Stability and Hopf Criteria in a Malware Dissemination Model for Wireless Sensor Networks with Distributed Recovery Delays

A delayed malware dissemination model for wireless sensor networks is extended by replacing the single fixed anti-virus cycle time with distributed recovery memories. Two normalized gamma kernels are considered: a one-stage exponential memory (weak kernel) and a two-stage gamma memory (strong kernel), both representing heterogeneous antivirus removal times across carrier and infectious nodes. The linear chain trick transforms the corresponding integro-differential systems into six- and eight-dimensional ordinary differential equation models. For these distributed-delay formulations, we derive the common disease-free equilibrium and its local stability conditions, prove that the endemic equilibrium and the basic reproduction number are preserved, and show that the inherited delayed-removal mechanism does not generate a positively invariant nonnegative orthant; in its place, we establish the strongest valid conditional boundedness and continuation result. Local endemic stability is reduced to a Routh–Hurwitz problem, and Hopf thresholds are obtained from simple purely imaginary roots together with transversality. For the reference parameter set, full spectral verification and first-Lyapunov-coefficient evaluation show that the first weak- and strong-kernel Hopf points are subcritical. Numerical simulations illustrate convergence below threshold and loss of equilibrium stability above it, while comparison with the discrete-delay benchmark shows that a broader recovery-time distribution postpones instability.