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Modelling the Spread of Infectious Diseases in ComplexMetapopulations

Published online by Cambridge University Press:  08 April 2010

J. Saldaña
J. Saldaña*
Affiliation:
Departament d’Informàtica i Matemàtica Aplicada Universitat de Girona, 17071 Girona Catalonia, Spain
*
*E-mail: joan.saldana@udg.edu
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Abstract

Two main approaches have been considered for modelling the dynamics of the SIS model oncomplex metapopulations, i.e, networks of populations connected by migratory flows whoseconfigurations are described in terms of the connectivity distribution of nodes (patches)and the conditional probabilities of connections among classes of nodes sharing the samedegree. In the first approach migration and transmission/recovery process alternatesequentially, and, in the second one, both processes occur simultaneously. Here we followthe second approach and give a necessary and sufficient condition for the instability ofthe disease-free equilibrium in generic networks under the assumption of limited (orfrequency-dependent) transmission. Moreover, for uncorrelated networks and under theassumption of non-limited (or density-dependent) transmission, we give a bounding intervalfor the dominant eigenvalue of the Jacobian matrix of the model equations around thedisease-free equilibrium. Finally, for this latter case, we study numerically theprevalence of the infection across the metapopulation as a function of the patchconnectivity.

Keywords

metapopulationsinfectious diseasesreaction-diffusion processes
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 6: Ecology (Part 2) , 2010 , pp. 22 - 37
Copyright
© EDP Sciences, 2010

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References

Anderson, J.. A secular equation for the eigenvalues of a diagonal matrix perturbation . Linear Algebra Appl. 246 (1996), 49-70.CrossRefGoogle Scholar
Baronchelli, A., Catanzaro, M., Pastor-Satorras, R.. Bosonic reaction-diffusion processes on scale-free networks . Phys. Rev. E 78 (2008), 016111 CrossRefGoogle ScholarPubMed
A. Berman, R.J. Plemmons. Nonnegative matrices in the mathematical sciences. SIAM, Classics in Applied Mathematics 9, Philadelphia, PA, 1994.
Boguñá, M., Pastor-Satorras, R.. Epidemic spreading in correlated complex networks . Phys.Rev.E 66 (2002), 047104 CrossRefGoogle Scholar
Colizza, V., Pastor-Satorras, R. Vespignani., A. Reaction-diffusion processes and metapopulation models in heterogeneous networks . Nat. Phys. 3 (2007), 276–282.CrossRefGoogle Scholar
Colizza, V., Vespignani, A.. Invasion Threshold in Heterogeneous Metapopulation Networks . Phys. Rev. Lett. 99 (2007), 148701 CrossRefGoogle ScholarPubMed
Colizza, V. Vespignani., A. Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations . 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 R 0 as a predictor of disease invasion in structured populations . J. R. Soc. Interface, 4 (2007), 315-324.CrossRefGoogle Scholar
Fall, A., Iggidr, A., Sallet, G. Tewa, J.J.. Epidemiological models and Lyapunov functions . Math. Model. Nat. Phenom. 2 (2007), 62–83.CrossRefGoogle Scholar
Hufnagel, L., Brockmann, D. Geisel., T. Forecast and control of epidemics in a globalized world . PNAS 101 (2004), 15124–15129.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 . Phys. Rev. E 80 (2009) 041920. CrossRefGoogle ScholarPubMed
M. J. Keeling, P. Rohani. Modeling infectious diseases in humans and animals. Princeton University Press, 2008.
Li, J., Zou, X.. Dynamics of an epidemic model with non-local infections for diseases with latency over a patchy environment . J. Math. Biol. (2009) DOI 10.1007/s00285-009-0280-9 Google Scholar
Liebovitch, L. S. Schwartz., I. B. Migration induced epidemics: dynamics of flux-based multipatch models . Phys. Lett. A 332 (2004), 256–267.CrossRefGoogle Scholar
Newman, M. E. J., Strogatz, S. H., Watts, D. J.. Random graphs with arbitrary degree distributions and their applications . Phys. Rev. E 64 (2001), 026118 CrossRefGoogle ScholarPubMed
Newman, M. E. J.. Mixing patterns in networks . Phys. Rev. E 67 (2003), 026126 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 . Phys. Lett. A 372 (2008), 5017–5025.CrossRefGoogle Scholar
Rvachev, L.A. Longini., I.M. A mathematical model for the global spread of influenza . Math. Biosci. 75 (1985), 3-22.CrossRefGoogle Scholar
Saldaña, J.. Continuous-time formulation of reaction-diffusion processes on heterogeneous metapopulations . Phys. Rev. E 78 (2008), 012902 CrossRefGoogle ScholarPubMed
Wang, W. Zhao., X.-Q. An epidemic model in a patchy environment . Math. Biosci. 190 (2004), 97–112.CrossRefGoogle Scholar