Epidemic dynamics on higher-dimensional small world networks

Abstract

Dimension governs dynamical processes on networks. The social and technological networks which we encounter in everyday life span a wide range of dimensions, but studies of spreading on finite-dimensional networks are usually restricted to one or two dimensions. To facilitate investigation of the impact of dimension on spreading processes, we define a flexible higher-dimensional small world network model and characterize the dependence of its structural properties on dimension. Subsequently, we derive mean field, pair approximation, intertwined continuous Markov chain and probabilistic discrete Markov chain models of a COVID-19-inspired susceptible-exposed-infected-removed (SEIR) epidemic process with quarantine and isolation strategies, and for each model identify the basic reproduction number , which determines whether an introduced infinitesimal level of infection in an initially susceptible population will shrink or grow. We apply these four continuous state models, together with discrete state Monte Carlo simulations, to analyse how spreading varies with model parameters. Both network properties and the outcome of Monte Carlo simulations vary substantially with dimension or rewiring rate, but predictions of continuous state models change only slightly. A different trend appears for epidemic model parameters: as these vary, the outcomes of Monte Carlo change less than those of continuous state methods. Furthermore, under a wide range of conditions, the four continuous state approximations present similar deviations from the outcome of Monte Carlo simulations. This bias is usually least when using the pair approximation model, varies only slightly with network size, and decreases with dimension or rewiring rate. Finally, we characterize the discrepancies between Monte Carlo and continuous state models by simultaneously considering network efficiency and network size.

Correlation dimension in empirical networks

“…Here we consider regular D-dimensional toroidal lattices with even degree k 2D and size N = L D , where L is an integer [9]. 3 The desired network size N 0 may not be a power of D; we realize a network of size about N 0 by realizing one of size N = N 0 1/D D .…”

“…Next we characterize dimension of a wide selection of empirical networks with nontrivial scaling intervals, as well as higher-dimensional generalizations [9] of the small-world network model [47]. We estimate correlation dimension D and upper cutoff s max by either finding the final minimum of the KS distance or minimizing the negative log-likelihood while fitting a power law to the correlation c(s).…”

“…Dimension is a fundamental property of networks, determining structural characteristics [1][2][3][4], governing dynamical processes [3][4][5][6][7] such as epidemic spreading [8,9], and supporting investigations of transportation [10], communication [11], biology [12,13], energy distribution [14], percolation [15], software systems [16], and fundamental physics [17,18]. Network correlation dimension [19] defines the scaling of separation between pairs of nodes so, compared to other formulations of dimension, is of particularly clear relevance to many dynamical processes on networks.…”

“…We employ lattices mentioned but not explored in Ref [9],. which use a periodic version of the city block (or Manhattan or taxi cab) metric.…”

Epidemic Dynamics of Two-Pathogen Spreading for Pairwise Models

“…The key insight to capturing spatial information is to track pairs rather than individuals [12,[16][17][18][19][20][21][22][23][24][25][26]. Altmann first formulated a partner model to investigate the sexual disease and inferred the basic reproduction number based on a Markov process [16].…”