Global Dynamics at the Critical Threshold R0 = 1 in a Delayed Computer Virus Model With Saturated Incidence

This study investigates the propagation dynamics of computer viruses using a delayed susceptible–latent–breaking (S–L–B) compartmental model that incorporates a discrete latency delay and a saturated incidence rate. While global behavior away from the critical threshold is well documented in the literature, the dynamics at R0=1 remain delicate because the disease-free equilibrium is nonhyperbolic. We analyze this regime using a center-manifold reduction for retarded functional differential equations. The delayed characteristic equation is used to identify the threshold, and the local reduced dynamics have a transcritical normal form structure under the standard spectral nonresonance assumption. The resulting critical trajectory exhibits algebraic relaxation O1/t instead of exponential decay. The saturation coefficient enters the quadratic normal-form term with a fixed sign pattern, implying a forward threshold for all physically admissible κ≥0. The WannaCry incident is considered solely as a qualitative source of motivation, and no parameter calibration based on incident telemetry data has been conducted. Complementary delay-aware simulations and near-threshold sweeps show that the just-below-threshold regimes remain exponential but can decay slowly, whereas the delay mainly perturbs the amplitude in the critical tail for a nominal parameter set. From a cybersecurity standpoint, these findings suggest that adjusting defense mechanisms only to achieve R0⟶1− may still result in prolonged and transient infection dynamics. Consequently, effective mitigation strategies should aim to maintain a sufficiently safe margin below the unity.