$	fixed or absent	$m$	$T$ [d]	$\Delta t$ [d]	DINN tab.	DINN time
COVID-19 (SIDR)	4	3	3	–	100	500	5.05	8	20 min
HIV	3	8	9	$\mu_{T}$ , $N$	50	20	0.408	9	22 h
Smallpox	8	8	10	–	50	100	2.04	10	14 h
Tuberculosis (SLIT)	4	8	8	$c$	50	40	0.816	11	12 h
Pneumonia (SVCIR)	5	14	15	$k$ , $\tau$	50	100	2.04	12	41 h
Ebola (SEIHFR)	6	10	10	$\theta_{1}$ , $\delta_{1}$ , $\delta_{2}$	100	100	1.01	13	33 h

Table 1: The eleven DINN benchmark models: number of states

n

, estimated physical parameters

k

, length

p

of the coefficient vector

\bm{\theta}

, parameters fixed because they are not identifiable or absent from the model, number of samples

m

, time horizon

T

, sampling interval

\Delta t

, and the DINN table and training time reported by Shaier et al. (2022) .

Panel of Figure 1 — Figure 1: Conformalized physics-informed regression. The same fixed estimator (PIR or PIWR) is applied to the data and to a population of simulated epidemics with known parameters; the calibration scores of the population set the scale $\hat{q}$ of the simultaneous intervals.

		DINN		PIR-FD2		PIR-FD4		PIR-FD6		PIWR / PIWR-RACC
Par.	Actual	Found	% err.	Found	% err.	Found	% err.	Found	% err.	Found	% err.	90 % interval
noise level $\epsilon=0$
$\alpha$	0.191	0.1932	1.15	0.1892	0.946	0.1909	0.0389	0.191	0.00329	0.191	0.000105	[0.191, 0.191]
$\beta$	0.05	0.0501	0.2	0.04962	0.764	0.04999	0.0274	0.05	0.00212	0.05	0.000108	[0.049999, 0.050001]
$\gamma$	0.0294	0.0297	1.02	0.02913	0.907	0.02939	0.0365	0.0294	0.00304	0.0294	7.05e-05	[0.0294, 0.0294]
CPU time [ms]		$\approx$ 20 min		0.07		0.07		0.07		0.21

Table 2: COVID-19 (SIDR): estimates found by PIR-FD2, PIR-FD4, PIR-FD6 and PIWR with their % errors, and the PIWR-RACC 90 % simultaneous interval, without noise and with

1\,\%

noise. The CPU time is the median of 30 repeated fits of one data set ( section 5.9 ). DINN values and training time from Shaier et al. (2022) (noise-free data only).

Figure 2: COVID-19 (SIDR). (a) Data with

1\,\%

noise (markers) and the model integrated with the PIWR parameters (lines). (b) Calibration scores of PIWR with

1\,\%

noise, plain and adaptive, with the quantiles

\hat{q}

(dashed). (c) Errors of the estimates with their 90 % simultaneous intervals (plain for PIR, PIWR-RACC for PIWR) for both noise levels, next to the DINN errors; symmetric logarithmic axis.

Figure 3: Noise-free error against the sampling interval

\Delta t

on the same time horizon. Results of the COVID-19 case (left 3 sub-figures) with error of each parameter. Results of the Measles case with median error over its five parameters. The vertical line marks the DINN grid.

		$\epsilon=0$				$\epsilon=1\,\%$
Model	DINN	PIR-FD2	PIR-FD4	PIR-FD6	PIWR	PIR-FD2	PIR-FD4	PIR-FD6	PIWR
COVID-19	1.02	0.91	0.036	0.003	0.00011	1.2	0.49	0.44	0.46
HIV	0.779	20	7.1	0.32	1.1	35	51	43	22
Smallpox	5.1	39	35	8.2	6.9	29	24	5.6	6.5
Tuberculosis	3.63	11	3.9	2.3	0.043	15	8.6	6.9	8.5
Pneumonia	2.88	25	0.72	0.034	0.16	13	45	57	11

Table 3: Median % error at the literature values over the estimated parameters of each model. DINN: noise-free data, as reported by Shaier et al. (2022) within prescribed search ranges. Bold: most accurate PIR/PIWR estimator per noise level.

