Conditional Inverse Learning of Time-Varying Reproduction Numbers Inference

Lanlan Yu; Quan-Hui Liu; Haoyue Zheng; Xinfu Yang

cs.LGMar 18, 2026

Conditional Inverse Learning of Time-Varying Reproduction Numbers Inference

Authors: Lanlan Yu, Quan-Hui Liu, Haoyue Zheng, Xinfu Yang

Organizations: College of Computer Science, Sichuan University, Chengdu, China · Engineering Research Center of Machine Learning and Industry Intelligence, Ministry of Education, Sichuan University, Chengdu, China

Abstract

Estimating time-varying reproduction numbers from epidemic incidence data is a central task in infectious disease surveillance, yet it poses an inherently ill-posed inverse problem. Existing approaches often rely on strong structural assumptions derived from epidemiological models, which can limit their ability to adapt to non-stationary transmission dynamics induced by interventions or behavioral changes, leading to delayed detection of regime shifts and degraded estimation accuracy. In this work, we propose a Conditional Inverse Reproduction Learning framework (CIRL) that addresses the inverse problem by learning a {conditional mapping} from historical incidence patterns and explicit time information to latent reproduction numbers. Rather than imposing strongly enforced parametric constraints, CIRL softly integrates epidemiological structure with flexible likelihood-based statistical modeling, using the renewal equation as a forward operator to enforce dynamical consistency. The resulting framework combines epidemiologically grounded constraints with data-driven temporal representations, producing reproduction number estimates that are robust to observation noise while remaining responsive to abrupt transmission changes and zero-inflated incidence observations. Experiments on synthetic epidemics with controlled regime changes and real-world SARS and COVID-19 data demonstrate the effectiveness of the proposed approach.

Figures & tables

Figure 1: Overview of Conditional Inverse Reproduction Learning (CIRL) framework

Method	Single-step				Double-step				Double-step(masked)
Method	$\Delta_{\text{delay}}$	MDR	RMSE	MAE	$\Delta_{\text{delay}}$	MDR	RMSE	MAE	$\Delta_{\text{delay}}$	MDR	RMSE	MAE
EpiEstim	6.00 [5.00,6.00]	0.00	0.36 [0.31,0.42]	0.22 [0.19,0.26]	6.00[5.00,6.00]	0.00	0.40 [0.37,0.49]	0.27 [0.24,0.31]	6.00 [-9.00,12.00]	0.00	0.83[0.57,1.76]	0.50[0.38,0.74]
EpiNow2	8.00[7.00,11.00]	0.00	0.35 [0.32,0.39]	0.22 [0.20,0.27]	47.00[27.50,49.00]	0.01	0.45 [0.37,0.59]	0.36[0.27,0.47]	52.00[50.00,55.00]	0.03	0.57 [0.52,0.62]	0.49 [0.45,0.52]
UDENet	-39.00[-39.50,7.00]	0.97	0.60[0.47,2.04]	0.52[0.40,1.86]	-29.00[-29.20,-28.00]	0.97	1.53[0.55,2.17]	1.35[0.51,2.03]	$\sim$	1.00	0.61[0.55,1.93]	0.56[0.51,1.75]
CIRL(ours)	1.00 [0.00,1.00]	0.01	0.38[0.32, 0.48]	0.28 [0.23,0.33]	1.00 [0.00,2.00]	0.01	0.48[0.36,0.56]	0.33 [0.26,0.39]	5.00 [2.00,9.00]	0.02	0.51 [0.44,0.58]	0.36 [0.31,0.42]

Table 1: Quantitative comparison of different methods on synthetic benchmarks over 100 simulations. For continuous metrics (RMSE, MAE, and

\Delta_{\text{delay}}

), results are reported as Median [

Q_{1},Q_{3}

]. Bold values indicate the best performance in each column, and underlined values indicate the second-best.

Figure 2: Median and interquartile range of the inferred

R_{t}

over 100 simulations. Solid lines denote the median estimate across 100 simulations, while shaded regions indicate the interquartile range (25%–75%). (a)

R_{t}

estimation across single-step change scenarios, while (b) estimation on double-step change scenarios. .

Panel of Figure 3 — Figure 3: Estimated reproduction number trajectories under different observation noise conditions. Both panels show estimates obtained from a single synthetic epidemic with smooth oscillation dynamics. (a) corresponds to incidence generated from a Poisson process, while (b) illustrates the same epidemic after introducing proportional zero-inflated mask in the observations. The comparison highlights the robustness of the inferred transmission dynamics to observation sparsity.

