Conditional Inverse Learning of Time-Varying Reproduction Numbers Inference

Conditional In verse Lear ning of Time-V arying Reproduction Numbers Infer ence Lanlan Y u 1 2 Quan-Hui Liu 1 2 Haoyue Zheng 1 2 Xinfu Y ang 1 2 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 in verse problem. Existing ap- proaches often rely on strong structural assump- tions deri ved from epidemiological models, which can limit their ability to adapt to non-stationary transmission dynamics induced by interventions or behavioral changes, leading to delayed detec- tion of regime shifts and de graded estimation ac- curacy . In this work, we propose a Conditional In- verse Reproduction Learning frame work (CIRL) that addresses the in verse problem by learning a conditional mapping from historical incidence pat- terns and explicit time information to latent repro- duction numbers. Rather than imposing strongly enforced parametric constraints, CIRL softly in- tegrates epidemiological structure with fle xible likelihood-based statistical modeling, using the rene wal equation as a forward operator to enforce dynamical consistency . The resulting framew ork combines epidemiologically grounded constraints with data-driven temporal representations, produc- ing reproduction number estimates that are robust to observ ation noise while remaining responsi ve to abrupt transmission changes and zero-inflated incidence observations. Experiments on synthetic epidemics with controlled regime changes and real-world SARS and CO VID-19 data demon- strate the effecti veness of the proposed approach. 1. Introduction The estimation of the time-v arying reproduction number, R t , is fundamental to characterizing the dynamics of infec- tious disease outbreaks and guiding public health interv en- tions ( W allinga & T eunis , 2004 ; Fraser et al. , 2011 ; Gostic 1 College of Computer Science, Sichuan Uni versity , Chengdu, China 2 Engineering Research Center of Machine Learning and In- dustry Intelligence, Ministry of Education, Sichuan Uni versity , Chengdu, China. Correspondence to: Quan-Hui Liu < quan- huiliu@scu.edu.cn > . Pr eprint. Marc h 19, 2026. et al. , 2020 ; T revisin et al. , 2023 ). V alues of R t abov e or be- low the critical threshold of 1 indicate expanding or declin- ing epidemics, respecti vely ( W allinga & Lipsitch , 2007 ; Liu et al. , 2020 ). From a mechanistic perspecti ve, the transmis- sion dynamics is go verned by the rene wal process ( K ermack & McKendrick , 1927 ). The expected incidence at time t , denoted as E [ I t ] , is modeled as the product of the instanta- neous reproduction number and the total infectiousness of the population ( Cori et al. , 2013 ): E [ I t ] = R t ∞ X τ =1 I t − τ w ( τ ) , (1) where I t denotes the number of ne wly infected indi viduals in discrete time t , the time-varying reproduction number R t denotes the ratio of the number of new infections generated at time step t , and w ( τ ) represents the serial (or generation) interval distrib ution, satisfying P ∞ τ =1 w ( τ ) = 1 . This equa- tion defines a forward operator mapping latent transmission dynamics to the observed cases ( Dai et al. , 2023 ). Since R t is unobserved, inferring it from noisy and poten- tially delayed incidence data I = { I t } T t =1 constitutes a classic ill-posed in verse pr oblem ( Benning & Burger , 2018 ; Parag , 2021 ). Multiple latent trajectories may explain the same observ ations, and small perturbations in the data can induce large v ariations in inferred dynamics. It is therefore necessary to constrain the solution space in order to recov er meaningful latent transmission trajectories. ( W allinga & T eunis , 2004 ; Cori et al. , 2013 ; Abbott et al. , 2020 ). T o mitigate this ill-posedness, standard approaches typi- cally impose har d constraints —rigid structural assumptions designed to force a unique solution ( W allinga & T eunis , 2004 ; Cori et al. , 2013 ; Abbott et al. , 2020 ; Song & Xiao , 2022 ). Ho wever , recent theoretical analyses indicate that hard-constrained or purely deterministic R t estimation can be problematic, since rigid assumptions are incompatible with non-stationary transmission dynamics ( Ghosh et al. , 2023 ), especially in scenarios with early missing observa- tions and sudden shifts dri ven by interventions ( Dai et al. , 2023 ; Lau et al. , 2021 ). For instance, the local stationarity assumption inherent in sliding-windo w methods (e.g., Epi- Estim) fundamentally contradicts the instantaneous changes of intervention-induced discontinuities ( Cori et al. , 2013 ; Parag , 2021 ). Similarly , neural methods rooted in dif feren- tial equations often implicitly enforce e xponential genera- 1 Conditional In verse Learning of Time-V arying Repr oduction Numbers Inference tion intervals (memoryless dynamics) ( Song & Xiao , 2022 ), div erging from the non-Marko vian and context-dependent nature of real transmission processes( Pang et al. , 2025 ). When transmission dynamics change abruptly , such struc- tural mismatches can cause model to conflate transient re- porting artifacts with genuine epidemiological signals, lead- ing to detection lags and smoothed trajectories ( Nash et al. , 2022 ). This failure mode mirrors con vergence pathologies frequently observed in scientific machine learning when physical laws are enforced on imperfectly observed sys- tems ( Karniadakis et al. , 2021 ; Lu et al. , 2021 ). Second, deterministic frame works are ill-equipped to han- dle stochasticity arising from missing observations, leading to poorly calibrated uncertainty ( W ang & W alker , 2025 ). Observation processes in realistic surveillance are distorted by structural zeros (e.g., due to limited diagnostic capacity) and stochastic delays, which rigid or purely deterministic optimization tends to misinterpret as meaningful transmis- sion changes ( Parag , 2021 ; Abbott et al. , 2020 ; Lison et al. , 2024 ). While probabilistic frame works like EpiNo w2 ( Ab- bott et al. , 2020 ) introduce smoothness priors to mitigate noise, they typically lack explicit mechanisms to model these systematic reporting failures. This limitation results in poorly calibrated uncertainty quantification, particularly in low-incidence re gimes