Figure 4: Median % error at the literature values for all models, without noise (a) and with

1\,\%

noise (b).

	$\epsilon=0$				$\epsilon=1\,\%$
Model	PIR-FD6	PIWR	PIWR-RACC	cov. [%]	PIR-FD6	PIWR	PIWR-RACC	cov. [%]	time [s]
COVID-19	0.0097	0.00062	0.0012	89.0	4.5	2.1	1.9	89.1	9
HIV	2e+02	17	13	89.1	1.5e+03	4e+02	4e+02	88.8	10
Smallpox	80	40	40	89.6	1.5e+02	54	64	90.4	19
Tuberculosis	7.7	0.32	0.36	88.6	25	24	23	90.2	15
Pneumonia	0.14	0.44	0.43	90.0	1.2e+02	64	61	88.2	14

Table 4: Median half-width (in % of the actual value) of the 90 % simultaneous intervals at the literature values for PIR-FD6 (plain), PIWR (plain) and PIWR-RACC, the empirical PIWR-RACC coverage on 2,000 test problems, and the total time of the conformal pipeline per model. The pipeline covers four estimators, both noise levels, the tuning, 999 calibration and 2,000 test problems, and 1,000 noise realizations.

Figure 5: (a, b) Median half-width of the 90 % simultaneous intervals at the literature values for PIR-FD6 (plain), PIWR (plain) and PIWR-RACC. (c) Empirical coverage on 2,000 test problems for all models and noise levels, plain (filled) and adaptive (open); circles

\epsilon=0

, diamonds

\epsilon=1\,\%

. Target: 90 %.

		Nielsen et al. (2025)		$\epsilon=0$			$\epsilon=1\,\%$
Par.	Actual	PIR	PINN	PIR-FD2	PIR-FD6	PIWR	PIR-FD6	PIWR	PIWR-RACC 90 % interval
$\gamma_{1},\gamma_{2},\gamma_{3}$ known (setting of Nielsen et al.)
$\beta$	0.5	0.0378	0.248	0.0579	3.36e-06	1.76e-06	0.0562	0.0674	[0.497, 0.504]
$\phi_{1}$	0.05	0.111	1.86	0.149	1.4e-06	1.07e-05	0.595	0.444	[0.049, 0.0506]
$\phi_{2}$	0.05	0.0236	3.24	0.173	5.45e-06	1.83e-05	1.59	0.565	[0.0489, 0.0505]
$\theta$	0.1	0.0222	9.51	0.224	1.16e-05	3.19e-05	0.978	0.767	[0.0979, 0.104]

Table 5: S3I3R model ( Nielsen et al., 2025 ) : % errors of the estimates and PIWR-RACC 90 % simultaneous intervals (

\epsilon=1\,\%

). PIR and PINN: errors reported in Tables 1 and 4 of Nielsen et al. (2025) for noise-free data with

\gamma_{1,2,3}

known.

Method	$\beta$ [1/day]	$\gamma$ [1/day]	$\mathcal{R}_{0}$	coverage [%]	test err. [%]	fit [ms]
PIR-FD2	1.81 [-5.98, 9.61]	0.4 [-1.14, 1.94]	4.54 [-31.7, 40.8]	95.6 (94.7–96.4)	55.4	0.13
PIR-FD6	2.41 [-7.15, 12]	0.332 [-0.848, 1.51]	7.26 [-45.9, 60.5]	95.9 (94.8–96.9)	57.8	0.13
PIWR	1.7 [-1.63, 5.02]	0.483 [-0.356, 1.32]	3.51 [-9.39, 16.4]	95.7 (95.2–96.4)	32.6	0.2
PIWR-RACC	1.7 [-1.36, 4.75]	0.483 [-0.289, 1.26]	3.51 [-8.36, 15.4]	95.2 (94.3–95.5)	32.6	0.2
PIWR-VP	1.81 [1.09, 2.52]	0.456 [0.388, 0.524]	3.96 [2.89, 5.04]	94.4 (93.3–96.4)	7.56	2.3
PIWR-VP-RACC	1.81 [1.15, 2.47]	0.456 [0.393, 0.519]	3.96 [2.97, 4.96]	94.4 (93.3–96.6)	7.56	2.3

Table 6: Boarding-school influenza: estimates with 95 % simultaneous conformal intervals, mean coverage (range over five calibration/test splits), median % error on the test problems, and the time per fit, against the posterior mean and 95 % credible intervals of the Stan SIR model ( Grinsztajn et al., 2021 ) .

