+
- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

  • Free to Read

Robustness and fragility of the susceptible-infected-susceptible epidemic models on complex networks

Wesley Cota1, Angélica S. Mata2, and Silvio C. Ferreira1,3

  • 1Departamento de Física, Universidade Federal de Viçosa, 36570-900 Viçosa, Minas Gerais, Brazil
  • 2Departamento de Física, Universidade Federal de Lavras, 37200-000 Lavras, Minas Gerais, Brazil
  • 3National Institute of Science and Technology for Complex Systems, Brazil
PDF

Phys. Rev. E 98, 012310 – Published 18 July, 2018

DOI: https://doi.org/10.1103/PhysRevE.98.012310

Abstract

We analyze two alterations of the standard susceptible-infected-susceptible (SIS) dynamics that preserve the central properties of spontaneous healing and infection capacity of a vertex increasing unlimitedly with its degree. All models have the same epidemic thresholds in mean-field theories but depending on the network properties, simulations yield a dual scenario, in which the epidemic thresholds of the modified SIS models can be either dramatically altered or remain unchanged in comparison with the standard dynamics. For uncorrelated synthetic networks having a power-law degree distribution with exponent γ<5/2, the SIS dynamics are robust exhibiting essentially the same outcomes for all investigated models. A threshold in better agreement with the heterogeneous rather than quenched mean-field theory is observed in the modified dynamics for exponent γ>5/2. Differences are more remarkable for γ>3, where a finite threshold is found in the modified models in contrast with the vanishing threshold of the original one. This duality is elucidated in terms of epidemic lifespan on star graphs. We verify that the activation of the modified SIS models is triggered in the innermost component of the network given by a k-core decomposition for γ<3 while it happens only for γ<5/2 in the standard model. For γ>3, the activation in the modified dynamics is collective involving essentially the whole network while it is triggered by hubs in the standard SIS. The duality also appears in the finite-size scaling of the critical quantities where mean-field behaviors are observed for the modified but not for the original dynamics. Our results feed the discussions about the most proper conceptions of epidemic models to describe real systems and the choices of the most suitable theoretical approaches to deal with these models.

Physics Subject Headings (PhySH)

Article Text

References (67)

