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

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

This study proposes a Fractional-Order Physics-Informed Neural Network (FPINN) framework for solving inverse parameter estimation problems in both fractional SIR and augmented SEIR epidemiological models and demonstrates that the proposed method accurately reconstructs epidemic trajectories and captures the influence of memory effects on disease evolution.

Boosted training strategy for physics-informed neural networks in modeling non-linear computer virus dynamics with vertical transmission

Computer-virus propagation in networked systems is often modeled by epidemic-type derivative equations. Still, standard quantitative solvers can become computationally demanding and may exhibit small stability for long-term predictions. This work develops a Physics-Informed Neural Network (PINN) based framework for an e-epidemic SI1I2R model of computer-virus spread that incorporates a possibly transmissible class, an amply transmissible class, and direct transmission, allowing nodes to be initially compromised without contact. The proposed PINN uses a multilayer, amply connected architecture with Swish-ReLU activations to approximate the solution to the coupled nonlinear common differential equations and to enforce the governing dynamics through a physics-based loss. We use standard error metrics such as Mean Squared Error (MSE), Theil’s Inequality Coefficient (TIC), and Mean Absolute Deviation (MAD) to compare the model's performance against a reference ODE Solver. We also examine training and validation loss profiles across several trials. A grid-based sensitivity analysis of key parameters (the infection rate and the progression rate from I1 to I2) is conducted to determine how they affect the peak sizes of infected and recovered populations. The results show that the PINN accurately reproduces the reference SI1I2R dynamics with low errors across all compartments. Swish-ReLU is the most stable and accurate training function of the ones that were tested. These results indicate that PINNs are a reliable and possibly resource-efficient way to model and predict how computer viruses spread. They also show that they could be useful for future AI-based epidemic modeling in cybersecurity.

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

Experiments with synthetic data and COVID-19 surveillance data show that SUC–PINN recovers plausible hidden infection trajectories, yields stable parameter estimates, and provides accurate short-term forecasts, support SUC–PINN as a practical computational approach for inverse modeling and prediction in partially observed epidemic dynamics.

Failure-informed PINNs for a multi-strain SVEIR epidemic model

Tightly coupled multi-compartment epidemic models tend to expose a weakness of standard Physics-Informed Neural Networks (PINNs). When collocation points are placed uniformly, the network spends much of its capacity on smooth regions while leaving the sharp transients poorly resolved. We work around this by pairing Failure-Informed PINNs (FI-PINNs) with a Self-Adaptive Importance Sampling (SAIS) refinement strategy, and we apply the combination to a nine-equation Susceptible/Vaccinated/Exposed/Infected/Recovered (SVEIR) model that follows three viral strains together with a vaccination compartment. The idea behind the construction is simple. We build a residual-based limit-state function, estimate the failure probability associated with it, and let the network steer its own sampling toward the time intervals where the governing equations are not yet satisfied to within a prescribed tolerance. On the same temporal domain, with identical initial conditions and the same network architecture, SAIS-enhanced FI-PINNs reach a relative \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document} error of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.04\times 10^{-6}$$\end{document}, against \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$7.36\times 10^{-3}$$\end{document} for uniform sampling and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3.42\times 10^{-4}$$\end{document} for Residual-Based Adaptive Refinement (RAR) [Lu et. al,. 63:208-228, 2021], an improvement of two to three orders of magnitude. The benefits go beyond raw accuracy. The failure-probability estimate gives an interpretable stopping criterion, and the truncated Gaussian proposal of SAIS keeps the sampling-bias risk under control, a risk that affects purely greedy residual-based schemes. The framework is general enough to be transported to other coupled compartmental systems and to data-assimilation settings in which partial observations are available. 92D30 , 65L05 , 68T07 , 00A71

Semi-analytical hierarchical Bayesian inference of nonlinear model structure in stochastic dynamics: Applied to compartmental models of infectious diseases

A Bayesian computational framework for parsimonious inference in stochastic nonlinear dynamical systems is presented. This framework enables the concurrent estimation of system states, time-varying parameters, time-invariant parameters, and the optimal sparsity structure of the model parameters. Because differential equation-based models are often simplified mechanistic or phenomenological representations, robust inference from noisy measurement data requires explicit treatment of model error and uncertainty. Model error and time-varying parameters can be represented as random processes, enabling inference while making minimal assumptions about the underlying sources of discrepancy and variability. Adopting stochastic differential equation representations affords the model significant flexibility, but can also render it susceptible to overfitting during statistical inversion, where the inferred model may track noise rather than the underlying signal. To alleviate the effects of overfitting and to enable the discovery of the optimal sparse representation of the time-invariant parameters, a Bayesian sparse learning algorithm is embedded within the framework. This sparse learning framework adopts an approximate hierarchical Bayesian setting defined by a series of semi-analytical expressions. The model structure inference framework is validated using a stochastic compartmental model for tracking and forecasting active cases of an infectious disease. Compartmental models describe population-level infectious disease dynamics through interactions among population fractions grouped by disease state. Mathematically, such models consist of a system of coupled ordinary differential equations. This example adopts an expressive compartmental model that includes multiple possible interactions between disease states, motivated by early uncertainty surrounding COVID-19 reinfection dynamics and their implications for long-term epidemic forecasting. The sparse learning exercise permits the inference of a priori unknown epidemiological dynamics from simulated public health data, discovering the nested compartmental model that optimizes the trade-off between average data-fit and model complexity. It is shown that inducing sparsity among the model parameters eliminates redundant interactions between compartments, equivalently revealing the optimal coupling structure between differential equations.

Physics-Informed Neural Networks for Fractional SEIR Models with Power Mittag-Leffler Kernels