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

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.

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.

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

Dynamical Analysis and SDEPINN-Based Modeling of a Fractional-Order Stochastic SIS Epidemic Model

This paper proposes a novel susceptible–infected–susceptible (SIS) epidemic model incorporating a fractional-order term and white-noise perturbations. Several analytical results concerning its dynamical behavior are established. Firstly, we prove the existence and uniqueness of model solutions using the Carathéodory approximation. Secondly, the extinction of disease is rigorously proven based on the properties of the quadratic function. Meanwhile, the Ulam–Hyers stability of the model is derived by using stochastic Gronwall-type inequalities and the stochastic analysis techniques. Furthermore, the solution of the proposed fractional-order stochastic model can be approximated by that of the corresponding averaged model under suitable averaging conditions. Then, numerical simulations are conducted to illustrate the effects of key parameters on disease extinction, stability, and long-term dynamical behavior. To improve its adaptability to real-world epidemic dynamics, this paper embeds dynamical constraints into neural networks and constructs a stochastic physics-informed identification framework for seasonal infectious diseases. Empirical results based on monthly influenza data from Xinjiang show that the proposed framework can capture epidemic trends, seasonal peaks, and fitting uncertainty. These results provide a theoretically grounded and practically applicable approach for infectious disease modeling under memory effects and stochastic perturbations.

A Discrete Fractional-Order Model of Diabetes–Cardiovascular Dynamics with Reservoir Computing–Based Forecasting

This study presents a novel discrete fractional-order mathematical framework for modeling and forecasting the coupled progression of diabetes and cardiovascular complications, with emphasis on population dynamics in Saudi Arabia. Using the discrete Caputo fractional operator, the model captures memory effects and long-term disease dependence not represented by classical integer-order systems. The population is divided into five interacting compartments: susceptible, exposed, diabetic without major complications, diabetic with severe complications, and cardiovascular-affected individuals. A rigorous qualitative analysis establishes equilibrium points and examines their local stability under fractional discrete dynamics. Stability regions are identified in parameter space, clarifying the mechanisms governing disease persistence and progression. A feedback-based control strategy is proposed to restore stability of the disease-free equilibrium when destabilization occurs. To enhance predictive performance, a Reservoir Computing framework is integrated and compared with Long Short-Term Memory networks, demonstrating improved accuracy and lower computational cost. Numerical simulations validate theoretical findings and highlight key epidemiological influences. Future work will extend the model by incorporating optimal control strategies aimed at minimizing infections through targeted interventions such as vaccination, improved hygiene, and environmental sanitation. The integration of fractional-order dynamics with control theory offers valuable insights for public health decision-making and epidemic mitigation.

A Fractional Physics-Informed Neural Network with Ensemble Learning for Dengue Forecasting