Infections on Temporal Networks—A Matrix-Based Approach

We extend the concept of accessibility in temporal networks to model infections with a finite infectious period such as the susceptible-infected-recovered (SIR) model. This approach is entirely based on elementary matrix operations and unifies the disease and network dynamics within one algebraic framework. We demonstrate the potential of this formalism for three examples of networks with high temporal resolution: networks of social contacts, sexual contacts, and livestock-trade. Our investigations provide a new methodological framework that can be used, for instance, to estimate the epidemic threshold, a quantity that determines disease parameters, for which a large-scale outbreak can be expected.

[1]  Christophe Fraser,et al.  HIV-1 transmission, by stage of infection. , 2008, The Journal of infectious diseases.

[2]  Herbert W. Hethcote,et al.  The Mathematics of Infectious Diseases , 2000, SIAM Rev..

[3]  M. S. Bartlett,et al.  Some Evolutionary Stochastic Processes , 1949 .

[4]  Petter Holme,et al.  Simulated Epidemics in an Empirical Spatiotemporal Network of 50,185 Sexual Contacts , 2010, PLoS Comput. Biol..

[5]  Mark C. Parsons,et al.  Communicability across evolving networks. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Christos Faloutsos,et al.  Epidemic spreading in real networks: an eigenvalue viewpoint , 2003, 22nd International Symposium on Reliable Distributed Systems, 2003. Proceedings..

[7]  Alessandro Vespignani,et al.  The GLEaMviz computational tool, a publicly available software to explore realistic epidemic spreading scenarios at the global scale , 2011, BMC infectious diseases.

[8]  D. Helbing,et al.  The Hidden Geometry of Complex, Network-Driven Contagion Phenomena , 2013, Science.

[9]  Matt J. Keeling,et al.  Networks and the Epidemiology of Infectious Disease , 2010, Interdisciplinary perspectives on infectious diseases.

[10]  M. Keeling,et al.  Networks and epidemic models , 2005, Journal of The Royal Society Interface.

[11]  L. Sander,et al.  Percolation on heterogeneous networks as a model for epidemics. , 2002, Mathematical biosciences.

[12]  A. Barrat,et al.  Simulation of an SEIR infectious disease model on the dynamic contact network of conference attendees , 2011, BMC medicine.

[13]  A. Barabasi,et al.  Evolution of the social network of scientific collaborations , 2001, cond-mat/0104162.

[14]  Jari Saramäki,et al.  Temporal Networks , 2011, Encyclopedia of Social Network Analysis and Mining.

[15]  Igor M. Sokolov,et al.  Unfolding accessibility provides a macroscopic approach to temporal networks , 2012, Physical review letters.

[16]  Matt J Keeling,et al.  Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[17]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[18]  Piotr Sapiezynski,et al.  Measuring Large-Scale Social Networks with High Resolution , 2014, PloS one.

[19]  R. Luce.  A note on Boolean matrix theory , 1952 .

[20]  Shlomo Havlin,et al.  Improving immunization strategies. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  C. Staubach,et al.  Epidemiology of classical swine fever in Germany in the 1990s. , 2000, Veterinary microbiology.

[22]  Nicola Santoro,et al.  Time-varying graphs and dynamic networks , 2010, Int. J. Parallel Emergent Distributed Syst..

[23]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[24]  Alessandro Vespignani,et al.  Immunization of complex networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Andrei Z. Broder,et al.  Graph structure in the Web , 2000, Comput. Networks.

[26]  M. Konschake,et al.  On the Robustness of In- and Out-Components in a Temporal Network , 2013, PloS one.

[27]  V. Colizza,et al.  Analytical computation of the epidemic threshold on temporal networks , 2014, 1406.4815.

[28]  Sergey Melnik,et al.  Accuracy of mean-field theory for dynamics on real-world networks. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Gwendolyn Halford,et al.  CSIS Website: Center for Strategic and International Studies , 2000 .

[30]  S. Kirkpatrick.  Percolation and Conduction , 1973 .

[31]  I. M. Sokolov,et al.  Epidemics, disorder, and percolation , 2003, cond-mat/0301394.

[32]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .

[33]  B. Bollobás.  The evolution of random graphs , 1984 .

[34]  Leo Katz,et al.  A new status index derived from sociometric analysis , 1953 .

[35]  Alain Barrat,et al.  Optimizing surveillance for livestock disease spreading through animal movements , 2012, Journal of The Royal Society Interface.

[36]  P. Grassberger.  On the critical behavior of the general epidemic process and dynamical percolation , 1983 .

[37]  M. Newman.  Spread of epidemic disease on networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  David Lazer,et al.  Inferring friendship network structure by using mobile phone data , 2009, Proceedings of the National Academy of Sciences.

[39]  Alessandro Vespignani,et al.  Reaction–diffusion processes and metapopulation models in heterogeneous networks , 2007, cond-mat/0703129.

[40]  Piet Van Mieghem,et al.  Exact Markovian SIR and SIS epidemics on networks and an upper bound for the epidemic threshold , 2014, 1402.1731.

[41]  I. Kiss,et al.  The network of sheep movements within Great Britain: network properties and their implications for infectious disease spread , 2006, Journal of The Royal Society Interface.

[42]  A. Barabasi,et al.  Lethality and centrality in protein networks , 2001, Nature.

[43]  Luis E C Rocha,et al.  Information dynamics shape the sexual networks of Internet-mediated prostitution , 2010, Proceedings of the National Academy of Sciences.