Bifurcation in Polymorphic Virus Dynamics: An Adaptive ODE Approach With Lyapunov Stability Analysis

ABSTRACT

Polymorphic malware poses a persistent cybersecurity challenge due to its ability to dynamically alter its code structure while preserving malicious functionality, thereby evading signature‐based defenses. We develop an adaptive multi‐variant ordinary differential equation (ODE) framework that captures the co‐evolutionary dynamics between polymorphic malware and responsive defense mechanisms, integrating mutation‐driven variant switching with a feedback‐driven antivirus response. Using the next‐generation matrix approach, we derive the basic reproduction number  and show that crossing  induces a transcritical bifurcation separating eradication and persistence regimes. Global asymptotic stability of the disease‐free equilibrium is proved via the Castillo–Chavez–Feng–Huang framework. Endemic equilibria are analyzed through analytical arguments and reproducible numerical simulations, demonstrating that adaptive mutation pressure can sustain persistence even under strong defensive responses. The framework provides actionable theoretical insights for designing system‐level cybersecurity strategies.