Figure 6: Boarding-school influenza. (a) Number of boys in bed, the SIR trajectory with the PIWR-VP parameters, the envelope of the trajectories whose parameters lie in the PIWR-VP-RACC 95 % set, and the trajectory at the Stan posterior mean. (b–d) 95 % intervals for

\beta

\gamma

and

\mathcal{R}_{0}

, where the dotted line is the Stan posterior mean. The finite-difference and free-coefficient PIWR intervals extend beyond the axes.

$\gamma_{1},\gamma_{2},\gamma_{3}$	$\beta$	$\phi_{1}$	$\phi_{2}$	$\theta$	nominal	within $\pm 20\,\%$
	90 % half-width [%]				coverage [%], test $\gamma_{i}$
estimated	1.8	34	33	2.4	89.3 (own population)
known exactly	0.68	1.6	1.6	2.7	90.9 (90.1–92.0)	2.4 (2.2–2.6)
known to $\pm 5\,\%$	4.1	9.4	9.6	17	100.0 (100.0–100.0)	59.0 (52.6–65.5)
known to $\pm 20\,\%$	8.2	19	19	33	100.0 (100.0–100.0)	89.9 (88.2–90.8)

Table 7: S3I3R with the recovery rates

\gamma_{1},\gamma_{2},\gamma_{3}

fixed at their nominal values in the estimator (PIWR,

1\,\%

noise): median half-width (in % of the actual value) of the 90 % simultaneous intervals at the literature values for different calibrations of

\gamma_{i}

, and mean coverage (range over five calibration/test splits) on test epidemics with the nominal

\gamma_{i}

and with

\gamma_{i}

within

\pm 20\,\%

. First row: all seven rates estimated ( table 5 ), coverage on its own calibration population.

	$\gamma$ used in				coverage [%], test $\gamma$
Variant	estimator	calibration	$\beta$ [1/day]	$\mathcal{R}_{0}$	$0.5$	$[0.4,0.6]$
$\gamma$ estimated (PIWR-VP-RACC)	–	$[0.228,0.684]$	1.81 $[1.15,2.47]$	3.96 $[2.97,4.96]$	94.4 (own population)
$\gamma=0.5$ known, plain	$0.5$	$0.5$	2.25 $[1.24,3.27]$	4.51 $[2.48,6.54]$	94.8 (93.8–95.8)	90.5 (88.2–92.6)
$\gamma=0.5$ known, RACC	$0.5$	$0.5$	2.25 $[0.925,3.59]$	4.51 $[1.85,7.17]$	95.1 (94.0–96.4)	94.2 (92.8–95.0)
$\gamma=0.5$ known, plain	$0.5$	$[0.4,0.6]$	2.25 $[1.14,3.37]$	4.51 $[2.28,6.74]$	97.1 (96.9–97.8)	95.2 (94.5–95.8)
$\gamma=0.5$ known, RACC	$0.5$	$[0.4,0.6]$	2.25 $[0.909,3.6]$	4.51 $[1.82,7.2]$	95.6 (94.7–96.2)	95.1 (94.5–95.7)

Table 8: Boarding-school influenza with prior knowledge of the recovery rate: estimates and 95 % simultaneous conformal intervals for

\beta

and

\mathcal{R}_{0}

, and mean coverage (range over five calibration/test splits) on test epidemics with

\gamma=0.5

and with

\gamma\in[0.4,0.6]

. First row:

\gamma

estimated ( table 6 ); its calibration range of

\gamma

is shown, and its coverage is measured on its own calibration population.

