  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 $\Phi_S(t)$, namely, the probability that a given neighbor of a node
is susceptible at time $t$, 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 $t_c$ 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 $t_c$. Our theoretical results are confirmed by
extensive simulations of the SIR process.
