Virus Spread in Networks

Abstract

The influence of the network characteristics on the virus spread is analyzed in a new-the -intertwined Markov chain-model, whose only approximation lies in the application of mean field theory. The mean field approximation is quantified in detail. The -intertwined model has been compared with the exact 2 -state Markov model and with previously proposed "homogeneous" or "local" models. The sharp epidemic threshold , which is a consequence of mean field theory, is rigorously shown to be equal to = 1 ( max ( )), where max ( ) is the largest eigenvalue-the spectral radius-of the adjacency matrix . A continued fraction expansion of the steady-state infection probability at node is presented as well as several upper bounds.Index Terms-Epidemic threshold, Markov theory, mean field theory, spectral radius, virus spread.

A stochastic SIS epidemic model with heterogeneous contacts

“…Van Mieghem and Cator [35] generalize the graph of Simon et al [29] by adding a nodal component to the infection; i.e., λ i = ǫ for nodes i ∈ N , with ǫ > 0. Similarly to Van Mieghem et al [34], a lexicographical ordering of states leads Van Mieghem and Cator [35] to a bipartite graph and a recursive structure for the rate matrix. However, this ordering does not discriminate the various levels and therefore a nested fractal-type structure appears that, though interesting, it cannot be exploited from a computational point of view.…”

“…Since the adjacency matrix A is not assumed to be necessarily symmetric, Equation (1) allows us to reflect heterogeneous contacts, and it can be seen as an elaborate version of Van Mieghem et al [34,Equation (4)]. We express the state space S in terms of levels as ∪ N k=0 l(k), where the kth level is given by l(k) = {x ∈ S : #I(x) = k} for values k ∈ {0, 1, ..., N }.…”

“…The graph of Simon et al [29], and Van Mieghem et al [34] amounts to the values λ i = 0 and γ −1 i = γ −1 for nodes i ∈ N , and β i,j = β if (i, j) ∈ L, that is, β i,j = 0 when i and j are not connected by an edge. For this undirected graph, Simon et al [29] group the 2 N states of S in levels l(k) as in our approach, but they do not specify any concrete labeling for the N k states of l(k), and they use lumping to derive exact models where the number of equations compared to the original system is significantly reduced in such a way that the lumped system is either a reduced but exact version of the original one, or an approximation that in the limit of large graphs (N → ∞) becomes exact.…”

“…In addition, disease-causing external rates λ i ≥ 0 and mean recovery times γ −1 i are assumed to depend on node i. Unlike the exact 2 N -state Markov chain analyzed by Van Mieghem et al [34,Section III], our model with heterogeneous contacts may be thought of as a directed network, that is, a graph in which each edge has a direction, pointing from one node to another in such a way that the adjacency matrix may be asymmetric and disease-causing internal/external infection rates and recovery rates are not necessarily homogeneous.…”

“…Our approach complements and extends previous studies on the SIS type dynamics with heterogeneous contacts. The earlier papers by Simon et al [29], Van Mieghem et al [34], and Van Mieghem and Cator [35] focus on undirected graphs where heterogeneities are related to the position of the nodes within the contact network. The descriptor of interest in Simon et al [29] is the mean transient number of infectives and susceptibles in the contact network.…”