In this work, we study the evolution of the susceptible individuals during the spread of an epidemic modeled by the susceptible-infected-recovered (SIR) process spreading on the top of complex networks. Using an edge-based compartmental approach and percolation tools, we find that a time-dependent quantity , namely, the probability that a given neighbor of a node is susceptible at time , is the control parameter of a node void percolation process involving those nodes on the network not-reached by the disease. We show that there exists a critical time above which the giant susceptible component is destroyed. As a consequence, in order to preserve a macroscopic connected fraction of the network composed by healthy individuals which guarantee its functionality, any mitigation strategy should be implemented before this critical time . Our theoretical results are confirmed by extensive simulations of the SIR process.
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