  1. A.-L. Barabási and M. Pósfai, Network Science (Cambridge University Press, Cambridge, 2016).
  2. R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys. 74, 47 (2002).
  3. A. Barrat, M. Barthélemy, and A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, Cambridge, 2008).
  4. R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys. 87, 925 (2015).
  5. S. Chatterjee and R. Durrett, Contact processes on random graphs with power law degree distributions have critical value 0, Ann. Probab. 37, 2332 (2009).
  6. R. Durrett, Some features of the spread of epidemics and information on a random graph, Proc. Natl. Acad. Sci. USA 107, 4491 (2010).
  7. C. Castellano and R. Pastor-Satorras, Thresholds for Epidemic Spreading in Networks, Phys. Rev. Lett. 105, 218701 (2010).
  8. J. P. Gleeson, Binary-State Dynamics on Complex Networks: Pair Approximation and Beyond, Phys. Rev. X 3, 021004 (2013).
  9. M. Kitsak, L. K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H. E. Stanley, and H. A. Makse, Identification of influential spreaders in complex networks, Nat. Phys. 6, 888 (2010).
  10. M. E. J. Newman, The spread of epidemic disease on networks, Phys. Rev. E 66, 016128 (2002).
  11. M. Boguñá, C. Castellano, and R. Pastor-Satorras, Nature of the Epidemic Threshold for the Susceptible-Infected-Susceptible Dynamics in Networks, Phys. Rev. Lett. 111, 068701 (2013).
  12. S. C. Ferreira, R. S. Sander, and R. Pastor-Satorras, Collective versus hub activation of epidemic phases on networks, Phys. Rev. E 93, 032314 (2016).
  13. A. S. Mata and S. C. Ferreira, Multiple transitions of the susceptible-infected-susceptible epidemic model on complex networks, Phys. Rev. E 91, 012816 (2015).
  14. G. F. de Arruda, E. Cozzo, T. P. Peixoto, F. A. Rodrigues, and Y. Moreno, Disease Localization in Multilayer Networks, Phys. Rev. X 7, 011014 (2017).
  15. G. St-Onge, J.-G. Young, E. Laurence, C. Murphy, and L. J. Dubé, Phase transition of the susceptible-infected-susceptible dynamics on time-varying configuration model networks, Phys. Rev. E 97, 022305 (2018).
  16. C.-R. Cai, Z.-X. Wu, M. Z. Q. Chen, P. Holme, and J.-Y. Guan, Solving the Dynamic Correlation Problem of the Susceptible-Infected-Susceptible Model on Networks, Phys. Rev. Lett. 116, 258301 (2016).
  17. Z.-W. Wei, H. Liao, H.-F. Zhang, J.-R. Xie, B.-H. Wang, and G.-L. Chen, Localized-endemic state transition in the susceptible-infected-susceptible model on networks (2017), arXiv:1704.02925.
  18. C. Castellano and R. Pastor-Satorras, Relating Topological Determinants of Complex Networks to Their Spectral Properties: Structural and Dynamical Effects, Phys. Rev. X 7, 041024 (2017).
  19. X.-H. Chen, S.-M. Cai, W. Wang, M. Tang, and H. E. Stanley, Predicting epidemic threshold of correlated networks: A comparison of methods, Phys. A: Stat. Mech. Appl. 505, 500 (2018).
  20. R. Pastor-Satorras and A. Vespignani, Epidemic Spreading in Scale-Free Networks, Phys. Rev. Lett. 86, 3200 (2001).
  21. R. Pastor-Satorras and A. Vespignani, Epidemic dynamics and endemic states in complex networks, Phys. Rev. E 63, 066117 (2001).
  22. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Critical phenomena in complex networks, Rev. Mod. Phys. 80, 1275 (2008).
  23. Y. Wang, D. Chakrabarti, C. Wang, and C. Faloutsos, Epidemic spreading in real networks: An eigenvalue viewpoint, in Proc. 22nd Int. Symp. Reliab. Distrib. Syst. (IEEE Computer Society, Piscataway, NJ, 2003), pp. 25–34.
  24. P. Van Mieghem, Epidemic phase transition of the SIS type in networks, Europhys. Lett. 97, 48004 (2012).
  25. D. Chakrabarti, Y. Wang, C. Wang, J. Leskovec, and C. Faloutsos, Epidemic thresholds in real networks, ACM Trans. Inf. Syst. Secur. 10, 1 (2008).
  26. S. C. Ferreira, C. Castellano, and R. Pastor-Satorras, Epidemic thresholds of the susceptible-infected-susceptible model on networks: A comparison of numerical and theoretical results, Phys. Rev. E 86, 041125 (2012).
  27. A. S. Mata and S. C. Ferreira, Pair quenched mean-field theory for the susceptible-infected-susceptible model on complex networks, Europhys. Lett. 103, 48003 (2013).
  28. A. S. Mata, R. S. Ferreira, and S. C. Ferreira, Heterogeneous pair-approximation for the contact process on complex networks, New J. Phys. 16, 053006 (2014).
  29. R. Anderson and R. May, Infectious Diseases of Humans: Dynamics and Control, Dynamics and Control (Oxford University Press, Oxford, 1992).
  30. M. Catanzaro, M. Boguñá, and R. Pastor-Satorras, Generation of uncorrelated random scale-free networks, Phys. Rev. E 71, 027103 (2005).
  31. W. Cota and S. C. Ferreira, Optimized Gillespie algorithms for the simulation of Markovian epidemic processes on large and heterogeneous networks, Comput. Phys. Commun. 219, 303 (2017).
  32. C. Castellano and R. Pastor-Satorras, Competing activation mechanisms in epidemics on networks, Sci. Rep. 2, 371 (2012).
  33. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, k-Core Organization of Complex Networks, Phys. Rev. Lett. 96, 040601 (2006).
  34. R. Pastor-Satorras and A. Vespignani, Epidemic dynamics in finite-size scale-free networks, Phys. Rev. E 65, 035108 (2002).
