PL EN


Preferences help
enabled [disable] Abstract
Number of results
Journal
2013 | 11 | 12 | 1662-1673
Article title

Finite size effects in epidemic spreading: the problem of overpopulated systems

Content
Title variants
Languages of publication
EN
Abstracts
EN
In this paper we analyze the impact of network size on the dynamics of epidemic spreading. In particular, we investigate the pace of infection in overpopulated systems. In order to do that, we design a model for epidemic spreading on a finite complex network with a restriction to at most one contamination per time step, which can serve as a model for sexually transmitted diseases spreading in some student communes. Because of the highly discrete character of the process, the analysis cannot use the continuous approximation widely exploited for most models. Using a discrete approach, we investigate the epidemic threshold and the quasi-stationary distribution. The main results are two theorems about the mixing time for the process: it scales like the logarithm of the network size and it is proportional to the inverse of the distance from the epidemic threshold.
Publisher

Journal
Year
Volume
11
Issue
12
Pages
1662-1673
Physical description
Dates
published
1 - 12 - 2013
online
20 - 12 - 2013
References
  • [1] R. M. Anderson, R. M. May, Infectious Diseases of Humans, Dynamics and Control (Oxford University Press, Oxford, 1992) [PubMed]
  • [2] M. E. J. Newman, SIAM Rev. 45, 167 (2003) http://dx.doi.org/10.1137/S003614450342480[Crossref]
  • [3] M. Boguñá, C. Castellano, R. Pastor-Satorras, Phys. Rev. Lett. 111, 068701 (2013) http://dx.doi.org/10.1103/PhysRevLett.111.068701[Crossref]
  • [4] C. Buono, F. Vazquez, P. A. Macri, L. A. Braunstein, Phys. Rev. E 88, 022813 (2013) http://dx.doi.org/10.1103/PhysRevE.88.022813[Crossref]
  • [5] A. S. Mata, S. C. Ferreira, Europhys. Lett. 103, 48003 (2013) http://dx.doi.org/10.1209/0295-5075/103/48003[Crossref]
  • [6] X.-L. Peng, X.-J. Xu, X. Fu, T. Zhou, Phys. Rev. E 87, 022813 (2013) http://dx.doi.org/10.1103/PhysRevE.87.022813[Crossref]
  • [7] A. S. Saumell-Mendiola, M. Ángeles Serrano, M. Boguñá, Phys. Rev. E 86, 026106 (2012) http://dx.doi.org/10.1103/PhysRevE.86.026106[Crossref]
  • [8] A. Barrat, M. Barthélemy, A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, Cambridge, 2008) http://dx.doi.org/10.1017/CBO9780511791383[Crossref]
  • [9] W. Ganczarek, arXiv:1307.5503 [physics.soc-ph]
  • [10] R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001) http://dx.doi.org/10.1103/PhysRevLett.86.3200[Crossref]
  • [11] R. Pastor-Satorras, A. Vespignani, Evolution and Structure of the Internet: A Statistical Physics Approach (Cambridge University Press, Cambridge, 2004) http://dx.doi.org/10.1017/CBO9780511610905[Crossref]
  • [12] M. Boguñá, R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 90, 028701 (2003) http://dx.doi.org/10.1103/PhysRevLett.90.028701[Crossref]
  • [13] S. Gómez, A. Arenas, J. Borge-Holthoefer, S. Meloni, Y. Moreno, Europhys. Lett. 89, 38009 (2010) http://dx.doi.org/10.1209/0295-5075/89/38009[Crossref]
  • [14] M. Boguñá, R. Pastor-Satorras, Phys. Rev. E 68, 036112 (2003) http://dx.doi.org/10.1103/PhysRevE.68.036112[Crossref]
  • [15] Y. Moreno, J. B. Gómez, A. F. Pacheco, Phys. Rev. E 68, 035103 (2003) http://dx.doi.org/10.1103/PhysRevE.68.035103[Crossref]
  • [16] A. V. Goltsev, S. N. Dorogovtsev, J. G. Oliveira, J. F. F. Mendes, Phys. Rev. Lett. 109, 128702 (2012) http://dx.doi.org/10.1103/PhysRevLett.109.128702[Crossref]
