Conditional Inverse Learning of Time-Varying Reproduction Numbers Inference

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 (6)

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