where the model produces danger- ously narrow confidence interv als around potentially biased estimates ( Lison et al. , 2024 ; Hettinger et al. , 2025 ). In this work, we advocate a shift from rigid structural as- sumptions tow ard a conditional inver se learning ( Arridge et al. , 2019 ; Ongie et al. , 2020 ) perspectiv e for estimat- ing time-varying reproduction numbers, which we term Conditional In verse Reproduction Learning (CIRL) . In- stead of imposing piecewise-constant dynamics or window- based smoothing constraints on R t , we reformulate the problem as learning a flexible conditional in verse mapping, R t = f θ ( t, H t ) , conditioned on the recent explicit time coordinate t and epidemiological history H t . Under this formulation, the admissible solution space is implicitly con- strained by the conditioning structure, rather than by hard- coded temporal assumptions. CIRL consists of two key components. (i) A Conditional In verse Mapping Network that parameterizes f θ ( · ) using a neural architecture conditioned on historical observations and time. (ii) A Statistical Observation and Consistency Module , which replaces deterministic fitting targets with a probabilistic objectiv e to accommodate the heterogene- ity and sparsity of real-world surveillance data, including ov er-dispersion, reporting artifacts, and early missing obser- vations. This soft consistenc y constraint suppresses noise- induced high-frequency oscillations while retaining sensi- tivity to abrupt transmission changes when supported by sufficient e vidence in the conditioning history . Since con- ditional in verse mapping depends only on past incidence, the frame work can be applicable in prospecti ve estimation settings, while naturally enabling short-term forward simu- lation as an auxiliary check of epidemiological consistency . Our key contrib utions are summarized as follows: • W e reformulate R t estimation as a pr obabilistic con- ditional inver se problem under the renew al equation, av oiding hard-coded temporal assumptions on R t . • W e introduce a likelihood-based consistency formu- lation that replaces deterministic fitting targets, en- abling principled handling of heterogeneous and sparse surveillance data. • W e empirically v alidate the proposed framework on synthetic epidemics and real epidemic data, demon- strating robust in v erse inference under under -reporting, missing observ ations, and abrupt transmission changes. 2. Problem F ormulation The time-v arying reproduction number ( R t ) quantifies the av erage number of secondary infections generated by in- fectious individuals at time t . W e formulate the epidemic dynamics as a discrete-time process with observed incidence I = { I t } T t =1 , where I t ∈ N denotes the number of newly re- ported cases at time t . The goal is to recover the underlying time-varying reproduction numbers, denoted as R = { R t } , which are treated as latent, time-v arying variables inferred from the observables I . Forward Renewal Equation. The relationship between latent transmission trajectories R and observed cases I is gov erned by the renewal equation: I t ∼ D ( λ t ) , (2) where D denotes a discrete count distribution with expecta- tion λ t . The term λ t denotes the conditional expectation: λ t = R t t − 1 X τ =1 I t − τ w τ , (3) where w τ is the empirical generation interval distribution with P τ w τ = 1 . This defines a forward mapping F : R → I , which is both man y-to-one and one-to-many . Gi ven the renew al intensity λ t , the observed incidence I t is modeled as a noisy count process centered at λ t , potentially subject to reporting noise and ov er-dispersion. Conditional In verse Mapping. Given observed incidence I and a known generation interv al distribution { w τ } , esti- mating R t corresponds to the in verse recovery under the renew al equation. This problem is inherently ill-posed, since 2 Conditional In verse Learning of Time-V arying Repr oduction Numbers Inference multiple R t trajectories can produce identical incidence se- quences. T o address this ambiguity , our objectiv e is to learn a conditional in verse mapping F − 1 : I → R that infers latent R t sequences consistent with the observed epidemic dynamics while le veraging past incidence as conditioning information, as follows: F − 1 : R t = f θ ( t, H t ) , (4) where f θ is a parameterized in verse mapping learned from the data, t denotes the explicit time coordinate, and H t = { I t − L , . . . , I t − 1 } defines a historical context (recepti ve field) of size L .Accordingly , Eq. ( 4 ) applies only to t > L , as the initial window lacks sufficient historical context to infer R t . It is worth emphasizing that our conditioning horizon L solely defines the temporal context av ailable to the conditional in verse mapping, and does not impose any piecewise-constant, local smoothness, or stability assump- tions on R t . Under this formulation, R t inference is formu- lated as a conditional inv erse learning problem grounded in epidemiological renew al dynamics. 3. Method W e propose a Conditional In verse Repr oduction Learning (CIRL) frame work that learns a parameterized conditional in verse mapping f θ associated with the renew al operator F to infer R t from observed incidence data. As illustrated in Figure 1 , f θ is conditioned on the retrospective epidemic history H t and the temporal coordinate t , and outputs latent transmission dynamics R , from which the expected inci- dence E [ I t ] is obtained via the renew al equation (Eq. ( 1 )). Based on the architecture of f θ and the strategy for learning its parameters, the CIRL frame work is or ganized into tw o complementary components: (i) the Conditional In verse Mapping Network , which maps historical incidence and temporal context to latent transmission dynamics, and (ii) the Statistical Observation and Consistency Module , which enforces consistency with observed incidence through a probabilistic rene wal-based formulation. T ogether, these components enable flexible, data-driven inference of R t while respecting epidemiological dynamics. 