Weak and Strong Gamma Distributed Delays in a Patch-Enabled SLBP Computer-Virus Model

A patch-enabled delayed SLBP computer-virus model is extended by replacing the fixed delay in the breaking-out class with a distributed memory. Two gamma kernels are considered: the weak (single-stage) kernel, which encodes an exponentially fading memory, and the strong (two-stage) kernel, which encodes a memory peaked at a positive past time. The linear-chain trick converts the integro-differential equations into finite-dimensional ODE systems of dimension five and six, respectively, yielding polynomial characteristic equations amenable to a Routh–Hurwitz and Hopf analysis. We then carry out a direct numerical comparison of the three formulations on a common parameter set. The discrete-delay model loses stability through a Hopf bifurcation at a critical delay; both gamma models retain stability up to a substantially larger mean memory time, the weak kernel being the most stabilising and the strong kernel intermediate between the weak kernel and the discrete delay. The smearing of the past contribution by the gamma kernels therefore delays the onset of oscillations by a sizeable factor at fixed mean memory.