Deep learning-based predictions and analysis of Norovirus dynamics using a fractional-order mathematical model

A computational study combining a fractional-order compartmental model with deep neural networks found that lower fractional orders were associated with greater stability of the model’s disease-free equilibrium, while higher vaccination coverage and treatment effectiveness produced lower predicted infection prevalence. The study examined norovirus transmission as a mathematical system rather than analyzing infections observed in a patient population.

The model divides the population into five epidemiological compartments: susceptible, exposed, infected, vaccinated and recovered individuals. Susceptible people enter the population at a constant recruitment rate and may become vaccinated, while protection from vaccination is represented by vaccine efficacy and may decline over time. Exposed individuals progress to the infectious compartment, and infected people either recover or experience disease-induced mortality. Natural mortality applies across the model’s compartments, and recovered individuals are assumed to have immunity against reinfection.

The authors replaced the classical first-order time derivatives with a Caputo–Fabrizio fractional derivative. In the mathematical formulation, this non-integer-order approach was intended to represent memory and nonlocal effects in disease transmission. The model included transmission, vaccination, waning vaccine protection, progression from exposure to infection, recovery, disease-related mortality and natural mortality as parameters affecting its predicted trajectories.

Fixed-point results were used to investigate whether the fractional-order model has solutions and whether those solutions are unique under specified mathematical assumptions. The existence analysis used Krasnoselskii’s fixed-point theorem, while additional contraction arguments were applied to establish uniqueness under stated conditions. The researchers also derived Ulam–Hyers stability results. These were properties of the mathematical formulation under its assumptions, not empirical observations of norovirus outbreaks.

For numerical simulation, the team implemented a three-step Adams–Bashforth scheme in MATLAB. Simulations were performed at different fractional orders and with varying transmission-related parameters. The reported mathematical findings indicated that the stability of the disease-free equilibrium increased as the fractional order decreased. The simulations also predicted that broad vaccination coverage and improved treatment effectiveness would substantially reduce infection prevalence. These are modeled effects rather than estimates of benefit from a vaccination or treatment program evaluated in people.

A feedforward deep neural network was then used to approximate the model’s five compartment trajectories over time. The network received time as its input and produced outputs for the susceptible, vaccinated, exposed, infected and recovered compartments. Its loss function incorporated the differential-equation residuals and initial conditions, allowing training without supplying an exact solution. Automatic differentiation calculated derivatives of network outputs, and the Adam optimizer was used with a reported learning rate of 10−2.

The network used a proposed activation function called NGNDG-AF, intended to provide nonlinear behavior while remaining smooth and responsive to small negative inputs. Models for the five compartments were trained over 1,000 epochs, and the authors applied Bayesian regularization to reduce residual error and select weights. Reported regression coefficients for the compartment-specific analyses were approximately one. The neural-network trajectories closely overlapped the corresponding fractional-order numerical solutions in the paper’s comparisons.

The reported mean squared errors differed among compartment and scenario combinations. Best values for the susceptible-compartment scenarios ranged from 543.8996 to 740.3281, while those for the infected-compartment scenarios ranged from 16.2253 to 45.1371. All reported networks completed 1,000 epochs without validation checks triggering early stopping, and the optimization diagnostics included finite parameter norms and sums of squared parameters. These results support the authors’ assessment that the trained networks approximated the numerical model under the tested configurations.

The study is limited by its reliance on mathematical simulations, selected parameter values and comparisons between computational solutions rather than validation against a documented norovirus outbreak dataset. The deep learning component acts as an approximation method for the fractional-order equations; it does not independently establish predictive performance in clinical or public-health settings. The authors proposed extending the modeling framework to spatial dynamics, stochastic effects and machine-learning-based parameter estimation, but those extensions were not evaluated in the reported work.

Subject of Research: Fractional-order norovirus transmission modeling and neural-network approximation

Article Title: Deep learning-based predictions and analysis of Norovirus dynamics using a fractional-order mathematical model

Article References: Rehman, B., & Ali, A. (2026). Deep learning-based predictions and analysis of Norovirus dynamics using a fractional-order mathematical model. PLOS One, 21(10), e0359983. https://doi.org/10.1371/journal.pone.0359983

Image Credits: AI Generated

DOI: 10.1371/journal.pone.0359983

Keywords: norovirus, fractional-order model, deep neural networks, epidemic modeling, compartmental model, numerical simulation, disease-free equilibrium, vaccination, treatment effectiveness, Deep, learning-based, predictions