Case 1: constant $\beta$
	PINN		PIR
Error	joint	split	FD2	FD6	PIWR
$S$	$2.8\cdot 10^{-3}$	$6.2\cdot 10^{-4}$	$6.1\cdot 10^{-5}$	$6.1\cdot 10^{-5}$	$1.6\cdot 10^{-4}$
$I$	$3.8\cdot 10^{-3}$	$8.4\cdot 10^{-4}$	$2.6\cdot 10^{-4}$ †	$2.6\cdot 10^{-4}$ †	$2.0\cdot 10^{-4}$
$R$	$3.3\cdot 10^{-3}$	$7.4\cdot 10^{-4}$	$4.9\cdot 10^{-5}$	$4.9\cdot 10^{-5}$	$1.7\cdot 10^{-4}$
$\beta$	$5.1\cdot 10^{-3}$	$3.6\cdot 10^{-4}$	$8.9\cdot 10^{-4}$	$5.6\cdot 10^{-5}$	$9.7\cdot 10^{-5}$

Table 9: Time-varying transmission ( Millevoi et al., 2024 ) : relative 2-norm errors at 90 equispaced times and CPU time per fit for the joint and split PINNs ( Millevoi et al., 2024 , Tables 2–4) and for PIR-FD2, PIR-FD6 and PIWR with a B-spline expansion of

\beta(t)

(medians over 100 noise realizations). † PIR uses the data for

I

;

S

and

R

follow from their integral. The reference curves of Cases 2–4 are digitized (see text); the hardware differs.

Figure 7: Time-varying transmission, Cases 2 and 3 of Millevoi et al. (2024) , one noise realization. (a, b) Poisson counts of the infected, exact solution and PIWR fit. (c, d) Reference

\beta(t)

and

\mathcal{R}_{t}

, the PIR-FD6 and PIWR estimates, and their

90\,\%

simultaneous RACC bands after day 20.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Par.	Actual	Found	% err.	Found	% err.	90 % interval	Found	% err.	90 % interval
		DINN		PIWR / PIWR-RACC, $\epsilon=0$			PIWR / PIWR-RACC, $\epsilon=0.01$
$\alpha$	0.191	0.1932	1.15	0.191	0.000105	[0.191, 0.191]	0.1908	0.124	[0.1882, 0.1934]
$\beta$	0.05	0.0501	0.2	0.05	0.000108	[0.05, 0.05]	0.04976	0.487	[0.04881, 0.05071]
$\gamma$	0.0294	0.0297	1.02	0.0294	7.05e-05	[0.0294, 0.0294]	0.02926	0.46	[0.02865, 0.02988]

Appendix

Table 10: COVID-19 (SIDR): PIWR estimates and PIWR-RACC 90 % simultaneous intervals, and the DINN values are from Shaier et al. (2022) .

		DINN		PIWR / PIWR-RACC, $\epsilon=0$			PIWR / PIWR-RACC, $\epsilon=0.01$
Par.	Actual	Found	% err.	Found	% err.	90 % interval	Found	% err.	90 % interval
$s$	10	10	0.00751	10.19	1.89	[7.79, 12.59]	16.77	67.7	[-89.76, 123.3]
$\mu_{I}$	0.26	0.2613	0.489	0.2602	0.0806	[0.2592, 0.2612]	0.2782	7	[0.2313, 0.3251]
$\mu_{b}$	0.24	0.2417	0.728	0.2405	0.214	[0.2339, 0.2472]	0.2545	6.04	[0.04948, 0.4595]
$\mu_{V}$	2.4	2.42	0.83	2.404	0.181	[2.345, 2.464]	2.536	5.66	[0.7169, 4.355]
$r$	0.03	0.03061	2.02	0.0232	22.7	[-0.05083, 0.09724]	-0.124	513	[-3.335, 3.088]

Appendix

Table 11: HIV: PIWR estimates and PIWR-RACC 90 % simultaneous intervals, and the DINN values are from Shaier et al. (2022) .