Method	$\Delta_{\text{delay}}$	MDR	RMSE	MAE
Full CIRL	5.00 [2.00,9.00]	0.02	0.51 [0.44,0.58]	0.36 [0.31,0.42]
w/o TCE	3.00 [1.00,7.75]	0.02	0.72[0.59,0.83]	0.48 [0.40,0.54]
w/o HIE	-9.00[-10.00,-7.00]	0.018	0.67 [0.66,0.68]	0.54[0.53,0.55]
w/ MSE	-7.00[-11.25,-3.00]	0.00	0.75[0.67,0.84]	0.52[0.45,0.61]
w/ Poisson	-10.00[-15.00,-5.00]	0.00	0.78[0.65,0.88]	0.51[0.43,0.58]

Table 2: Quantitative comparison on synthetic double-regime-change epidemics with proportional zero-inflated mask in the obserations. All regression metrics are reported as Median [

Q_{1},Q_{3}

] over 100 simulations.

Panel of Figure 4 — Figure 4: Evaluation during the 2003 SARS in Hong Kong. (a) Retrospective $R_{t}$ estimation by CIRL and baseline methods, with CIRL additionally generating short-term future $R_{t}$ trajectories. (b) Incidence fitting and short-horizon projection implied by CIRL. The comparison highlights the robustness of the inferred transmission dynamics to observation sparsity.

Panel of Figure 5 — Figure 5: Evaluation on the Ontario COVID-19 first wave. (a) Comparison of retrospective $R_{t}$ estimates. (b) Incidence fitting and implied projection. The results provide a qualitative assessment of the internal consistency of the learned conditional inverse mapping under noisy and non-stationary real-world observations.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 6 — Figure 6: Visual analysis of the solitary failure case. (a) shows the synthetic incidence data, characterized by extreme sparsity and low counts ( $y_{max}\leq 4$ ). (b) exhibits the estimated $R_{t}$ trajectory. Due to the indistinguishability between the weak transmission signal and background noise, the model conservatively maintained a flat trend, resulting in a missed detection of the $R_{t}$ shift.

Datasets	SARS 2003		COVID-19
Datasets	RMSE	MAE	RMSE	MAE
Fitting	8.62	5.05	45.34	33.58
Prediction	1.51	1.35	37.41	27.84

Appendix

Table 3: Evaluation of incidence reconstruction and short-term forecasting accuracy. RMSE and MAE are reported for both in-sample fitting and rolling prospective forecasts of incidence (

I_{t}

). The fitting error measures how well the inferred transmission dynamics reconstruct the observed data, while the rolling forecast error assesses the temporal consistency and generalization of the learned conditional inverse mapping.

Dataset Scenario	Comparison Target	CIRL(Ours)	EpiNow2	EpiEstim
Scenario-1	$\lambda_{true}$	10.04 [ 6.03,14.10]	8.45 [ 5.21,12.03]	10.26[ 6.31,14.06]
Scenario-2	$\lambda_{true}$	7.13[ 4.66,10.81]	5.93 [ 4.12, 8.45]	7.03 [ 4.88,10.44]
Scenario-3	$\lambda_{true}$	7.04 [ 4.86,10.39]	6.80 [ 4.49,10.66]	7.83[ 5.34,12.08]

Appendix

Table 4: Reconstruction Performance. RMSE of estimated incidence

\hat{I}_{t}

against the true Poisson rate

\lambda_{true}

. Values are reported as the Median

[Q_{1},Q_{3}]

Method	Fitting to Noisy Observations			Internal Consistency			Reconstruction
	(Target: $I_{obs}$ )			(Target: $\lambda_{renewal}$ )			(Target: $I_{raw}$ )
	Scenario-1	Scenario-2	Scenario-3	Scenario-1	Scenario-2	Scenario-3	Scenario-3
EpiEstim	0.00 [0.00,0.00]	0.00 [0.00,0.00]	0.00 [0.00,0.00]	4.41[3.37,5.87]	3.90[3.01,5.31]	4.76[3.56,7.50]	3.16 [1.90,5.19]
EpiNow2	3.91[2.87,4.79]	3.40[2.71,4.66]	4.53[3.26,7.17]	3.31 [2.11,4.51]	2.78 [1.99,3.75]	3.59 [2.35,5.77]	4.01[3.05,6.02]
CIRL(Ours)	1.94 [1.55,2.42]	1.89 [1.58,2.47]	3.67 [2.51, 5.86]	3.62 [2.66,5.35]	3.31 [2.35,4.85]	3.03 [2.23,4.32]	2.44 [1.91,3.06]

Appendix

Table 5: Reconstruction Metrics. Values are Median [

Q_{1},Q_{3}

]. Left (Fitting to Noisy Observations): Near-zero error on

I_{obs}

indicates overfitting to noise. Middle (Internal Consistency): Low error against

\lambda_{renewal}

confirms mathematical validity. Right (Reconstruction / Counterfactual Recovery): Evaluated solely on Scenario-3 , measuring the error against the unmasked truth (

I_{raw}

). Note that while baselines fit

I_{obs}

well, they fail to recover the hidden cases.

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

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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.

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