  35. J. Marro and R. Dickman, Nonequilibrium Phase Transitions in Lattice Models, Collection Alea-Saclay: Monographs and Texts in Statistical Physics (Cambridge University Press, Cambridge, 1999).
  36. R. Dickman and M. A. Burschka, Nonequilibrium critical poisoning in a single-species model, Phys. Lett. A 127, 132 (1988).
  37. L. Böttcher, H. J. Herrmann, and M. Henkel, Dynamical universality of the contact process, J. Phys. A: Math. Theor. 51, 125003 (2018).
  38. R. S. Sander, M. M. de Oliveira, and S. C. Ferreira, Quasi-stationary simulations of the directed percolation universality class in d=3 dimensions, J. Stat. Mech. (2009) P08011.
  39. M. M. de Oliveira, S. G. Alves, and S. C. Ferreira, Continuous and discontinuous absorbing-state phase transitions on Voronoi-Delaunay random lattices, Phys. Rev. E 93, 012110 (2016).
  40. T. E. Harris, Contact interactions on a lattice, Ann. Probab. 2, 969 (1974).
  41. D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys. 22, 403 (1976).
  42. R. Pastor-Satorras, A. Vázquez, and A. Vespignani, Dynamical and Correlation Properties of the Internet, Phys. Rev. Lett. 87, 258701 (2001).
  43. S. Morita, Six susceptible-infected-susceptible models on scale-free networks, Sci. Rep. 6, 22506 (2016).
  44. M. Boguñá and R. Pastor-Satorras, Epidemic spreading in correlated complex networks, Phys. Rev. E 66, 047104 (2002).
  45. M. Boguñá, C. Castellano, and R. Pastor-Satorras, Langevin approach for the dynamics of the contact process on annealed scale-free networks, Phys. Rev. E 79, 036110 (2009).
  46. R. S. Sander, G. S. Costa, and S. C. Ferreira, Sampling methods for the quasistationary regime of epidemic processes on regular and complex networks, Phys. Rev. E 94, 042308 (2016).
  47. M. Boguñá, R. Pastor-Satorras, and A. Vespignani, Cut-offs and finite-size effects in scale-free networks, Eur. Phys. J. B 38, 205 (2004).
  48. J. A. Hołyst, J. Sienkiewicz, A. Fronczak, P. Fronczak, and K. Suchecki, Universal scaling of distances in complex networks, Phys. Rev. E 72, 026108 (2005).
  49. M. Henkel, H. Hinrichsen, and S. Lübeck, Non-Equilibrium Phase Transitions: Volume 1: Absorbing Phase Transitions, Theoretical and Mathematical Physics (Springer, Netherlands, 2008).
  50. C. Castellano and R. Pastor-Satorras, Non-Mean-Field Behavior of the Contact Process on Scale-Free Networks, Phys. Rev. Lett. 96, 038701 (2006).
  51. C. Castellano and R. Pastor-Satorras, Routes to Thermodynamic Limit on Scale-Free Networks, Phys. Rev. Lett. 100, 148701 (2008).
  52. H. Hong, M. Ha, and H. Park, Finite-Size Scaling in Complex Networks, Phys. Rev. Lett. 98, 258701 (2007).
  53. J. D. Noh and H. Park, Critical behavior of the contact process in annealed scale-free networks, Phys. Rev. E 79, 056115 (2009).
  54. S. C. Ferreira, R. S. Ferreira, C. Castellano, and R. Pastor-Satorras, Quasistationary simulations of the contact process on quenched networks, Phys. Rev. E 84, 066102 (2011).
  55. S. C. Ferreira, R. S. Ferreira, and R. Pastor-Satorras, Quasistationary analysis of the contact process on annealed scale-free networks, Phys. Rev. E 83, 066113 (2011).
  56. R. S. Ferreira and S. C. Ferreira, Critical behavior of the contact process on small-world networks, Eur. Phys. J. B 86, 1 (2013).
  57. W. Cota, S. C. Ferreira, and G. Ódor, Griffiths effects of the susceptible-infected-susceptible epidemic model on random power-law networks, Phys. Rev. E 93, 032322 (2016).
  58. Y. Moreno, R. Pastor-Satorras, and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, Eur. Phys. J. B 26, 521 (2002).
  59. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Ising model on networks with an arbitrary distribution of connections, Phys. Rev. E 66, 016104 (2002).
  60. C. P. Herrero, Ising model in scale-free networks: A Monte Carlo simulation, Phys. Rev. E 69, 067109 (2004).
  61. F. A. Rodrigues, T. K. D. Peron, P. Ji, and J. Kurths, The Kuramoto model in complex networks, Phys. Rep. 610, 1 (2016).
  62. A. V. Goltsev, S. N. Dorogovtsev, J. G. Oliveira, and J. F. F. Mendes, Localization and Spreading of Diseases in Complex Networks, Phys. Rev. Lett. 109, 128702 (2012).
  63. H. K. Lee, P.-S. Shim, and J. D. Noh, Epidemic threshold of the susceptible-infected-susceptible model on complex networks, Phys. Rev. E 87, 062812 (2013).
  64. M. Granovetter, Threshold models of collective behavior, Am. J. Sociol. 83, 1420 (1978).
  65. D. Centola and M. Macy, Complex contagions and the weakness of long ties, Am. J. Sociol. 113, 702 (2007).
  66. E. Campbell and M. Salathé, Complex social contagion makes networks more vulnerable to disease outbreaks, Sci. Rep. 3, 1 (2013).
  67. M. Karsai, G. Iñiguez, K. Kaski, and J. Kertész, Complex contagion process in spreading of online innovation, J. R. Soc. Interface 11, 20140694 (2014).

Outline

Information

Phys. Rev. E 98, 012310– Published 18 July, 2018
  • Received 23 April 2018
Crossmark - Check for updates button

©2018 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation

点击 这是indexloc提供的php浏览器服务，不要输入任何密码和下载