This paper pairing Failure-Informed PINNs with a Self-Adaptive Importance Sampling (SAIS) refinement strategy, and applying the combination to a nine-equation Susceptible/Vaccinated/Exposed/Infected/Recovered (SVEIR) model that follows three viral strains together with a vaccination compartment, shows an improvement of two to three orders of magnitude.
Abstract
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
Abstract.
In the early stages of a newly emerged or reemerged disease, there is a rapid increase in new infections, which can potentially lead to a healthcare crisis due to constraints of available medical resources. It is important to note that the recovery period of these emerging diseases typically follows a Gamma distribution rather than being narrowly centered around the mean. In this study, we propose a susceptible-infectious-recovered (SIR) network model that incorporates a general recovery rate and a saturation treatment function. We establish the well-posedness and global stability of the disease-free equilibrium in the model by employing semigroup theory and the standard comparison principle, respectively. From an epidemiological perspective, when the delayed treatment effect exceeds a significant threshold, a phenomenon known as backward bifurcation emerges near the disease-free equilibrium. This assertion is supported by an updated version of the Lyapunov–Schmidt approach. Additionally, we conduct numerical simulations to investigate how network topology and non-Markovian processes affect the patterns of disease transmission.
Junyuan Yang, Maia Martcheva, Jun Zhang et al.· SIAM Journal on Applied Math...· 0 citations
Regional surveillance data reflect local transmission, reporting, seeding, and external infection pressure, which are difficult to identify separately. We introduce GeoID-PINN, a physics-informed neural network (PINN) for susceptible-infectious-recovered-deceased (SIRD) dynamics. The model represents spatial dependence with a row-stochastic source-composition matrix whose rows assign nonnegative source weights that sum to one. We regularize this matrix toward a spatial prior constructed from distance, adjacency, commuting, or lead-lag information. In a four-region simulation with known truth, a compatible distance prior gives source-composition error 0.099. The error rises to 0.159 without regularization and 0.577 under a strongly misspecified prior, while trajectory fit and transmission-scale estimates remain similar. Accurate trajectories therefore do not guarantee recovery of the regional dependence structure. We also evaluate GeoID-PINN retrospectively using COVID-19 data from 64 Louisiana counties. Relative to an autoregressive negative-binomial baseline, Forecast-Trained Geo-PINN reduces mean squared error (MSE) from 32,957 to 11,468 and mean absolute error (MAE) from 70.60 to 57.73. The baseline has lower negative log likelihood (NLL), 5.158 versus 5.346, indicating better distributional fit but worse point accuracy. In a controlled 15-county comparison, county adjacency reduces MSE by 6.85 percent and MAE by 3.1 percent. Similar performance across plausible priors supports structured regularization but not unique edge recovery. These results require prior-sensitivity and observation-model checks before interpretation.
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.
Fitri Cahyani, Abdurakhman Abdurakhman, Chyntia Meininda Anjanni· The eurasia proceedings of s...· 0 citations
Most previous computer virus propagation (CVP) models are smooth, meaning that their right-hand sides are continuously differentiable. However, recovery resources for compromised hosts are often limited, and the aggregate recovery rate may decrease once the number of bursting nodes exceeds a defense threshold. To describe this resource-constrained mechanism, this article proposes a nonsmooth susceptible–latent–bursting–susceptible (SLBS) model with a two-level recovery function and a Holling-II saturated infection rate. Well-posedness, positivity, and positive invariance of the feasible region are first proved. The basic reproduction number is derived by the next-generation matrix method, and its normalized sensitivity indices is provided. The virus-endemic equilibria are obtained by reducing the equilibrium equations to a strictly increasing scalar equation, with special attention to the threshold case at the nonsmooth switching surface. Local stability is established by piecewise linearization and explicit Routh–Hurwitz criteria. Finally, vector-graphic numerical simulations, convergence checks, and parameter robustness tests are reported. The results clarify how limited recovery capacity and saturated infection jointly affect hierarchical control of network viruses.
Yiran Chen, Ning Liao, Xiaofan Yang et al.· Mathematics· 0 citations
We develop an exact finite-population stochastic framework for SIR epidemics evolving under Markovian switching between intervention regimes. The epidemic state is augmented by a finite phase component, allowing transmission, recovery, and direct immunity-acquisition rates to depend on the active regime. Phase-transition intensities may depend on the current epidemic state, so that policy escalation can react to the number of infectious individuals. Exploiting the monotonicity of the susceptible compartment, we derive level-wise recursions for the joint Laplace--Stieltjes transform and probability generating function of the extinction time and the number of infections generated before extinction. These recursions yield the infection-count distribution, conditional extinction-time transforms, and mixed moments linking epidemic duration and infection burden, while replacing a large global linear system with small phase-level solves. The framework is illustrated using weekly mpox incidence data from Luxembourg. A baseline one-phase SIR model is calibrated by maximum likelihood under a Poisson observation model. The calibrated baseline is then used for conditional comparisons of fixed control regimes, early versus delayed strict intervention, vaccination-supported control, and state-dependent escalation. The results show how switching mechanisms affect both the total number of infected individuals and the extinction time, including their dispersion. Since the switching mechanisms are specified rather than estimated from the intervention history, the results are conditional model-based comparisons rather than estimates of the historical effects of interventions in Luxembourg.
Vasileios E. Papageorgiou, Irène Votsi, Samis Trevezas· 0 citations
A Physics-Informed Neural Network 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 is developed.
Jamshaid Ul Rahman, Shanza Shabeer, Noreen Mustafa et al.· Discover Artificial Intellig...· 0 citations