dorsal/arxiv
View SchemaSecond look at the spread of epidemics on networks
| Authors | Eben Kenah, James M. Robins |
|---|---|
| Categories | |
| ArXiv ID | q-bio/0610057 |
| URL | https://arxiv.org/abs/q-bio/0610057 |
| DOI | 10.1103/PhysRevE.76.036113 |
| Journal | Physical Review E 76: 036113, September 2007 |
Abstract
In an important paper, M.E.J. Newman claimed that a general network-based stochastic Susceptible-Infectious-Removed (SIR) epidemic model is isomorphic to a bond percolation model, where the bonds are the edges of the contact network and the bond occupation probability is equal to the marginal probability of transmission from an infected node to a susceptible neighbor. In this paper, we show that this isomorphism is incorrect and define a semi-directed random network we call the epidemic percolation network that is exactly isomorphic to the SIR epidemic model in any finite population. In the limit of a large population, (i) the distribution of (self-limited) outbreak sizes is identical to the size distribution of (small) out-components, (ii) the epidemic threshold corresponds to the phase transition where a giant strongly-connected component appears, (iii) the probability of a large epidemic is equal to the probability that an initial infection occurs in the giant in-component, and (iv) the relative final size of an epidemic is equal to the proportion of the network contained in the giant out-component. For the SIR model considered by Newman, we show that the epidemic percolation network predicts the same mean outbreak size below the epidemic threshold, the same epidemic threshold, and the same final size of an epidemic as the bond percolation model. However, the bond percolation model fails to predict the correct outbreak size distribution and probability of an epidemic when there is a nondegenerate infectious period distribution. We confirm our findings by comparing predictions from percolation networks and bond percolation models to the results of simulations. In an appendix, we show that an isomorphism to an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model.
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"abstract": "In an important paper, M.E.J. Newman claimed that a general network-based\nstochastic Susceptible-Infectious-Removed (SIR) epidemic model is isomorphic to\na bond percolation model, where the bonds are the edges of the contact network\nand the bond occupation probability is equal to the marginal probability of\ntransmission from an infected node to a susceptible neighbor. In this paper, we\nshow that this isomorphism is incorrect and define a semi-directed random\nnetwork we call the epidemic percolation network that is exactly isomorphic to\nthe SIR epidemic model in any finite population. In the limit of a large\npopulation, (i) the distribution of (self-limited) outbreak sizes is identical\nto the size distribution of (small) out-components, (ii) the epidemic threshold\ncorresponds to the phase transition where a giant strongly-connected component\nappears, (iii) the probability of a large epidemic is equal to the probability\nthat an initial infection occurs in the giant in-component, and (iv) the\nrelative final size of an epidemic is equal to the proportion of the network\ncontained in the giant out-component. For the SIR model considered by Newman,\nwe show that the epidemic percolation network predicts the same mean outbreak\nsize below the epidemic threshold, the same epidemic threshold, and the same\nfinal size of an epidemic as the bond percolation model. However, the bond\npercolation model fails to predict the correct outbreak size distribution and\nprobability of an epidemic when there is a nondegenerate infectious period\ndistribution. We confirm our findings by comparing predictions from percolation\nnetworks and bond percolation models to the results of simulations. In an\nappendix, we show that an isomorphism to an epidemic percolation network can be\ndefined for any time-homogeneous stochastic SIR model.",
"arxiv_id": "q-bio/0610057",
"authors": [
"Eben Kenah",
"James M. Robins"
],
"categories": [
"q-bio.QM",
"cond-mat.stat-mech",
"math.PR"
],
"doi": "10.1103/PhysRevE.76.036113",
"journal_ref": "Physical Review E 76: 036113, September 2007",
"title": "Second look at the spread of epidemics on networks",
"url": "https://arxiv.org/abs/q-bio/0610057"
},
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