Stochastic simulation adds memory and delay to mathematical model of computer-virus spread

A mathematical modeling study developed and numerically tested a stochastic fractional-order delay model for computer-virus propagation in a population connected to the internet. The research partitions computers into uninfected, latent-infected and actively compromised groups, then represents historical effects with a Caputo fractional derivative, delayed transmission with a time lag, and random fluctuations with Brownian perturbations. Numerical experiments indicated that each of the fractional order, transmission delay and noise intensity changed the modeled trajectories of the three computer populations.

The model represents the computer population as a normalized total. Uninfected computers can enter the system, and computers can leave it or be removed. New infections enter the latent group and may progress to the active “seizing” group, while latent and active computers can also be cured. Transmission depends on both latent and active infected computers and is evaluated at a delayed time, with an exponential attenuation factor associated with the delay. The analytical results apply to the deterministic fractional-order delay system obtained by setting the stochastic intensities to zero.

For that deterministic system, the researchers used a fractional-integral formulation and the Banach fixed-point theorem to establish a unique local solution. They then extended the argument through continuation on the feasible region. The authors also proved that solutions starting from nonnegative histories remain nonnegative, with the uninfected population remaining positive under the stated initial condition, and that every nonnegative solution is uniformly bounded. These proofs concern the deterministic model rather than the fully stochastic system.

The analysis identified two equilibria: a computer-virus-free equilibrium and a computer-virus-present equilibrium. A next-generation matrix calculation produced a basic reproduction number, with the virus-free equilibrium associated with the subcritical case below a threshold of one and the virus-present equilibrium appearing when the threshold is exceeded. The researchers derived local and global stability conditions for both equilibria in the deterministic fractional-order delay model. They explicitly did not claim almost-sure, mean-square or other probabilistic stochastic stability from these analytical results.

Sensitivity analysis of the reproduction number indicated that contact rates and infection probabilities had the largest positive normalized forward sensitivity indices. Parameters representing intervention measures and recovery had negative indices in the analysis, while the fractional order also affected sensitivity. These findings are properties of the model under its specified assumptions and parameter choices; the study did not estimate intervention effects from observed malware outcomes.

To simulate the stochastic model, the authors developed a stochastic non-standard finite difference scheme. The method retains the model’s fractional-memory term, evaluates transmission at the delayed time level and applies the same Brownian increment to the three equations. Positivity and boundedness were proved for the deterministic version of the scheme. Because Brownian increments can be negative, the researchers did not claim unconditional positivity or boundedness for the fully stochastic discretization.

Simulation experiments varied the fractional order while holding other settings fixed, and the resulting trajectories showed that memory changed the transient evolution of the uninfected, latent and active computer populations. Separate experiments varied the transmission delay with the fractional order fixed, showing that delay altered both the timing of the infection input and its effective magnitude through the attenuation factor. Combined simulations varied both parameters. Under the tested settings, trajectories remained bounded over the simulation interval.

Increasing the stochastic intensity produced greater pathwise variation in all three modeled populations, while the deterministic case followed a smoother trajectory. In a common-path comparison, the stochastic non-standard finite difference scheme and the fractional Euler–Maruyama method generated qualitatively similar trajectories. The Euler–Maruyama method had slightly smaller maximum absolute error and root-mean-square error in the particular experiment analyzed. The non-standard scheme nevertheless showed comparatively controlled behavior as the time step increased, an observation that the authors characterized as time-step sensitivity rather than proof of convergence.

The study used parameter values drawn from fitted or estimated sources and values adopted from prior literature. It also cited global malware and cyberattack statistics from 2020 through 2025 as context, including reported growth in malware samples and cyberattacks, but did not use those time series to calibrate or statistically validate the model. The principal limitations were the absence of a complete real-world time-series calibration, numerical rather than rigorous treatment of stochastic stability and the absence of a general convergence proof for the proposed scheme. The authors identified data-driven estimation, rigorous stochastic analysis, network heterogeneity and optimal-control extensions as possible future work.

Subject of Research: Stochastic fractional-order delay modeling and numerical simulation of computer-virus propagation

Article Title: A stochastic fractional model for computer virus propagation with memory effects and exponential delay

Article References: Raza, A., Lampart, M., Shafique, U., Baleanu, D., & Abd Elwahab, M. E. (2026). A stochastic fractional model for computer virus propagation with memory effects and exponential delay. PLOS One, 21(10), e0360254. https://doi.org/10.1371/journal.pone.0360254

Image Credits: AI Generated

DOI: 10.1371/journal.pone.0360254

Keywords: computer-virus model, fractional differential equations, delay dynamics, stochastic modeling, Brownian perturbations, numerical simulation, reproduction number, sensitivity analysis, cybersecurity, stochastic, fractional, model