In this paper, we propose an extended discrete-time computer virus model by integrating a memristor map to enhance the system’s dimensionality. The original three-dimensional virus model is transformed into a four-dimensional memristor-based discrete map, enabling the exploration of richer dynamical behaviors. The chaotic dynamics of the model are investigated using phase portraits, bifurcation diagrams, and the maximum Lyapunov exponent. To capture memory effects inherent in networked systems, the integer-order memristor model is further generalized to the fractional-order domain. The nonlinear dynamics and complexity of the resulting fractional-order system are analyzed through \(C_0\) complexity and Approximate Entropy (ApEn). Numerical results demonstrate the emergence of complex chaotic behavior and highlight the impact of fractional-order parameters on the system dynamics. These findings provide a theoretical basis for developing more robust strategies in computer virus detection and control.