# Modelling the Spread of Infectious Diseases in Complex Metapopulations

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 5, Issue: 6, page 22-37
- ISSN: 0973-5348

## Access Full Article

top## Abstract

top## How to cite

topSaldaña, J.. "Modelling the Spread of Infectious Diseases in Complex Metapopulations." Mathematical Modelling of Natural Phenomena 5.6 (2010): 22-37. <http://eudml.org/doc/197644>.

@article{Saldaña2010,

abstract = {Two main approaches have been considered for modelling the dynamics of the SIS model on
complex metapopulations, i.e, networks of populations connected by migratory flows whose
configurations are described in terms of the connectivity distribution of nodes (patches)
and the conditional probabilities of connections among classes of nodes sharing the same
degree. In the first approach migration and transmission/recovery process alternate
sequentially, and, in the second one, both processes occur simultaneously. Here we follow
the second approach and give a necessary and sufficient condition for the instability of
the disease-free equilibrium in generic networks under the assumption of limited (or
frequency-dependent) transmission. Moreover, for uncorrelated networks and under the
assumption of non-limited (or density-dependent) transmission, we give a bounding interval
for the dominant eigenvalue of the Jacobian matrix of the model equations around the
disease-free equilibrium. Finally, for this latter case, we study numerically the
prevalence of the infection across the metapopulation as a function of the patch
connectivity.},

author = {Saldaña, J.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {metapopulations; infectious diseases; reaction-diffusion processes},

language = {eng},

month = {4},

number = {6},

pages = {22-37},

publisher = {EDP Sciences},

title = {Modelling the Spread of Infectious Diseases in Complex Metapopulations},

url = {http://eudml.org/doc/197644},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Saldaña, J.

TI - Modelling the Spread of Infectious Diseases in Complex Metapopulations

JO - Mathematical Modelling of Natural Phenomena

DA - 2010/4//

PB - EDP Sciences

VL - 5

IS - 6

SP - 22

EP - 37

AB - Two main approaches have been considered for modelling the dynamics of the SIS model on
complex metapopulations, i.e, networks of populations connected by migratory flows whose
configurations are described in terms of the connectivity distribution of nodes (patches)
and the conditional probabilities of connections among classes of nodes sharing the same
degree. In the first approach migration and transmission/recovery process alternate
sequentially, and, in the second one, both processes occur simultaneously. Here we follow
the second approach and give a necessary and sufficient condition for the instability of
the disease-free equilibrium in generic networks under the assumption of limited (or
frequency-dependent) transmission. Moreover, for uncorrelated networks and under the
assumption of non-limited (or density-dependent) transmission, we give a bounding interval
for the dominant eigenvalue of the Jacobian matrix of the model equations around the
disease-free equilibrium. Finally, for this latter case, we study numerically the
prevalence of the infection across the metapopulation as a function of the patch
connectivity.

LA - eng

KW - metapopulations; infectious diseases; reaction-diffusion processes

UR - http://eudml.org/doc/197644

ER -

## References

top- J. Anderson. A secular equation for the eigenvalues of a diagonal matrix perturbation. Linear Algebra Appl.246 (1996), 49-70. Zbl0861.15006
- A. Baronchelli, M. Catanzaro, R. Pastor-Satorras. Bosonic reaction-diffusion processes on scale-free networks. Phys. Rev. E78 (2008), 016111
- A. Berman, R.J. Plemmons. Nonnegative matrices in the mathematical sciences. SIAM, Classics in Applied Mathematics 9, Philadelphia, PA, 1994. Zbl0815.15016
- M. Boguñá, R. Pastor-Satorras. Epidemic spreading in correlated complex networks. Phys.Rev.E66 (2002), 047104
- V. Colizza, R. Pastor-Satorras A. Vespignani.Reaction-diffusion processes and metapopulation models in heterogeneous networks. Nat. Phys.3 (2007), 276–282.
- V. Colizza, A. Vespignani. Invasion Threshold in Heterogeneous Metapopulation Networks. Phys. Rev. Lett.99 (2007), 148701
- V. Colizza A. Vespignani.Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations. J. theor. Biol.251 (2008), 450–467.
- P. C. Cross, P. L. F. Johnson, J. O. Lloyd-Smith W. M. Getz.Utility of R0 as a predictor of disease invasion in structured populations. J. R. Soc. Interface, 4 (2007), 315-324.
- A. Fall, A. Iggidr, G. Sallet J.J. Tewa. Epidemiological models and Lyapunov functions. Math. Model. Nat. Phenom.2 (2007), 62–83. Zbl1337.92206
- L. Hufnagel, D. Brockmann T. Geisel.Forecast and control of epidemics in a globalized world. PNAS101 (2004), 15124–15129.
- D. Juher, J. Ripoll, J. Saldaña. Analysis and Monte-Carlo simulations of a model for the spread of infectious diseases in heterogeneous metapopulations. Phys. Rev. E80 (2009) 041920.
- M. J. Keeling, P. Rohani. Modeling infectious diseases in humans and animals. Princeton University Press, 2008. Zbl1279.92038
- J. Li, X. Zou. Dynamics of an epidemic model with non-local infections for diseases with latency over a patchy environment. J. Math. Biol. (2009) DOI Zbl1198.92040URI10.1007/s00285-009-0280-9
- L. S. Liebovitch I. B. Schwartz.Migration induced epidemics: dynamics of flux-based multipatch models. Phys. Lett. A332 (2004), 256–267. Zbl1123.92309
- M. E. J. Newman, S. H. Strogatz, D. J. Watts. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E64 (2001), 026118
- M. E. J. Newman. Mixing patterns in networks. Phys. Rev. E67 (2003), 026126 Zbl1151.91746
- Y.-A. Rho, L. S. Liebovitch I. B. Schwartz.Dynamical response of multi-patch, flux-based models to the input of infected people: Epidemic response to initiated events. Phys. Lett. A372 (2008), 5017–5025. Zbl1221.92071
- L.A. Rvachev I.M. Longini.A mathematical model for the global spread of influenza. Math. Biosci.75 (1985), 3-22. Zbl0567.92017
- J. Saldaña. Continuous-time formulation of reaction-diffusion processes on heterogeneous metapopulations. Phys. Rev. E78 (2008), 012902
- W. Wang X.-Q. Zhao.An epidemic model in a patchy environment. Math. Biosci.190 (2004), 97–112. Zbl1048.92030

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.