Temporal percolation of the susceptible network in an epidemic spreading

Abstract

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 ΦS(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.

Figure 1

Equivalence between and . as a function of () obtained in Refs. , and as a function of (solid line) obtained from Eqs. (3)–(2) and (12)–(13) with and mean connectivity 4.07 in the giant component for (A) a ER network with and (B) SF network with , and . In the insets we show as a function of from the simulations (symbols) and from Eqs. (3)–(2) and (12)–(13) (solid line) for () and (). (Color online).

Figure 2. Schematic of the behavior of Eq. (12) for

. From the initial condition , and , satisfies Eq. (12). For we have two solutions that correspond to . When reaches the maximum of the function , , the giant susceptible component is destroyed. The dashed lines are used as a guide to show the possible solutions of Eq. (12).

Figure 3

Time evolution of for and () and mean connectivity in the giant component for (A) a ER network with () and (B) a SF networks with , minimal connectivity and (). The symbols correspond to the simulations with the time shifted to when % of the individuals are infected, and the solid lines correspond to the theoretical solutions (blue solid line) of Eqs. (12)–(13). In the insets we show the size of the second biggest susceptible cluster (red solid line) and the evolution of (black solid line) obtained from simulations. The value of (dashed line) was obtained from Eq. (16). has been amplified by a factor of 50 in order to show it on the same scale as the rest of the curves. The simulations are averaged over 1000 network realizations with . (Color online).