3.1. Conditional In verse Mapping Network W e instantiate f θ using a modular neural architecture de- signed to capture non-stationarity , temporal dependence, and multi-scale epidemic dynamics. The conditional in verse mapping f θ ( t c , H t c ) is implemented as a modular neural architecture that decomposes the inference task into dis- tinct functional components. Specifically , the architecture consists of: (i) a temporal embedding module that encodes explicit time information, (ii) a historical incidence encoder that extracts multi-scale features from past observ ations, (iii) a fusion module that produces a latent reproduction number R t c at time t c . T emporal Coordinate Embedding (TCE). The explicit time index t c is embedded via a coordinate-based encoding, e t c = ϕ time ( ϕ FF ( t c )) , (5) where ϕ time ( · ) is a learnable embedding function imple- mented as a multilayer perceptron (MLP). Here, ϕ FF ( t c ) is a Fourier feature mapping about time t c : ϕ FF ( t c ) =  cos( απ t c ) sin( απ t c )  , (6) where α is a user-defined, non-trainable frequency vector . This mapping addresses spectral bias, enabling the model to efficiently capture both high-frequency v ariations and long-range temporal dependencies, going beyond local pat- terns to model global temporal effects and long-term non- stationarity ( Ning et al. , 2023 ). Historical Incidence Encoder (HIE). T o extract infor- mativ e representations from the historical incidence context H t c , we employ a sequence encoder: h t c = ϕ hist  [ H t c ; ∆ H t c ]  , (7) where ϕ hist is designed to capture epidemic dynamics at multiple temporal scales, via multi-scale T emporal Con vo- lutions Network (TCN) ( Szegedy et al. , 2015 ; Bai , 2018 ) with different recepti ve fields. In particular, to augment the representational capacity of the temporal encoder, we incor - porate the first-order temporal difference ∆ H t c ← { ∆ I t = I t − I t − 1 , t ≤ t c } as a secondary input channel. This explic- itly provides the model with instantaneous growth dynamics, enabling the TCN to ef fecti vely distinguish between dif fer- ent epidemic phases (e.g., peak deceleration vs. early-stage acceleration) ev en when absolute incidence magnitudes are comparable. T o capture diverse temporal dynamics, we use an Inception- style con volutional block with tw o parallel TCN branches employing different kernel sizes (e.g., K = { 3 , 7 } ). The short-term branch detects rapid fluctuations and localized outbreaks, while the long-term branch smooths noise and tracks stable trends. Both branches use dilated causal con vo- lutions for a large recepti ve field while preserving temporal causality , generating multi-scale feature maps F short t c and F long t c . The concatenated multi-scale features are projected into a sequence of embeddings, which are then processed by a T ransformer Encoder ( V aswani et al. , 2017 ): h t c = A ttn  [ F short t c ; F long t c ]  . (8) By leveraging the self-attention mechanism, the model learns to selectiv ely weight specific historical time points that are most informativ e for the current R t c state, effec- tiv ely capturing non-local temporal dependencies that are often missed by purely con volutional filters. 3 Conditional In verse Learning of Time-V arying Repr oduction Numbers Inference Conditional Inverse Mapping Network Statistical Observation and Consistency Module Historical Incidence Enco der Te m p o r a l C o o r d i n a t e E m b e d d i n g t Incidence Data ... I t-1 I t-2 I t-τ ... t-τ t-2 t-1 t I t H t Δ H t Short - kernel TCN Long - kernel TCN Attention Fourier Mapping Linear Layer Conditional Inverse Inference R t π t Renewal Equatio n Zero - Inflated Poisson (ZIP) likelihood Generation Interval Distributi on λ t H t Δ H t F igure 1. Overvie w of Conditional In verse Reproduction Learning (CIRL) framew ork Conditional In verse Inference of R t The temporal coor- dinate embedding e t c and historical incidence feature h t c are integrated through a nonlinear fusion operator: z t c = ϕ fusion ( e t c , h t c ) , (9) where ϕ fusion ( · ) is implemented as feature concatenation follo wed by a Cross-Attention module ( V aswani et al. , 2017 ) to capture interactions between global temporal context and local historical dynamics. The latent reproduction number is then obtained as R t c = f θ ( t c , H t c ) = ϕ out ( z t c ; t c , H t c , θ ) , (10) where ϕ out ( · ) is implemented as an MLP followed by a positivity-preserving acti vation function. This design in- troduces a low-dimensional bottleneck between historical observations and transmission dynamics, which regularizes the in verse mapping and improv es estimation stability . 3.2. Statistical Observation and Consistency Module Giv en the conditional in verse mapping in Eq. ( ( 10 ) ), here, we describe ho w the parameters θ of f θ ( t c , H t c ) are learned from observed incidence using a physics-informed statistical objecti ve, enabling accurate inference of the latent reproduc- tion numbers. T o this end, the Statistical Observation and Consistency Module defines a probabilistic, rene wal-based likelihood that enforces coherence between the inferred R t trajectory and observed incidence, while robustly handling ov er-dispersion, reporting noise, and observation gaps via a zero-inflated objectiv e. By optimizing the network parame- ters under this objectiv e, the frame work learns a conditional in verse mapping that is both epidemiologically consistent and flexible enough to capture comple x temporal transmis- sion patterns. Renewal-based Zero-Inflated Poisson Observation Model. As defined Eq. ( 3 ) in the problem formulation, the renew al equation acts as a forward operator that maps the la- tent reproduction number R t and historical incidence to the expected number of new cases λ t . This forward mapping enables us to enforce statistical consistency between the inferred transmission dynamics and the observed epidemic curve. Rather than treating the rene wal equation as a hard constraint, we incorporate it as a fully dif ferentiable struc- tural component within the learning objecti ve. This design facilitates gradient flo w through epidemiological laws, guid- ing the model toward physically plausible solutions without compromising optimization flexibility . T o explicitly account for the stochasticity , reporting noise, and zero-inflation char- acteristic of surveillance data, we model the observed in- cidence I t using a robust probabilistic count distrib ution conditioned on the renew al expectation λ t (Eq. ( 3 )). Real-world surv eillance data is characterized by tw o dis- tinct types of noise: ov er-dispersion (v ariance exceeding the mean) and zero-inflation (excess zeros due to under- reporting or lo w transmission periods). Standard Poisson objectiv es fail to capture these dynamics. Therefore, we model the observed incidence I t using a Zero-Inflated Pois- son (ZIP) distrib ution ( Lambert , 1992 ). The probability mass function is defined as: P Z I P ( I t | λ t , π t ) = ( π t + (1 − π t ) e − λ t , if I t = 0 (1 − π t ) λ I t t e − λ t I t ! , if I t > 0 . (11) Notably , unlike standard ZIP models that assume a static π , we estimate π t as a time-v arying latent parameter via a parallel network branch (activ ated by Sigmoid), allowing the model to dynamically account for structural reporting failures. By optimizing the likelihood of the observed data under this distribution, our model creates a robust b uffer against observation noise, prev enting the ”over -fitting to noise” problem common in deterministic inv ersion. W e formulate the inference of R t as a regularized maximum likelihood estimation problem, yielding the data fidelity term L obs = − log P Z I P ( I t | λ t , π t ) , (12) which encourages the inferred reproduction numbers to gen- 4 Conditional In verse Learning of Time-V arying Reproduction Numbers Inference erate incidence trajectories that are statistically aligned with observations. T emporal Smoothness Regularization of R t . T o miti- gate the inherent ill-posedness of inferring R t from noisy incidence, we impose a smoothness regularization ( Rudin et al. , 1992 ) on the inferred reproduction number sequence. Specifically , we penalize abrupt changes in R t by applying a robust loss to its first-order temporal dif ferences: L smooth = X t ℓ Huber ( R t , R t − 1 ) , (13) where ℓ Huber ( · ) denotes the Huber loss ( Huber , 1992 ), serv- ing as a differentiable approximation of T otal V ariation (TV) regularization ( Rudin et al. , 1992 ). This formulation sup- presses spurious high-frequency fluctuations while remain- ing tolerant to minor stochastic variations and preserving sensitivity to genuine re gime shifts. Overall T raining Objective. The final learning objecti ve integrates observation consistency with physics-informed regularization: L = L obs + ω L smooth , (14) where ω control the relati ve strength of temporal smooth- ness. All components are fully dif ferentiable, enabling end- to-end optimization of the conditional in verse operator via gradient-based methods. 3.3. Inference and Usage After training, the learned in verse mapping f θ ( t, H t ) en- ables retrospectiv e estimation of R for the training window and real-time inference for incoming data ( t ≥ T ). Ad- ditionally , the corresponding incidence trajectories can be reconstructed by recursi vely applying the rene wal equation (Eq. ( 1 )) with the inferred rates. 4. Experimental Evaluation 4.1. Experimental Setup Datasets. T o evaluate the robustness and generalization capability of our framew ork, we conducted experiments on synthetic benchmarks and two di verse real-world epidemic scenarios. • Synthetic datasets were generated by sampling in- cidence counts from a Poisson distribution ( Dai et al. , 2023 ), using a SARS-like generation interval (Mean= 8 . 0 , SD= 3 . 0 ) under two R t profiles: single- step and double-step changes, with a lo w seed of 2 infected indi viduals ( I 0 = 2 ). Furthermore, we applied a proportional zer o-inflated mask (pre-peak p = 0 . 3 , post-peak p = 0 . 05 ) to simulate realistic surv eillance ev olution and stress-test the model against structural sparsity . • Real-world datasets consisted of (i) SARS 2003 (Hong K ong) ( Cori et al. , 2009 ), characterized by data sparsity and a long generation interval ( Lipsitch et al. , 2003 ), and (ii) CO VID-19 (Ontario) ( Song & Xiao , 2022 ), representing a dense, high-noise scenario with a short generation interval ( Knight & Mishra , 2020 ). Evaluation Metrics. Synthetic experiments are e valuated using RMSE as the primary accuracy metric, MAE as a robustness measure, and detection delay to quantify respon- siv eness to regime changes: RMSE = s 1 T ′ X t ∈T  ˆ R t − R ⋆ t  2 , (15) MAE = 1 T ′ X t ∈T    ˆ R t − R ⋆ t    , (16) where ˆ R t denotes the inferred reproduction number , T is the set of valid time t ≥ L , and T ′ = |T | . Beyond pointwise accurac y , we evaluated the responsive- ness of inferred reproduction numbers to shifts in transmis- sion dynamics. For scenarios with known change points (e.g., abrupt or multiple regime shifts), we measure the detection delay as ∆ delay = ˆ t cp − t cp , (17) where t cp denotes the first true change point, and ˆ t cp is the first index where the inferred ˆ R t crosses the epidemic threshold R t = 1 . T o assess methods’ ability to detect regime shifts, we report the Missed Detection Rate (MDR), defined as the fraction of simulations in which ˆ R t fails to cross the threshold at the true change point. T o emulate the genesis of real-world epidemics, which typi- cally originate from a few inde x cases, all simulations were initialized with a low seed of I 0 = 2 . This realistic ini- tialization naturally predisposes the system to stochastic extinctions and high variance. Therefore, we summarize per - formance metrics using the median and interquartile interv al ( [ Q 1 , Q 3 ] ) rather than the mean and standard deviation. For real epidemic data, where R ⋆ t is unobserved, we do not report regression-based accuracy metrics. Instead, we assessed qualitati ve consistenc y , temporal plausibility , and forward v alidation through short-term incidence forecasting (10 days). Specifically , we e v aluate the predictiv e perfor- mance of the implied incidence ˆ I t + h (Eq. ( 1 ) ) using stan- dard metrics (e.g., RMSE and MAE) over short horizons, serving as an auxiliary check on the inferred transmission dynamics. 