Par.	Actual	Found	% err.	Found	% err.	90 % interval	Found	% err.	90 % interval
		DINN		PIWR / PIWR-RACC, $\epsilon=0$			PIWR / PIWR-RACC, $\epsilon=0.01$
$\chi_{1}$	0.06	0.0554	7.72	0.05598	6.7	[0.03761, 0.07436]	0.05614	6.43	[0.02666, 0.08563]
$\chi_{2}$	0.04	0.038	4.92	0.04017	0.418	[0.02454, 0.05579]	0.04233	5.83	[0.01734, 0.06732]
$\epsilon_{1}$	0.975	0.9839	0.909	1.046	7.3	[0.6453, 1.447]	1.039	6.58	[0.3989, 1.679]
$\epsilon_{2}$	0.3	0.2841	5.28	0.3224	7.48	[-1.355, 2]	0.1372	54.3	[-2.54, 2.815]
$\rho$	0.975	0.9759	0.0882	0.9767	0.17	[0.9543, 0.999]	0.9762	0.126	[0.9396, 1.013]

Appendix

Table 12: Smallpox: PIWR estimates and PIWR-RACC 90 % simultaneous intervals, and the DINN values are from Shaier et al. (2022) .

		DINN		PIWR / PIWR-RACC, $\epsilon=0$			PIWR / PIWR-RACC, $\epsilon=0.01$
Par.	Actual	Found	% err.	Found	% err.	90 % interval	Found	% err.	90 % interval
$\delta$	500	509.5	1.86	500.2	0.0343	[499.4, 501]	484.8	3.04	[435.1, 534.5]
$\beta$	13	12.54	3.63	13.01	0.0432	[12.98, 13.03]	12.49	3.92	[11, 13.98]
$\mu$	0.143	0.1474	3.01	0.1429	0.037	[0.1428, 0.1431]	0.1458	1.98	[0.1335, 0.1582]
$k$	0.5	0.5396	7.34	0.5001	0.0164	[0.4934, 0.5068]	0.5489	9.77	[0.125, 0.9728]
$r_{1}$	2	1.989	0.542	2.002	0.0819	[1.994, 2.009]	1.829	8.53	[1.377, 2.281]

Appendix

Table 13: Tuberculosis (SLIT): PIWR estimates and PIWR-RACC 90 % simultaneous intervals, and the DINN values are from Shaier et al. (2022) .

		DINN		PIWR / PIWR-RACC, $\epsilon=0$			PIWR / PIWR-RACC, $\epsilon=0.01$
Par.	Actual	Found	% err.	Found	% err.	90 % interval	Found	% err.	90 % interval
$\pi$	0.01	0.0098	2	0.06304	530	[-0.116, 0.242]	1.839	1.83e+04	[-23.99, 27.67]
$\lambda$	0.1	0.099	0.962	0.09996	0.036	[0.09987, 0.1001]	0.09767	2.33	[0.08431, 0.111]
$\epsilon$	0.002	0.0022	11.7	0.002238	11.9	[0.001662, 0.002813]	0.0204	920	[-0.0651, 0.1059]
$\phi$	0.0025	0.0027	7.49	0.002465	1.4	[0.002362, 0.002568]	-0.002245	190	[-0.01684, 0.01235]
$\chi$	0.001	0.0011	6.74	0.000959	4.1	[0.0007445, 0.001173]	-0.002748	375	[-0.03821, 0.03272]

Appendix

Table 14: Pneumonia (SVCIR): PIWR estimates and PIWR-RACC 90 % simultaneous intervals, and the DINN values are from Shaier et al. (2022) .

		DINN		PIWR / PIWR-RACC, $\epsilon=0$			PIWR / PIWR-RACC, $\epsilon=0.01$
Par.	Actual	Found	% err.	Found	% err.	90 % interval	Found	% err.	90 % interval
$\beta_{1}$	3.532	3.559	0.762	3.53	0.0598	[3.526, 3.534]	4.182	18.4	[2.779, 5.585]
$\beta_{h}$	0.012	0.0129	7.81	-0.02367	297	[-0.08456, 0.03722]	0.5563	4.54e+03	[-12.45, 13.57]
$\beta_{f}$	0.462	0.4638	0.398	0.8687	88	[0.17, 1.567]	-23.29	5.14e+03	[-187.1, 140.5]
$\alpha$	0.08333	0.0866	3.93	0.08333	3.41e-05	[0.08333, 0.08333]	0.08334	0.0134	[0.08032, 0.08637]
$\gamma_{h}$	0.2381	0.2471	3.79	0.2381	9.95e-06	[0.2381, 0.2381]	0.2379	0.0721	[0.2243, 0.2516]

Appendix

Table 15: Ebola (SEIHFR): PIWR estimates and PIWR-RACC 90 % simultaneous intervals, and the DINN values are from Shaier et al. (2022) .

