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We provide an analytical framework for evaluating the performance of automatic patching systems. We use it to quantify the speed of patch or alert dissemination required for worm containment. Motivated by scalability and trust issues, we consider a hierarchical system where network hosts are organized into subnets, each containing a patch server (termed superhost). Patches are disseminated to superhosts through an overlay connecting them and, after verification, to end hosts within subnets. The analytical framework accommodates a variety of overlays through the novel abstraction of a minimum broadcast curve. It also accommodates filtering of scans across subnets. The framework provides quantitative estimates that can guide system designers in dimensioning automatic patching systems. The results are obtained mathematically and verified by simulation.