5 Conditional In verse Learning of Time-V arying Reproduction Numbers Inference Baselines. T o strictly e valuate the proposed frame work, we compare it with three representative methods, catego- rized by their strategy to solve the ill-posed in verse problem of R t estimation: • EpiEstim ( Cori et al. , 2013 ) is a widely used Bayesian framew ork that estimates R t by locally in verting the renew al equation under a sliding window assumption. W ithin each window , R t is assumed to be constant and inferred using a Poisson likelihood with a Gamma prior . • EpiNow2 ( Abbott et al. , 2020 ) extends the renewal equation framew ork by incorporating reporting delays, observ ation noise, and time-varying reproduction num- bers within a Bayesian state-space model. It models latent infections and R t jointly through hierarchical probabilistic inference, providing posterior uncertainty estimates. • Deep Learning–based Estimation (UDENet) ( Song & Xiao , 2022 ) parameterizes R t using neural networks trained on simulated epidemic data. This model learns a direct mapping from the recent incidence to repro- duction numbers without explicit enforcement of the renew al equation during inference. All baseline methods are provided with the same incidence data and generation interval distrib ution. 4.2. Perf ormance on Synthetic Benchmarks T o systematically ev aluate the proposed method under con- trolled transmission dynamics, we design a set of synthetic epidemic benchmarks with kno wn time-varying reproduc- tion numbers. Specifically , we consider abrupt regime- change scenarios with single or double-step changes and zero-inflated observation regimes, for which we generate 100 stochastic simulations with identical parameters and assess estimation accuracy (RMSE and MAE), regime shift detection rate (MDR), and shift detection delay ( ∆ delay ). Supplementary analysis of incidence reconstruction is pro- vided in Appendix C . Single regime-change scenario. W e e xamine stochastic settings characterized by single-step R t shifts, designed to emulate the abrupt discontinuities in transmission dynam- ics triggered by stringent non-pharmaceutical interventions (NPIs), such as city-wide lockdo wns. Figure 2 re veals dis- tinct failure modes inherent to the baseline methods, trace- able to their rigid structural priors. UDENet fails to de- tect shifts (MDR=97% in T able 1 ) due to its unrealistic memoryless generation interval formulation. EpiEstim is fundamentally limited by its rigid block-constant assump- tion, which presumes R t remains strictly constant within 0 20 40 60 80 100 Time 0 1 2 3 4 5 R t R t : 2 . 5 → 0 . 9 UDENet CIRL(Ours) EpiEstim EpiNow2 True F igure 2. Median and interquartile range of the inferred R t ov er 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. . the sliding window (e.g., τ = 7 ). This assumption acts as a temporal smoothing filter, rendering the method in- capable of resolving instantaneous shifts and leading to a significant median delay of 6 time steps in T able 1 . Sim- ilarly , EpiNow2, due to its reliance on Gaussian Process smoothing priors to enforce temporal continuity , tends to ov er-re gularize abrupt changes, producing overly smooth trajectories. Con versely , our data-driv en approach a voids these structural biases, achieving near -instantaneous detec- tion (median delay = 1 step), while maintaining compet- itiv e accuracy (RMSE/MAE within 0.03/0.06 of the best baseline). In this case, we also note a single failure in- stance (MDR= 0 . 01 ) where CIRL interpreted the shift as noise. In greater detail, the post-hoc analysis of Appendix A rev eals that this specific failure occurred in an e xtreme ’lo w- information’ regime: the simulation was characterized by a zero-proportion exceeding 50% and a peak incidence of only 4 . 0 20 40 60 80 100 Time 0 1 2 3 4 5 R t R t : 2 . 5 → 0 . 9 → 1 . 8 → 0 . 9 UDENet CIRL(Ours) EpiEstim EpiNow2 True (a) Poisson-generated incidence 0 20 40 60 80 100 Time 0 1 2 3 4 5 R t R t : 2 . 5 → 0 . 9 → 1 . 8 → 0 . 9 UDENet CIRL(Ours) EpiEstim EpiNow2 True (b) Poisson incidence with mask F igure 3. Estimated reproduction number trajectories under dif- ferent observ ation 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 compari- son highlights the robustness of the inferred transmission dynamics to observation sparsity . 6 Conditional In verse Learning of Time-V arying Reproduction Numbers Inference T able 1. Quantitativ e comparison of different methods on synthetic benchmarks o ver 100 simulations. For continuous metrics (RMSE, MAE, and ∆ delay ), results are reported as Median [ Q 1 , Q 3 ]. Bold v alues indicate the best performance in each column, and underlined values indicate the second-best. Method Single-step Double-step Double-step(masked) ∆ delay MDR RMSE MAE ∆ delay MDR RMSE MAE ∆ 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] ∼ 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] Double regime-change scenario. W e further extend the ev aluation to a double-step regime, designed to emulate the dynamics of stringent intervention (precipitating a sharp drop in R t ) follo wed by policy relaxation (leading to viral resurgence). The results corroborate our single-step analy- sis, as sho wn in Figure 3a . In this complex setting, UDENet remains effecti vely blind to these changes (MDR=97%), with its median trajectory f ailing to re gister the shifts en- tirely . Among the other baselines, EpiNow2 consistently ov er-smooths transitions, while EpiEstim exhibits a system- atic detection lag at ev ery change point. In stark contrast, our method captures e very structural shift with a median ∆ delay of only 1 time step. Although this responsi veness comes with a ne gligible trade-off in median precision (a gap of < 0 . 