Stochastic fractional order model for the computational analysis of computer virus.

This work presents a novel mathematical framework for analyzing the propagation dynamics of computer viruses by formulating a fractional-order model. The classical integer-order differential model of computer virus spread is reformulated using Caputo fractional derivatives, yielding a fractional computer virus model that captures the inherent memory and persistence characteristics of digital infection processes. A comprehensive analytical investigation is conducted, including the verification of fundamental properties such as positivity and boundedness of the system. The existence and uniqueness of the solutions are rigorously established using the Banach fixed-point theorem. The model exhibits two equilibrium states whose global stability is thoroughly analyzed. To incorporate the stochastic behavior of networked systems, such as fluctuating traffic, random user activity, and unpredictable system responses, the fractional computer virus model is extended into a stochastic fractional computer virus model by introducing white noise terms. Unlike previous studies, which often neglect the combined impact of stochasticity and memory, this research provides a rigorous treatment of both, ensuring the unique solvability of the stochastic fractional computer virus model. A Grunwald-Letnikov-based nonstandard finite difference scheme is developed to obtain reliable numerical approximations of the model while preserving essential qualitative features such as solution positivity and boundedness. Numerical simulations, based on realistic test scenarios, support the theoretical findings and illustrate the complex dynamics introduced by both fractional-order behavior and stochastic influences. This study provides a robust and realistic framework for understanding and predicting the spread of computer viruses in complex digital environments.