Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-13T04:13:36.822Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral Properties of the Connectivity Matrix and theSIS-epidemic Threshold for Mid-size Metapopulations

Published online by Cambridge University Press:  24 April 2014

D. Juher  and
V. Mañosa
D. Juher*
Affiliation:
Dept. Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, 17071, Girona, Spain
V. Mañosa
Affiliation:
Dept. Matemàtica Aplicada III, Universitat Politècnica de Catalunya, 08222, Terrassa, Spain
*
⋆Corresponding author. E-mail: david.juher@udg.edu
Get access

Abstract

We consider the spread of an infectious disease on a heterogeneous metapopulation definedby any (correlated or uncorrelated) network. The infection evolves under transmission,recovery and migration mechanisms. We study some spectral properties of a connectivitymatrix arising from the continuous-time equations of the model. In particular we show thatthe classical sufficient condition of instability for the disease-free equilibrium, wellknown for the particular case of uncorrelated networks, works also for the general case.We give also an alternative condition that yields a more accurate estimation of theepidemic threshold for correlated (either assortative or dissortative) networks.

Keywords

SIS epidemiccomplex networks
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 2: Epidemics models on networks , 2014 , pp. 108 - 120
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, R.M., May, R.M.. Directly transmitted infectious-diseases: control by vaccination. Science, 215 (1982), 1053–1060. CrossRefGoogle ScholarPubMed
Barabási, A., Albert, R.. Emergence of scaling in random networks. Science, 86 (1999), 509–512. Google Scholar
A. Berman, R.J. Plemmons. Nonnegative matrices in the Mathematical Sciences. Classics in Applied Mathematics, 9 SIAM (1994).
Boguñá, M., Pastor-Satorras, R.. Epidemic spreading in correlated complex networks. Physical Review E, 66 (4) (2002), 047104. CrossRefGoogle Scholar
Colizza, V., Vespignani, A.. Invasion threshold in Heterogeneous metapopulation models. Physical Review Letters, 99 (2007), 148701. CrossRefGoogle Scholar
Colizza, V., Pastor-Satorras, R., Vespignani, A.. Reaction-diffusion processes and metapopulation models in heterogeneous networks. Nature Physics, 3 (2007), 276–282. CrossRefGoogle Scholar
Colizza, V., Vespignani, A.. Epidemic modeling in metapopulation systems with heterogeneous coupling pattern. J. Theor. Biol, 251 (2008), 450–467. CrossRefGoogle ScholarPubMed
Cross, P. C., Johnson, P. L. F., Lloyd-Smith, J. O., Getz, W. M.. Utility of R0 as a predictor of disease invasion in structured populations. J. R. Soc. Interface, 4 (2007), 4:315–24. CrossRefGoogle Scholar
K. Dietz. Overall population patterns in the transmission cycle of infectious disease agents, in Population Biology of Infectious Diseases (ed. R.M. Anderson, R.M. May), Dahlem Workshop Reports Vol. 25, 87-102. Springer (1982).
Eames, K. T. D., Keeling, M. J.. Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases. PNAS, 99 (2002), 13330–13335. CrossRefGoogle ScholarPubMed
Fukś, H., Lawniczak, A.T., Duchesne, R.. Effects of population mixing on the spread of SIR epidemics. The European Physical Journal B, 50 issue 1 (2006), 209–214. CrossRefGoogle Scholar
Garcia-Domingo, J.L., Juher, D., Saldaña, J.. Degree correlations in growing networks with deletion of nodes. Physica D, 237 (2008), 640–651. CrossRefGoogle Scholar
R.A. Horn, C.R. Johnson. Matrix Analysis, Cambridge Univ. Press (1990).
Hufnagel, L., Brockmann, D., Geisel, T.. Forecast and control of epidemics in a globalized world. PNAS, 101 (2004), 15124. CrossRefGoogle Scholar
Juher, D., Ripoll, J., Saldaña, J.. Analysis and Monte Carlo simulations of a model for the spread of infectious diseases in heterogeneous metapopulations. Physical Review E, 80 (2009), 041920. CrossRefGoogle Scholar
Keeling, M. J.. The effects of local spatial structure on epidemiological invasions. Proc. R. Soc. Lond. B, 266 (1999), 859–869. CrossRefGoogle ScholarPubMed
McCallum, H., Barlow, N., Hone, J.. How should pathogen transmission be modelled?. Trends in Ecology & Evolution, 16 (2001), 295–300. CrossRefGoogle ScholarPubMed
May, R. M., Lloyd, A. L.. Infection dynamics on scale-free networks. Physical Review E, 64 (2001), 066112. CrossRefGoogle ScholarPubMed
Newman, M. E. J.. Spread of epidemic disease on networks, Physical Review E. 66 (2002), 016128. CrossRefGoogle Scholar
Newman, M. E. J.. Assortative mixing in networks, Phys. Rev. Lett. 89 (2002), 208701. CrossRefGoogle ScholarPubMed
Rho, Y.A., Liebovitch, L. S., Schwartz, I. B.. Dynamical response of multi-patch, flux-based models to the input of infected people: Epidemic response to initiated events. Physics Letters A, 372 (2008), 5017. CrossRefGoogle Scholar
Saldaña, J.. Continuous-time formulation of reaction-diffusion processes on heterogeneous metapopulations. Physical Review E, 78 (2008), 012902. CrossRefGoogle Scholar
Saldaña, J.. Modelling the spread of infectious diseases in complex metapopulations. Mathematical Modelling of Natural Phenomema, 5 (2010), No. 6, 22–37. CrossRefGoogle Scholar
H. R. Thieme. Mathematics in population biology. Princeton Univ. Press (2003).
Xulvi-Brunet, R., Sokolov, I.M.. Changing correlations in networks: assortativity and dissortativity. Acta Physica Polonica B. 36 (2005), 1431–1455. Google Scholar