Skip to main content Accessibility help
×
Home

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

  • D. Juher (a1) and V. Mañosa (a2)

Abstract

We consider the spread of an infectious disease on a heterogeneous metapopulation defined by any (correlated or uncorrelated) network. The infection evolves under transmission, recovery and migration mechanisms. We study some spectral properties of a connectivity matrix arising from the continuous-time equations of the model. In particular we show that the classical sufficient condition of instability for the disease-free equilibrium, well known 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 the epidemic threshold for correlated (either assortative or dissortative) networks.

Copyright

Corresponding author

Corresponding author. E-mail: david.juher@udg.edu

References

Hide All
[1] Anderson, R.M., May, R.M.. Directly transmitted infectious-diseases: control by vaccination. Science, 215 (1982), 10531060.
[2] Barabási, A., Albert, R.. Emergence of scaling in random networks. Science, 86 (1999), 509512.
[3] A. Berman, R.J. Plemmons. Nonnegative matrices in the Mathematical Sciences. Classics in Applied Mathematics, 9 SIAM (1994).
[4] Boguñá, M., Pastor-Satorras, R.. Epidemic spreading in correlated complex networks. Physical Review E, 66 (4) (2002), 047104.
[5] Colizza, V., Vespignani, A.. Invasion threshold in Heterogeneous metapopulation models. Physical Review Letters, 99 (2007), 148701.
[6] Colizza, V., Pastor-Satorras, R., Vespignani, A.. Reaction-diffusion processes and metapopulation models in heterogeneous networks. Nature Physics, 3 (2007), 276282.
[7] Colizza, V., Vespignani, A.. Epidemic modeling in metapopulation systems with heterogeneous coupling pattern. J. Theor. Biol, 251 (2008), 450467.
[8] 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), 4:31524.
[9] 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).
[10] Eames, K. T. D., Keeling, M. J.. Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases. PNAS, 99 (2002), 1333013335.
[11] 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), 209214.
[12] Garcia-Domingo, J.L., Juher, D., Saldaña, J.. Degree correlations in growing networks with deletion of nodes. Physica D, 237 (2008), 640651.
[13] R.A. Horn, C.R. Johnson. Matrix Analysis, Cambridge Univ. Press (1990).
[14] Hufnagel, L., Brockmann, D., Geisel, T.. Forecast and control of epidemics in a globalized world. PNAS, 101 (2004), 15124.
[15] 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.
[16] Keeling, M. J.. The effects of local spatial structure on epidemiological invasions. Proc. R. Soc. Lond. B, 266 (1999), 859869.
[17] McCallum, H., Barlow, N., Hone, J.. How should pathogen transmission be modelled?. Trends in Ecology & Evolution, 16 (2001), 295300.
[18] May, R. M., Lloyd, A. L.. Infection dynamics on scale-free networks. Physical Review E, 64 (2001), 066112.
[19] Newman, M. E. J.. Spread of epidemic disease on networks, Physical Review E. 66 (2002), 016128.
[20] Newman, M. E. J.. Assortative mixing in networks, Phys. Rev. Lett. 89 (2002), 208701.
[21] 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.
[22] Saldaña, J.. Continuous-time formulation of reaction-diffusion processes on heterogeneous metapopulations. Physical Review E, 78 (2008), 012902.
[23] Saldaña, J.. Modelling the spread of infectious diseases in complex metapopulations. Mathematical Modelling of Natural Phenomema, 5 (2010), No. 6, 2237.
[24] H. R. Thieme. Mathematics in population biology. Princeton Univ. Press (2003).
[25] Xulvi-Brunet, R., Sokolov, I.M.. Changing correlations in networks: assortativity and dissortativity. Acta Physica Polonica B. 36 (2005), 14311455.

Keywords

Related content

Powered by UNSILO

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

  • D. Juher (a1) and V. Mañosa (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.