Comparative Numerical Analysis of MSGDTM and Jacobi Spectral Collocation Methods for a Fractional-Order Computer Virus Model

This paper presents a new fractional-order model that captures the propagation dynamics of computer viruses within digital networks by incorporating memory effects and nonlocal interactions via the Caputo derivative. The network is partitioned into four functional compartments: susceptible, latently infected, breaking-out (actively infectious), and recovered computers. The analytical framework establishes the fundamental mathematical properties of the model—existence, uniqueness, nonnegativity, and boundedness of solutions—ensuring its well-posedness. The stability of the infection-free and endemic equilibria is investigated using Matignon’s theorem for fractional-order systems, Lyapunov functionals, and the fractional LaSalle invariance principle. The basic reproduction number R0 is derived via the next-generation matrix approach to determine threshold conditions governing the transition between virus eradication and persistence. To approximate the fractional dynamics, two robust numerical algorithms are implemented and compared: the multistep generalized differential transform method (MSGDTM) and the Jacobi spectral collocation scheme (JSCS). The model is extended to a stochastic framework to account for random fluctuations inherent in real-world network environments, with both deterministic and stochastic solvers evaluated through Monte Carlo simulations. The stochastic analysis reveals key phenomena including noise-induced transitions, enhanced extinction probability, and memory–noise interplay that are not captured by the deterministic model. Numerical experiments reveal that the fractional order strongly influences infection intensity and system memory, with smaller orders yielding delayed peaks and prolonged transients. While MSGDTM provides computational simplicity and flexibility for nonlinear problems, JSCS attains superior spectral accuracy and convergence efficiency. The comparative results highlight the role of fractional modeling in representing long-memory effects in cyber-epidemics and offer quantitative guidance for designing antivirus strategies in memory-dependent network environments.