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      SIR epidemics on evolving graphs

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          Abstract

          We consider evoSIR, a variant of the SIR model, on Erd\H os-Renyi random graphs in which susceptibles with an infected neighbor break that connection at rate \(\rho\) and rewire to a randomly chosen individual. We compute the critical infection rate \(\lambda_c\) and the probability of a large epidemic by showing that they are the same for the delSIR model in which \(S-I\) connections are deleted instead of rewired. The final size of a large delSIR epidemic has a continuous transition. Simulations suggest that the final size of a large evoSIR epidemic is discontinuous at \(\lambda_c\).

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          A critical point for random graphs with a given degree sequence

          Michael Molloy, Bruce Reed (1995)
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            Random graphs with arbitrary degree distributions and their applications

            D. Watts, S Strogatz, M. Newman (2000)
            Recent work on the structure of social networks and the internet has focussed attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the world-wide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.
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              The spread of epidemic disease on networks

              M. Newman (2002)
              The study of social networks, and in particular the spread of disease on networks, has attracted considerable recent attention in the physics community. In this paper, we show that a large class of standard epidemiological models, the so-called susceptible/infective/removed (SIR) models can be solved exactly on a wide variety of networks. In addition to the standard but unrealistic case of fixed infectiveness time and fixed and uncorrelated probability of transmission between all pairs of individuals, we solve cases in which times and probabilities are non-uniform and correlated. We also consider one simple case of an epidemic in a structured population, that of a sexually transmitted disease in a population divided into men and women. We confirm the correctness of our exact solutions with numerical simulations of SIR epidemics on networks.
              Article 0 comments Cited 318 times – based on 0 reviews
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                Author and article information

                Journal
                Publication date Created: 19 January 2019
                Article
                ArXiV ID: 1901.06568
                SO-VID: cf330e67-4635-47e6-b44e-f3cc4e0ca9cf
                License:

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                ACM classes 60K35
                Comments 24 pages, 12 pictures
                Categories math.PR

                ScienceOpen disciplines: Probability
                Data availability:
                ScienceOpen disciplines: Probability

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