08 in RMSE compared to EpiEstim in T able 1 ), our method provides significantly higher estimation certainty , evidenced by a tighter Interquartile Range (IQR) compared to EpiNow2. Zero-Inflated Observ ation Regimes. Real-world surveil- lance, particularly in early outbreak stages, is often plagued by under-reporting due to limited diagnostic capacity and asymptomatic transmission. T o emulate this, we introduce a state-dependent sparsity mechanism into the double-step scenario, applying a proportional zero-inflate mask. Un- der this stress test, UDENet collapses (MDR=100%), and EpiNow2 fails to identify clear change points. EpiEstim confirms its structural limitation, lagging by a median of 6 steps. Notably , the high density of zeros in the early phase creates an identifiability bottleneck for the initial R t → 0 transition, affecti ng both our method and EpiEstim, sho wn in Figure 3b . Howe ver , whereas EpiEstim produces highly unstable estimates (lar ge IQR for RMSE in T able 1 ), our framew ork demonstrates remarkable resilience. This ro- bustness arises from our Ph ysics-Informed Statistical Learn- ing module, which mitigates the influence of observ ation artifacts and enables more reliable learning of the latent transmission dynamics, resulting in consistently strong per- formance across RMSE, MAE, and ∆ delay . 4.3. Ablation Study T o assess the contribution of each model component un- der realistic surveillance conditions, we conducted an abla- T able 2. Quantitative comparison on synthetic double-regime- change epidemics with proportional zero-inflated mask in the ob- serations. All regression metrics are reported as Median [ Q 1 , Q 3 ] ov er 100 simulations. Method ∆ 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/ P oisson -10.00[-15.00,-5.00] 0.00 0.78[0.65,0.88] 0.51[0.43,0.58] tion study on datasets with proportional zero-inflated mask, rev ealing their complementary roles in shaping the accu- racy–responsi veness trade-of f. Removing the T emporal Coordinate Embedding (TCE) mod- ule results in the lowest latency (median ∆ delay = 3 . 00 ), suggesting a highly reactive model. Howe ver , this comes at the cost of over -reacting to noise: while it achiev es the second-best MAE, its RMSE deteriorates significantly to 0 . 72 . This discrepancy (decent MAE vs. poor RMSE) in- dicates that without TCE, the model suff ers from sporadic large errors/spikes, confirming TCE’ s role in suppressing extreme outliers. Con versely , excluding the Historical Incidence Encoder (HIE) leads to a reduction in IQR of the delay , implying a more rigid response pattern. While this variant achie ves the second-best RMSE and an MAE of 0 . 54 , it fails to attain the optimal precision of the full model. This suggests that HIE introduces necessary temporal flexibility , allowing the model to adaptiv ely adjust its lag for different scenarios rather than defaulting to a fixed (lo w-variance) b ut subopti- mal response. Furthermore, we find that replacing the ZIP-based likelihood of the Renewal-based Zer o-Inflated P oisson Observation Model with standard MSE-based (using Mean Square Error loss) or Poisson-based (using Poisson lik elihood) variants leads to consistent performance degradation, with RMSE rising to 0.75 and 0.78, respectiv ely . This indicates that the ZIP likelihood functions beyond regularization, acting as a principled observation model that coherently aligns the learning objectiv e with the sparse and zero-inflated data- generating process. Consequently , the proposed architecture does not rely on a single module b ut lev erages the synergy 7 Conditional In verse Learning of Time-V arying Reproduction Numbers Inference of all components to achie ve a robust equilibrium between timeliness and accuracy . 4.4. Empirical V alidation on Real-W orld Epidemic Dynamics Finally , gi ven the absence of ground truth R in real-world settings, we assess the performance of our frame work by focusing on epidemiological plausibility and internal consis- tency . W e train the CIRL model on the observed period to retrospectiv ely infer and extrapolate R trajectories (10-day horizon), benchmarking our R estimates against retrospec- tiv e baseline estimates obtained from the full time series, and assessing the alignment between model-generated inci- dence and reported case counts. 20 40 60 80 Time 0 2 4 6 8 10 12 14 R t UDENet CIRL(Ours) EpiEstim EpiNow2 (a) R t trajectory 0 20 40 60 80 Time 0 20 40 60 80 100 Number of Cases Observed Cases Fitted Cases Predicted Cases (b) Fitted & Projected incidence F igure 4. Evaluation during the 2003 SARS in Hong K ong. (a) Retrospectiv e 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 transmis- sion dynamics to observation sparsity . SARS 2003 (Hong Kong, China). W e utilized the canon- ical daily incidence dataset of the 2003 Hong K ong SARS outbreak (February–June 2003), sourced from the EpiEstim R package ( Cori et al. , 2013 ). Our inferred R t trajectory closely follows that of EpiEstim, as shown in Figure 4a , capturing the two prominent peaks identified by EpiEstim, albeit with slightly reduced fluctuations. During the late containment phase, our model’ s prospecti ve forecasts con- sistently indicate R t < 1 . 0 , in agreement with all compari- son methods, reflecting the effecti ve control of the outbreak. Furthermore, The model demonstrates high predicti ve con- sistency for incidence (Figure 4b ), with RMSE and MAE below 2 (Appendix B ), indicating that the inferred transmis- sion dynamics reliably reconstruct the observed epidemic trajectory . CO VID-19 (Ontario, Canada). W e analyzed the first wa ve of the CO VID-19 epidemic in Ontario, Canada (Jan- uary–June 2020), using the comparison benchmarks re- ported in UDENet ( Song & Xiao , 2022 ). As shown in Fig- ure 5a , our inferred R t trajectory closely follows the o verall trend of EpiNow2, while e xhibiting variability comparable to that of EpiEstim but with slightly reduced fluctuation 2020-03-01 2020-03-15 2020-04-01 2020-04-15 2020-05-01 2020-05-15 2020-06-01 2020-06-15 Time 0 2 4 6 8 10 12 R t UDENet CIRL(Ours) EpiEstim EpiNow2 0.50 0.75 1.00 (a) R t trajectory 2020-03-01 2020-03-15 2020-04-01 2020-04-15 2020-05-01 2020-05-15 2020-06-01 2020-06-15 Time 0 100 200 300 400 500 600 Number of Cases Observed Cases Fitted Cases Predicted Cases (b) Fitted & Projected incidence F igure 5. Ev aluation on the Ontario COVID-19 first wave. (a) Comparison of retrospecti ve R t estimates. (b) Incidence fitting and implied projection. The results provide a qualitative assessment of the internal consistency of the learned conditional in verse mapping under noisy and non-stationary real-world observ ations. magnitude. Following epidemic control in May , all meth- ods estimate R t to be near , yet below , the critical threshold of 1.0. Notably , over the final ten days, our predicted R t remains close to the EpiNow2 estimates while stabilizing nearer to 1.0 to ward the terminal stage, reflecting a conser - vati ve characterization of residual transmission dynamics. Meanwhile, the predicted incidence ( I t ) produced by our model closely matches the observed trend (Figure 5b ), with both RMSE and MAE smaller than those obtained from direct fitting (Appendix B ), suggesting that the learned con- ditional in verse mapping does not exhibit o verfitting. 