Unified Preventive and Reactive Cyber Defense Dynamics Is Still Globally Convergent

A class of the preventive and reactive cyber defense dynamics has recently been proven to be <italic>globally convergent</italic>, meaning that the dynamics <italic>always</italic> converges to a unique equilibrium whose location only depends on the values of the model parameters (but not the initial state of the dynamics). In this paper, we unify the aforementioned class of preventive and reactive cyber defense dynamics models and the closely related class of <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-intertwined epidemic models into a single framework. We prove that the unified dynamics is still globally convergent under some mild conditions, which are naturally satisfied by the two specific classes of dynamics models mentioned above and are inevitable when analyzing a more general framework. We also characterize the convergence speed of the unified dynamics. As a corollary, we obtain that the <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-intertwined epidemic model and its extension are globally convergent, together with a <italic>full</italic> characterization on their convergence speed, which is only partially addressed in the literature.