Modelling the Spread of Infectious Diseases in Complex Metapopulations

J. Saldaña

Mathematical Modelling of Natural Phenomena (2010)

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

Abstract

top
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.

How to cite

top

Saldaña, 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
  1. J. Anderson. A secular equation for the eigenvalues of a diagonal matrix perturbation. Linear Algebra Appl.246 (1996), 49-70. Zbl0861.15006
  2. A. Baronchelli, M. Catanzaro, R. Pastor-Satorras. Bosonic reaction-diffusion processes on scale-free networks. Phys. Rev. E78 (2008), 016111 
  3. A. Berman, R.J. Plemmons. Nonnegative matrices in the mathematical sciences. SIAM, Classics in Applied Mathematics 9, Philadelphia, PA, 1994.  Zbl0815.15016
  4. M. Boguñá, R. Pastor-Satorras. Epidemic spreading in correlated complex networks. Phys.Rev.E66 (2002), 047104 
  5. V. Colizza, R. Pastor-Satorras A. Vespignani.Reaction-diffusion processes and metapopulation models in heterogeneous networks. Nat. Phys.3 (2007), 276–282. 
  6. V. Colizza, A. Vespignani. Invasion Threshold in Heterogeneous Metapopulation Networks. Phys. Rev. Lett.99 (2007), 148701 
  7. V. Colizza A. Vespignani.Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations. J. theor. Biol.251 (2008), 450–467. 
  8. 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. 
  9. A. Fall, A. Iggidr, G. Sallet J.J. Tewa. Epidemiological models and Lyapunov functions. Math. Model. Nat. Phenom.2 (2007), 62–83. Zbl1337.92206
  10. L. Hufnagel, D. Brockmann T. Geisel.Forecast and control of epidemics in a globalized world. PNAS101 (2004), 15124–15129. 
  11. 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.  
  12. M. J. Keeling, P. Rohani. Modeling infectious diseases in humans and animals. Princeton University Press, 2008.  Zbl1279.92038
  13. 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
  14. L. S. Liebovitch I. B. Schwartz.Migration induced epidemics: dynamics of flux-based multipatch models. Phys. Lett. A332 (2004), 256–267. Zbl1123.92309
  15. M. E. J. Newman, S. H. Strogatz, D. J. Watts. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E64 (2001), 026118 
  16. M. E. J. Newman. Mixing patterns in networks. Phys. Rev. E67 (2003), 026126 Zbl1151.91746
  17. 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
  18. L.A. Rvachev I.M. Longini.A mathematical model for the global spread of influenza. Math. Biosci.75 (1985), 3-22. Zbl0567.92017
  19. J. Saldaña. Continuous-time formulation of reaction-diffusion processes on heterogeneous metapopulations. Phys. Rev. E78 (2008), 012902 
  20. W. Wang X.-Q. Zhao.An epidemic model in a patchy environment. Math. Biosci.190 (2004), 97–112. Zbl1048.92030

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.