5. Conclusion W e presented a Conditional In verse Reproduction Learning (CIRL) frame work for estimating time-varying reproduc- tion numbers from epidemic incidence data. By formulating the task as a conditional in verse problem, CIRL learns a direct mapping from historical incidence patterns and time information to latent reproduction numbers, a voiding strong parametric assumptions or piecewise-constant constraints commonly imposed in e xisting approaches. The proposed framew ork integrates epidemiological consistency through the rene wal equation as a forward operator , while employ- ing a Zero-Inflated Poisson observation model to account for inherent reporting noise and e xcess zeros ubiquitous in real-world surveillance. W ithout imposing parametric or piecewise-constant assumptions on the temporal e volution of R t , the learned mapping remains responsi ve to abrupt transmission changes. Ev aluations across synthetic and em- pirical benchmarks confirm CIRL ’ s precise R t inference, which is further corroborated by high-fidelity short-term forward simulations. 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Bayesian data augmentation for partially observed stochastic compartmental models. Bayesian Analysis , 20(1):107–130, 2025. 10 Conditional In verse Learning of Time-V arying Reproduction Numbers Inference A. F ailure Mode Analysis In Section 4 , we reported a single failure instance out of 100 simulations (MDR = 0.01). T o provide complete transparency , we visualized the incidenc data and model estimation for this specific instance of single-step regime shift scenario in Figure 6 . 0 20 40 60 80 100 Time 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Infection count (a) Infection counts 0 20 40 60 80 100 Time 0.5 1.0 1.5 2.0 2.5 3.0 R t EpiEstim EpiNow2 CIRL True R t (b) T ime-varying reproduction number F igure 6. V isual analysis of the solitary f ailure case. (a) shows the synthetic incidence data, characterized by extreme sparsity and low counts ( y max ≤ 4 ). (b) exhibits the estimated R t trajectory . Due to the indistinguishability between the weak transmission signal and background noise, the model conservati vely maintained a flat trend, resulting in a missed detection of the R t shift. As illustrated in Figure 6 , this simulation represents an extreme edge case characterized by critical data sparsity . The maximum daily incidence count recorded is only 3, with a zero-proportion exceeding 50%. In such a lo w-count regime, the signal-to-noise ratio is severely degraded, making the mathematical distinction between stochastic zeros and a true reduction in transmissibility ambiguous. Consequently , the CIRL model predicts a flat trend (blue line), failing to capture the ground-truth step change (red dashed line) at t = 40 . This indicates that in the absence of strong data evidence, the model defaults to a conserv ative estimate rather than ”hallucinating” a re gime shift. Howe ver , a comparativ e analysis of baseline behaviors highlights the relativ e robustness of our approach ev en in this failure case. While CIRL misses the detection, standard methods exhibit more detrimental failure modes. EpiEstim (green line) displays extreme v olatility , oscillating violently between R t values of approximately 0.5 and 3.0. This behavior confirms that standard rene wal methods are prone to o verfitting stochastic noise in low-incidence settings. Similarly , EpiNo w2 (purple line) produces an incorrect linear drift, failing to identify the abrupt dynamics. Thus, while quantitativ ely categorized as a miss for CIRL, this qualitati ve examination suggests that our framew ork maintains a higher degree of safety and stability compared to baselines, av oiding noise-induced volatility when the data quality is insuf ficient for reliable inference. B. Additional Results on Incidence Fitting and Prediction In addition to the main e valuations on the reproduction number R t , we report quantitati ve results on incidence fitting and short-term prediction on real-world datasets. Specifically , we compute RMSE and MAE between the observed incidence and the corresponding fitted and predicted incidence ˆ I t produced by our model, as shown in T able 3 . These results are provided in the appendix for completeness, as the primary focus of this w ork is the inference and forecasting of latent transmission dynamics rather than direct optimization for incidence-le vel accuracy . Ne vertheless, accurate incidence reconstruction and prediction serve as an important consistency check, ensuring that the inferred and predicted R t jointly explains both the observ ed and future incidence patterns. 11 Conditional In verse Learning of Time-V arying Reproduction Numbers Inference T able 3. Evaluation of incidence reconstruction and short-term forecasting accuracy . RMSE and MAE are reported for both in-sample fitting and rolling prospectiv e 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 in verse mapping. Datasets SARS 2003 COVID-19 RMSE MAE RMSE MAE Fitting 8.62 5.05 45.34 33.58 Prediction 1.51 1.35 37.41 27.84 C. Reconstructing Latent Dynamics from Noisy Obser vations In this section, we provide a detailed analysis of the reconstructed incidence ( ˆ I t ), which is the intermediate outputs generated during the R t estimation process described in the main text. C.1. Evaluation Scenarios and Metrics W e analyze the model outputs across the three synthetic scenarios of the main experiments: 1. Scenario-1 : A single abrupt change in transmissibility . 2. Scenario-2 : Double-step shifts in R t . 