		DINN		PIWR / PIWR-RACC, $\epsilon=0$			PIWR / PIWR-RACC, $\epsilon=0.01$
Par.	Actual	Found	% err.	Found	% err.	90 % interval	Found	% err.	90 % interval
$\pi_{h}$	10	9.932	0.683	10	0.000997	[9.992, 10.01]	9.964	0.357	[6.964, 12.96]
$\pi_{v}$	30	29.85	0.486	30	0.00301	[29.97, 30.03]	29.85	0.499	[20.53, 39.17]
$\lambda_{h}$	0.0553	0.0552	0.27	0.0553	1.24e-05	[0.05524, 0.05536]	0.0558	0.907	[0.03434, 0.07727]
$\delta_{h}$	0.99	0.9643	2.6	0.9919	0.194	[0.9895, 0.9944]	0.5389	45.6	[-0.009542, 1.087]
$\delta_{v}$	0.057	0.0567	0.529	0.057	0.00302	[0.05695, 0.05705]	0.05665	0.606	[0.0393, 0.07401]

Appendix

Table 16: Dengue: PIWR estimates and PIWR-RACC 90 % simultaneous intervals, and the DINN values are from Shaier et al. (2022) .

Par.	Actual	Found	% err.	Found	% err.	90 % interval	Found	% err.	90 % interval
		DINN		PIWR / PIWR-RACC, $\epsilon=0$			PIWR / PIWR-RACC, $\epsilon=0.01$
$r$	0.003333	0.0034	1.2	-0.1327	4.08e+03	[-15.16, 14.89]	-0.1364	4.19e+03	[-0.4602, 0.1873]
$\mu$	0.001667	0.0017	0.575	-0.1343	8.16e+03	[-15.16, 14.89]	-0.1381	8.39e+03	[-0.4619, 0.1856]
$\kappa$	0.1	0.1025	2.54	0.0409	59.1	[-30.33, 30.42]	0.03519	64.8	[-0.7564, 0.8268]
$\eta_{a}$	0.5	0.5035	0.702	-1.957	491	[-582, 578.1]	0.4914	1.72	[-12.2, 13.18]
$\eta_{c}$	0.1	0.1024	2.45	0.06861	31.4	[-9.067, 9.204]	0.02374	76.3	[-0.1683, 0.2157]

Appendix

Table 17: Anthrax (SIAC): PIWR estimates and PIWR-RACC 90 % simultaneous intervals, and the DINN values are from Shaier et al. (2022) .

Par.	Actual	Found	% err.	Found	% err.	90 % interval	Found	% err.	90 % interval
		DINN		PIWR / PIWR-RACC, $\epsilon=0$			PIWR / PIWR-RACC, $\epsilon=0.01$
$\mu$	0.02	0.02	0.02	0.02001	0.0359	[0.01869, 0.02132]	0.02006	0.29	[0.01709, 0.02303]
$\alpha$	0.5	0.5018	0.36	0.5002	0.0361	[0.4681, 0.5323]	0.5027	0.542	[0.43, 0.5754]
$\gamma_{a}$	18	18.02	0.417	18.01	0.0469	[16.83, 19.19]	18.07	0.364	[15.42, 20.71]
$\gamma_{c}$	36	36.07	0.359	36	0.01	[29.2, 42.81]	33.28	7.56	[18.2, 48.35]
$\beta_{aa}$	40	40.25	0.627	40.01	0.0207	[37.08, 42.94]	40.59	1.48	[34.08, 47.11]

Appendix

Table 18: Polio: PIWR estimates and PIWR-RACC 90 % simultaneous intervals, and the DINN values are from Shaier et al. (2022) .

