Use of Enumerative Combinatorics for proving the applicability of an asymptotic stability result on discrete-time SIS epidemics in complex networks

Abstract

In this paper, we justify by the use of Enumerative Combinatorics, that the results obtained in \cite{Alarcon1}, where is analysed the complex dynamics of an epidemic model to identify the nodes that contribute the most to the propagation process and because of that are good candidates to be controlled in the network in order to stabilize the network to reach the extinction state, is applicable in almost all the cases. The model analysed was proposed in \cite{Gomez1} %et al. [Phys.Rev.E 84, 036105(2011)] and results obtained in \cite{Alarcon1} implies that it is not necessary to control all nodes, but only a minimal set of nodes if the topology of the network is not regular. This result could be important in the spirit of considering policies of isolation or quarantine of those nodes to be controlled. Simulation results were presented in \cite{Alarcon1} for large free-scale and regular networks, that corroborate the theoretical findings. In this article we justify the applicability of the controllability result obtained in \cite{Alarcon1} in almost all the cases by means of the use of Combinatorics. {\em Mathematics Subjects Classification}: 05A16,34H20,58E25 {\em Keywords}: Asymptotic Graph Enumeration Problems; Network control; virus spreading.