Susceptible-infected-susceptible model: A comparison ofN-intertwined and heterogeneous mean-field approximations

Abstract

We introduce the ε-susceptible-infected-susceptible (SIS) spreading model, which is taken as a benchmark for the comparison between the N-intertwined approximation and the Pastor-Satorras and Vespignani heterogeneous mean-field (HMF) approximation of the SIS model. The N-intertwined approximation, the HMF approximation, and the ε-SIS spreading model are compared for different graph types. We focus on the epidemic threshold and the steady-state fraction of infected nodes in networks with different degree distributions. Overall, the N-intertwined approximation is superior to the HMF approximation. The N-intertwined approximation is exactly the same as the HMF approximation in regular graphs. However, for some special graph types, such as the square lattice graph and the path graph, the two mean-field approximations are both very different from the ε-SIS spreading model.

Accuracy criterion for the mean-field approximation in susceptible-infected-susceptible epidemics on networks

“…Thus, MFA implicitly assumes "large degrees" in graphs when the graph size N grows. The latter hints that MFA is likely not accurate (i) in lattices or regular graphs with fixed degree r, not depending on N , as confirmed by simulations in, e.g., [25], and (ii) in small graphs, say N < 10.…”

“…In particular, for SIS epidemics described by (2), this question translates to "which graph (or set of graphs) minimizes R i in (4) for any node i ?" So far (e.g., [25]), the accuracy of MFA has been evaluated by comparing the MFA epidemic threshold τ (1) c (or metastable state fraction of infected nodes) with the exact SIS epidemic threshold τ c . Unfortunately, the epidemic threshold as an accuracy metric is difficult to compute precisely and does not provide much insight to determine for which graph MFA is accurate.…”

“…Scaling of max τ r T with the network size N Figure 3 shows the scaling of max τ r T versus the network size N for the complete graph K N , the path graph P N (a tree with maximum hop count N − 1), the star graph K 1,N−1 , and the square lattice, for N ranging from N = 10 up to N = 100. The data were obtained by long simulations, described in detail in [25] . The simulations for N = 10 were compared to the 2 N -state Markov chain solution and were more accurate than two significant digits, so that the simulations were indistinguishable from the Markov chain solution in a plot.…”

Stochastic descriptors in an SIR epidemic model for heterogeneous individuals in small networks

“…Different approaches have been followed in the literature, and they take some advantage of the particular network structures arising in several applications in reality, such as scale-free networks [32] in problems related to social networks and the Internet, or small-world networks, introduced by Watts and Strogatz [43], which can be considered as a half-way point between lattices and random graphs. In general, analysing the dynamics of the spread of an epidemic within a network is a difficult problem from the analytical point of view, so that it is usual to follow Monte Carlo simulating approaches, or other approximating techniques such as the mean-field approximation [31,35], where individuals are classified and analysed in terms of their degree distribution, or the recently proposed N-intertwined approximation [42], originally developed for a SIS (susceptible-infected-susceptible) epidemic model and recently adapted for its SIR counterpart [46]; see [25] for a detailed comparison between these methods.…”

GEMFsim: A stochastic simulator for the generalized epidemic modeling framework

“…Therefore, the Kolmogorov equation, which governs probability distribution of the Markov process, is a system of 2 N coupled differential equations which is not computationally tractable for large number of nodes. This necessitates application of approximations [6], [7], [8] or simulation in order to study the SIS model.…”