Par.	Actual	Found	% err.	Found	% err.	90 % interval	Found	% err.	90 % interval
		DINN		PIWR / PIWR-RACC, $\epsilon=0$			PIWR / PIWR-RACC, $\epsilon=0.01$
$\mu$	0.02	0.0225	12.7	0.01834	8.31	[0.01411, 0.02257]	0.01882	5.92	[-0.07752, 0.1152]
$\beta_{0}$	50.54	–	–	47.5	6.01	[43.23, 51.77]	48.66	3.71	[-30.79, 128.1]
$\beta_{1}$	0.28	0.27	3.57	0.09722	65.3	[-0.2567, 0.4511]	0.2006	28.4	[-7.031, 7.432]
$\gamma$	100	97	3	90.61	9.39	[75.66, 105.6]	29.45	70.5	[-203.7, 262.6]
$\sigma$	35.84	34.71	3.15	32.46	9.44	[27.09, 37.82]	10.53	70.6	[-73.46, 94.52]

Appendix

Table 19: Measles (seasonal SEI): PIWR estimates and PIWR-RACC 90 % simultaneous intervals, and the DINN values are from Shaier et al. (2022) .

		DINN		PIWR / PIWR-RACC, $\epsilon=0$			PIWR / PIWR-RACC, $\epsilon=0.01$
Par.	Actual	Found	% err.	Found	% err.	90 % interval	Found	% err.	90 % interval
$b$	0.4	0.4033	0.83	0.5677	41.9	[-39.45, 40.59]	-1.459	465	[-125.4, 122.5]
$c$	0.5	0.5015	0.309	0.4683	6.33	[0.416, 0.5207]	0.4627	7.47	[0.2723, 0.653]
$\eta$	0.1	0.0999	0.0687	0.09803	1.97	[0.08887, 0.1072]	0.09615	3.85	[0.06212, 0.1302]
$\beta$	0.05	0.0498	0.41	-0.07811	256	[-8.387, 8.23]	0.1867	273	[-25.54, 25.92]
$\kappa$	0.6	0.6033	0.549	-1.354	326	[-83.54, 80.84]	-0.3063	151	[-39.06, 38.45]

Appendix

Table 20: Zika: PIWR estimates and PIWR-RACC 90 % simultaneous intervals, and the DINN values are from Shaier et al. (2022) .

Spatial Epidemic Modeling

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GeoID-PINN: Identifiability-Aware Regional Epidemic Inference with Geographic Coupling
Jul 28, 2026Weixiong Hua, Fan BuEpidemic Forecasting Parameter Identifiability
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GeoID-PINN: Identifiability-Aware Regional Epidemic Inference with Geographic Coupling

Weixiong Hua, Fan Bu

Department of Biostatistics · University of Michigan · Ann Arbor, Michigan, USA

Epidemic Forecasting Parameter Identifiability Physics-Informed ML Spatial Epidemic Modeling

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Regional surveillance data reflect local transmission, reporting, seeding, and external infection pressure, which are difficult to identify separately. We introduce GeoID-PINN, a physics-informed neural network (PINN) for susceptible-infectious-recovered-deceased (SIRD) dynamics. The model represents spatial dependence with a row-stochastic source-composition matrix whose rows assign nonnegative source weights that sum to one. We regularize this matrix toward a spatial prior constructed from distance, adjacency, commuting, or lead-lag information. In a four-region simulation with known truth, a compatible distance prior gives source-composition error 0.099. The error rises to 0.159 without regularization and 0.577 under a strongly misspecified prior, while trajectory fit and transmission-scale estimates remain similar. Accurate trajectories therefore do not guarantee recovery of the regional dependence structure. We also evaluate GeoID-PINN retrospectively using COVID-19 data from 64 Louisiana counties. Relative to an autoregressive negative-binomial baseline, Forecast-Trained Geo-PINN reduces mean squared error (MSE) from 32,957 to 11,468 and mean absolute error (MAE) from 70.60 to 57.73. The baseline has lower negative log likelihood (NLL), 5.158 versus 5.346, indicating better distributional fit but worse point accuracy. In a controlled 15-county comparison, county adjacency reduces MSE by 6.85 percent and MAE by 3.1 percent. Similar performance across plausible priors supports structured regularization but not unique edge recovery. These results require prior-sensitivity and observation-model checks before interpretation.

Abstract

Spatial Epidemic Modeling

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