Incorporating Memory Traces in Computer Virus Models Using Caputo Fractional Calculus

Abstract

A fractional-order mathematical model is developed to investigate the propagation and control of computer viruses in a networked environment. The model divides the computer population into four compartments: susceptible, latently infected, breaking-out, and antivirus-capable systems. Unlike traditional integer-order models, the formulation incorporates memory effects via the Caputo fractional derivative, offering a more realistic representation of historical dependencies in digital virus transmission. Rigorous qualitative analyses are conducted, proving the existence, uniqueness, positivity, and boundedness of solutions. The basic reproduction number [Formula: see text] is derived and serves as a threshold parameter determining the global stability of both virus-free and endemic equilibria. A detailed bifurcation analysis confirms a forward transcritical bifurcation at [Formula: see text]. Numerical simulations for various fractional orders demonstrate that stronger memory effects (smaller fractional orders) delay virus spread and alter system dynamics. Stochastic simulations using the Euler–Maruyama method reveal the influence of random perturbations on infection outcomes, highlighting the necessity of incorporating noise in realistic cybersecurity models. Optimal control strategies—prevention, treatment, and screening—are proposed and evaluated using Pontryagin’s Maximum Principle. Sensitivity analyses identify the transmission rate [Formula: see text] and latency transition rate [Formula: see text] as critical parameters. The model provides a comprehensive framework for understanding and managing computer virus outbreaks with memory and uncertainty considerations.