Abstract
In this paper, we justify by the use of Enumerative Combinatorics, the applicability of an asymptotic stability result on Discrete-Time Epidemics in Complex Networks, where the complex dynamics of an epidemic model to identify the nodes that contribute the most to the propagation process are analyzed, and, because of that, are good candidates to be controlled in the network in order to stabilize the network to reach the extinction state. The epidemic model analyzed was proposed and published in 2011 by of Gómez et al. The asymptotic stability result obtained in the present article imply 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 using a refined version of the asymptotic stability result were presented in another paper of the second author for large free-scale and regular networks that corroborate the theoretical findings. In the present article, we justify the applicability of the controllability result obtained in the mentioned paper in almost all the cases by means of the use of Combinatorics.
Highlights
The main aim of the present article is to prove the applicability of an asymptotic stability result on discrete-time SIS epidemics in complex networks by the use of a combinatorial enumeration argument.Here SIS means that the nodes can be susceptible to contagion, can transit to a state of infection and once they are healthy they can become again susceptible to contagion.The bifurcation analysis that will be presented in Section 4 as well as the criterion for selection of control nodes that will be presented in Section 5 of the present article were unpublished results prior to the more general and refined criterion obtained in the article [1]
The second author, in collaboration with other colleagues, refined the selection criterion of nodes to be controlled obtained in Section 5 of the present article and tested by simulation that the criterion worked under the hypothesis of non-homogeneity, for scale-free type topologies in the article [1]
We have applied the techniques of combinatorial enumeration to prove the applicability of a selection result of nodes to be controlled in a regular network modeled by a graph under the hypothesis of homogeneity in the behavior of the nodes
Summary
The main aim of the present article is to prove the applicability of an asymptotic stability result on discrete-time SIS epidemics in complex networks by the use of a combinatorial enumeration argument. During the development of the criterion of selection of nodes to be controlled, presented in Section 5 of the present article, we faced the problem that in the case that the behavior transition and interaction of the nodes were homogenous and the network had a topology of regular type, that is to say that the nodes had the same degree, we would be forced to have to control all the nodes since we could not distinguish them Due to this drawback, we decided to show through an enumerative combinatorial argument that, even in the case of homogeneous behavior of the nodes, the criterion is applicable in most cases because the probability that a randomly generated graph will be regular tends to zero as the number of nodes.
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