Investigating the impact of memory effects on computer virus population dynamics: A fractal–fractional approach with numerical analysis

In this study, we investigate the impact of memory effects through the application of the fractal–fractional derivative operator on population dynamics in computer viruses. By utilizing numerical analysis techniques, we explore the influence of varying fractal–fractional operator orders on virus propagation and conduct a parameter study to examine the effects of fractal dimension and fractional order on computer virus spread. Furthermore, in order to comprehensively compare and analyze the outcomes from various perspectives, we introduce a novel application of the operational matrix technique to the proposed dynamical system, transforming the fractal–fractional model into the Caputo derivative form. Two different numerical techniques, Adams–Bashforth and Taylor operational matrix, have been employed to solve the two distinct forms of the model. The obtained results yielded highly consistent outcomes for the two numerical methods employed, despite their utilization of distinct derivative operators. The reported results highlight the memory effect associated with fractal–fractional derivatives, where past states and behaviors continue to influence the system. The ability to capture these memory effects through fractal–fractional derivatives can be considered a benefit, enabling a more comprehensive understanding, analysis and modeling of computer virus dynamics. Additionally, we investigate four variants to ensure numerical stability, employing both the Ulam–Hyers (UH) and Ulam–Hyers–Rassias (UHR) criteria.