Computer viruses remain one of the most significant threats to computer networks, capable of spreading rapidly, disrupting services, and causing severe economic and security damages. Mathematical modeling provides valuable tools for analyzing the dynamics of virus propagation and designing effective defense strategies. In this paper, we propose a novel discrete computer virus model that integrates the effects of a memristor, a nonlinear circuit element with inherent memory. The memristor introduces strong feedback and enhances the emergence of chaotic behavior in the system. To capture long-term memory and hereditary effects, we further extend the formulation to a fractional discrete framework based on Caputo fractional difference operators. The bifurcations and the onset of chaos are investigated. To assess the degree of complexity, we employ the \(C_{0}\) complexity algorithm and the spectral entropy (SE) measure, which confirm the presence of chaotic dynamics in the fractional memristor-based virus model. The results provide new insights into how memory and nonlinearity, at both the hardware and software levels, influence the spread of computer viruses, thereby offering a deeper theoretical foundation for the development of improved cybersecurity strategies.