How (Not) to Hybridize Neural and Mechanistic Models for Epidemiological Forecasting

Physics-Informed Neural Networks (PINNs) for Parameter Estimation in the SIRS-D Epidemiological Model

Infectious diseases exhibit complex and rapidly evolving transmission dynamics, requiring modeling approaches that can accurately capture these mechanisms. The SIRS-D compartmental model provides a suitable framework, as it incorporates temporary immunity and disease-induced mortality within the epidemic process. Accurate parameter estimation is essential for quantifying the transmission rate, recovery rate, waning immunity rate, and mortality rate, which collectively govern the system behavior. Among existing estimation methods, Physics-Informed Neural Networks (PINNs) offer significant advantages by integrating observational data with the underlying structure of differential equations, thereby preserving physical consistency while maintaining robustness under imperfect data conditions. In this study, PINNs are employed to estimate the parameters of the SIRS-D model using synthetic data generated through the fourth-order Runge–Kutta (RK4) method to ensure stable and consistent numerical solutions. To better represent real-world measurement conditions, 5% noise is added to the synthetic data, introducing realistic variability into the training process. The results demonstrate that PINNs successfully reconstruct the trajectories of S(t), I(t), R(t), and D(t) with low prediction errors. The model achieves MAE values of 0.0065 (S), 0.0067 (I), 0.0208 (R), and 0.0043 (D), with corresponding RMSE values of 0.0090, 0.0074, 0.0253, and 0.0058. Moreover, the estimated parameters closely match the true values, yielding ????????=0.5031, ????=0.0996, ????=0.0095, and ????=0.0149, demonstrating strong parameter identification capability. These findings confirm that PINNs constitute a reliable and accurate framework for analyzing infectious disease dynamics and offer promising potential for extension to more complex epidemiological models and real-world datasets.

Hierarchical Bayesian autoregressive smooth transition time series models

Public health policy and disease surveillance systems require accurate forecasting of infectious disease dynamics to support timely interventions and resource allocation. However, classical linear time-series models often fail to capture abrupt regime shifts, nonlinear transmission patterns, and heterogeneous reporting commonly observed in surveillance data. This study investigates the practical advantages of hierarchical Bayesian smooth transition autoregressive (BH-STAR) models, including logistic (LSTAR) and exponential (ESTAR) specifications. Performance is evaluated through extensive simulation studies under controlled nonlinear data-generating mechanisms, followed by an empirical application to COVID-19 surveillance data from 53 African countries collected between March 2020 and December 2022. Simulation studies revealed a key directional asymmetry in model misspecification: fitting a logistic transition function to data generated under exponential dynamics resulted in moderate, stable parameter bias, whereas fitting an exponential transition function to logistic dynamics induced severe, compounding bias. Despite this parameter confounding under model misspecification, predictive accuracy remained stable across both data-generating processes. In the empirical application, the BH-STAR models consistently outperformed linear alternatives in out-of-sample forecasting. The hierarchical logistic STAR (HLSTAR) model achieved the highest overall predictive accuracy, reducing validation errors to an MAE of 0.58 and a MdAPE of 12.8%, corresponding to country-level forecast error reductions of 30–50% and overall error reductions exceeding 50% relative to the standard autoregressive benchmark. Hierarchical Bayesian smooth transition autoregressive models provide accurate forecasts by accommodating nonlinear regime-switching dynamics while delivering robust uncertainty quantification. These features make them well suited for infectious disease surveillance and public health decision-making in resource-limited, high-uncertainty settings.

SUC–PINNs: A physics-informed neural networks approach to inverse problems in epidemic models with partial observability

Partially observed epidemic systems are difficult to analyze because confirmed cases provide only an indirect view of transmission and hidden infections are usually unobserved. We consider the susceptible–unidentified infected–confirmed (SUC) epidemic model and develop SUC–PINN, a physics-informed neural network framework for estimating hidden epidemic states, transmission and confirmation rates, and the initial unidentified population from confirmed-case time series alone. The method combines confirmed-case observations with SUC residuals and initial-condition constraints, so that the learned trajectories remain tied to the governing dynamics. Experiments with synthetic data and COVID-19 surveillance data from Indonesia, Thailand, India, and the Philippines show that SUC–PINN recovers plausible hidden infection trajectories, yields stable parameter estimates, and provides accurate short-term forecasts. These results support SUC–PINN as a practical computational approach for inverse modeling and prediction in partially observed epidemic dynamics.

MechGNN-Epi: Mechanistically Constrained Spatiotemporal Graph Learning for Regional Epidemic Forecasting

Mechanistic compartmental models such as SIR provide interpretability and enforce conservation of population, but their predictive performance can degrade in heterogeneous settings where transmission varies over time and across regions. Conversely, spatiotemporal graph neural networks (GNNs) can learn flexible spatial interactions from data, yet unconstrained predictors may yield epidemiologically inconsistent dynamics. We propose MechGNN-Epi, a hybrid framework that couples a spatiotemporal graph encoder with a differentiable SIR update. Rather than directly predicting future infections, the model learns region- and time-specific transmission and recovery parameters from graph-based latent representations and then propagates compartments through a mechanistic solver. This design yields epidemiologically constrained trajectories and produces region- and time-indexed parameter proxies that can be inspected as diagnostic signals, while not being guaranteed as causally identifiable mechanistic parameters. We evaluate the pipeline on U.S. state/territory COVID-19 surveillance data using strict chronological splits and report variability across multiple random seeds, multi-horizon rollouts, and ablation/sensitivity studies against graph and model variants.

Fractional physics-informed neural network approach to inverse problems in epidemiological modeling

Neural Time Series Forecasting: A Problem-Driven Survey from Data Challenges to Model Choice

Surveys of time series forecasting usually proceed by model family, from recurrent networks to Transformers and foundation models. That chronology is useful, but it gives limited guidance when the practical question is why a forecast fails. This article instead organizes the literature around five recurring difficulties: nonlinear and nonstationary behavior, contamination and structural breaks, uncertainty, long contexts and cross-variable dependence, and limited target-domain data. Studies published between 2014–2025 are compared through the assumptions they make, the settings in which they work, and the failure modes they leave unresolved. The review covers recurrent and probabilistic models, decomposition methods, robust and uncertainty-aware forecasting, Transformer variants, theory-guided methods, and foundation models. Quantitative results are used only when the underlying protocol is sufficiently clear. Across these families, the evidence supports a restrained workflow: begin with a strong simple baseline, identify the dominant source of error, and add complexity only when it addresses that source.