  • [17] H. K. Lee, P.-S. Shim, J. D. Noh, Phys. Rev. E 87, 062812 (2013) http://dx.doi.org/10.1103/PhysRevE.87.062812[Crossref]
  • [18] T. Petermann, P. De Los Rios, J. Theor. Biol. 229, 1 (2004) http://dx.doi.org/10.1016/j.jtbi.2004.02.017[Crossref]
  • [19] R. Pastor-Satorras, A. Vespignani, Phys. Rev. E 65, 035108(R) (2002) http://dx.doi.org/10.1103/PhysRevE.65.035108[Crossref]
  • [20] T. E. Harris, Ann. Probab. 2, 969 (1974) http://dx.doi.org/10.1214/aop/1176996493[Crossref]
  • [21] M. Boguñá, C. Castellano, R. Pastor-Satorras, Phys. Rev. E 79, 036110 (2009) http://dx.doi.org/10.1103/PhysRevE.79.036110[Crossref]
  • [22] Y. Li, D. Han, J. Stat. Phys. 153, 312 (2013) http://dx.doi.org/10.1007/s10955-013-0832-7[Crossref]
  • [23] G. Odor, EPJ Web of Conferences 44, 04005 (2013) http://dx.doi.org/10.1051/epjconf/20134404005[Crossref]
  • [24] A. Sinclair, Algorithms for random generation and counting: a Markov chain approach (Birkhauser Verlag, Boston-Basel-Berlin, 1993) http://dx.doi.org/10.1007/978-1-4612-0323-0[Crossref]
  • [25] E. N. Gilbert, Ann. Math. Stat. 30, 1141 (1959) http://dx.doi.org/10.1214/aoms/1177706098[Crossref]
  • [26] M. E. J. Newman, Phys. Rev. Lett. 89, 208701 (2002) http://dx.doi.org/10.1103/PhysRevLett.89.208701[Crossref]
  • [27] D. Shah, Found. Trends Net. 3, 1 (2009) http://dx.doi.org/10.1561/1300000014[Crossref]
  • [28] D. J. Watts, S. H. Strogatz, Nature 393, 440 (1998) http://dx.doi.org/10.1038/30918[Crossref]
  • [29] M. E. J. Newman, Networks: An introduction (Oxford University Press, Oxford, 2011)
  • [30] A. M. Yaglom, Dokl. Acad. Nauk SSSR 56, 795 (1947) (in Russian)
  • [31] J. N. Darroch, E. Seneta, J. Appl. Prob. 2, 88 (1965) http://dx.doi.org/10.2307/3211876[Crossref]
  • [32] E. Seneta, D. Vere-Jones, J. Appl. Prob. 3, 403 (1966) http://dx.doi.org/10.2307/3212128[Crossref]
  • [33] J. N. Darroch, E. Seneta, J. Appl. Prob. 4, 192 (1967) http://dx.doi.org/10.2307/3212311[Crossref]
  • [34] R. L. Tweedie, J. Appl. Probab. 35, 517 (1998) http://dx.doi.org/10.1239/jap/1032265201[Crossref]
  • [35] D. J. Daley, Ann. Math. Statist. 40, 532 (1969) http://dx.doi.org/10.1214/aoms/1177697721[Crossref]
  • [36] M. Buiculescu, J. Appl. Probab. 12, 60 (1975) http://dx.doi.org/10.2307/3212407[Crossref]
  • [37] E. A. van Doorn, P. K. Pollett, Eur. J. Oper. Res. 230, 1 (2013) http://dx.doi.org/10.1016/j.ejor.2013.01.032[Crossref]
  • [38] R. S. Sander, S. C. Ferreira, R. Pastor-Satorras, Phys. Rev. E 87, 022820 (2013) http://dx.doi.org/10.1103/PhysRevE.87.022820[Crossref]
  • [39] P. Brémaud, Markov Chains, Gibbs Fields, Monte Carlo Simulations and Queues (Springer-Verlag, New York, 1999)
  • [40] E. Seneta, Non-negative Matrices and Markov Chains (Springer Science+Business Media, New York, 2006)
  • [41] F. R. Gantmacher, Applications of the Theory of Matrices (Interscience Publishers, New York, 1959)
  • [42] L. B. Koralov, Y. G. Sinai, Theory of Probability and Random Processes (Springer-Verlag, Berlin, 2007) http://dx.doi.org/10.1007/978-3-540-68829-7[Crossref]
  • [43] L. A. Breyer, R. O. Roberts, Stoc. Proc. Appl. 84, 177 (1999) http://dx.doi.org/10.1016/S0304-4149(99)00018-6[Crossref]
Document Type
Publication order reference
Identifiers
YADDA identifier
bwmeta1.element.-psjd-doi-10_2478_s11534-013-0312-3
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.