3. Scenario-3 : The incidence observations of Scenario-2 with pr oportional zer o-inflated mask . T o rigorously assess the quality of the reconstructed ˆ I t , we compare it against distinct ground-truth targets deri ved directly from the simulation data generation process: • λ true : The time-v arying Poisson rate parameter used to generate the synthetic data ( Dai et al. , 2023 ). This represents the theoretical expectation of daily cases deri ved from the true R t . • I obs : The final synthetic data used for model training, which includes stochastic Poisson noise ( Scenario-1 and Scenario-2 ) and artificial reporting zeros ( Scenario-3 ). Comparison here assesses how well the model fits the training data with structural noise. • I raw : Specifically for Scenario-3 , these are the incidence counts from Scenario-2 . Comparing against this target ev aluates the model’ s ability to recov er the original case counts lost to reporting failures. • λ renew al : The e xpected incidence calculated using incidence counts revised by the standard rene wal equation under the true R t . Comparison against this target e valuates the mathematical consistenc y of the framew ork (i.e., verifying that the estimated R t and the reconstructed incidence are aligned via the epidemiological renew al process). C.2. Perf ormance Analysis T able 4 and T able 5 summarize the quantitativ e results. Below , we discuss the implications of these metrics regarding latent trend recov ery , overfitting risks, and rob ustness to reporting failures. T able 4. Reconstruction Performance. RMSE of estimated incidence ˆ I t against the true Poisson rate λ true . V alues are reported as the Median [ Q 1 , Q 3 ] . Dataset Scenario Comparison T arget CIRL(Ours) EpiNow2 EpiEstim Scenario-1 λ true 10.04[ 6.03,14.10] 8.45 [ 5.21,12.03] 10.26[ 6.31,14.06] Scenario-2 λ true 7.13[ 4.66,10.81] 5.93 [ 4.12, 8.45] 7.03[ 4.88,10.44] Scenario-3 λ true 7.04[ 4.86,10.39] 6.80 [ 4.49,10.66] 7.83[ 5.34,12.08] 12 Conditional In verse Learning of Time-V arying Reproduction Numbers Inference T able 5. Reconstruction Metrics. V alues are Median [ Q 1 , Q 3 ]. Left (Fitting to Noisy Observ ations): Near-zero error on I obs indicates ov erfitting to noise. Middle (Internal Consistency): Low error against λ renew al confirms mathematical v alidity . 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 reco ver the hidden cases. Method Fitting to Noisy Observations Internal Consistency Reconstruction (T arget: I obs ) (T arget: λ renew al ) (T arget: 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] Recovery of Latent Dynamics and Generalization. As detailed in T able 4 , the Gaussian Process-based baseline, EpiNow2, exhibits strong performance in standard scenarios, achie ving the lo west RMSE against the true latent intensity λ true due to its smoothness priors. Our proposed CIRL framework demonstrates competiti ve stability . In Scenario-2, CIRL achiev es a median RMSE of 7.13, which is statistically indistinguishable from EpiNow2’ s 5.93 and EpiEstim’ s 7.03 gi ven the substantial overlap in interquartile interv als. Howe ver , the key distinction lies in consistency . While EpiEstim performs comparably in Scenario-2, it yields higher errors in Scenario-1 (10.26 vs. 10.04) and Scenario-3 (7.83 vs. 7.04). This fluctuation indicates that EpiEstim’ s reliance on observ ations makes it sensitive to stochastic variations, whereas CIRL maintains a more robust performance profile across v arying degrees of complexity . The Overfitting T rap versus Rigid Underfitting. A critical diagnostic insight is provided by the ev aluation against noisy observations ( I obs ) in T able 5 (Left). EpiEstim achie ves a median RMSE of 0.00 across all scenarios. While mathematically perfect, this zero-error fit re veals sev ere o verfitting, confirming that the model is tracking stochastic Poisson noise and reporting artifacts rather than the epidemiological trend. In contrast, EpiNo w2 consistently exhibits the highest RMSE against observ ations across all scenarios. While this indicates a resistance to noise, the elev ated error in Scenario-3 (4.53 compared to CIRL ’ s 3.67) suggests that its smoothing mechanism becomes overly rigid in the presence of zero-inflated data, leading to underfitting. CIRL strikes a balance with RMSE v alues ranging from 1.89 to 3.67, fitting the data suf ficiently well to learn the trend while maintaining enough deviation to a void modeling the noise. Robustness to Sparsity and Counterfactual Reconstruction. The advantages of the proposed frame work become most pronounced in Scenario-3, where reporting failures introduce structural zeros. As sho wn in T able 5 (Middle), while EpiNow2 performs well in standard scenarios (Scenario-1 and Scenario-2), its internal consistency deteriorates in Scenario-3 (RMSE 3.59). In contrast, CIRL achie ves the lo west consistency error (RMSE 3.03) in this challenging setting, confirming that its inferred R t remains mathematically aligned with the reconstructed incidence ev en under sev ere data sparsity . Most importantly , the Counterf actual Reconstruction metric (T able 5 , Right) pro vides the definiti ve e vidence of our method’ s efficac y . When e valuated against the unmasked truth ( I raw ), EpiNow2 yields a high RMSE of 4.01, implying that its smoothing priors are biased do wnwards by the artificial zeros. CIRL, howe ver , achiev es a significantly lower RMSE of 2.44. By explicitly modeling the zero-inflation probability ( π t ), CIRL effecti vely identifies structural reporting failures and ”fills in” the missing cases, thereby recov ering the true underlying incidence that baselines fail to capture. Most importantly , the counterfactual reco very (reconstruction) metric (T able 5 , right) offers the most direct e vidence of the effecti veness of our method. When ev aluated against the unmasked truth ( I raw ), EpiNow2 yields a high RMSE of 4.01, implying that its smoothing priors are biased by the artificial zeros. CIRL, howe ver , achie ves a significantly lower RMSE of 2.44. By e xplicitly modeling the zero-inflation probability ( π t ), CIRL effecti vely identifies structural reporting failures and ”fills in” the missing cases, thereby recov ering the true underlying incidence that baselines fail to capture. 13