Modelling the Spread of Infectious Diseases in Complex Metapopulations

J. Saldaña¹

¹ Departament d’Informàtica i Matemàtica Aplicada Universitat de Girona, 17071 Girona Catalonia, Spain

Mathematical modelling of natural phenomena, Tome 5 (2010) no. 6, pp. 22-37.

Résumé

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.

Détail

DOI : 10.1051/mmnp/20105602

Bibliographie

[1] J. Anderson A secular equation for the eigenvalues of a diagonal matrix perturbation Linear Algebra Appl. 1996 49 70

[2] A. Baronchelli, M. Catanzaro, R. Pastor-Satorras Bosonic reaction-diffusion processes on scale-free networks Phys. Rev. E 2008

[3] A. Berman, R.J. Plemmons. Nonnegative matrices in the mathematical sciences. SIAM, Classics in Applied Mathematics 9, Philadelphia, PA, 1994.

[4] M. Boguñá, R. Pastor-Satorras Epidemic spreading in correlated complex networks Phys.Rev.E 2002

[5] V. Colizza, R. Pastor-Satorras, A. Vespignani. Reaction-diffusion processes and metapopulation models in heterogeneous networks Nat. Phys. 2007 276 282

[6] V. Colizza, A. Vespignani Invasion Threshold in Heterogeneous Metapopulation Networks Phys. Rev. Lett. 2007

[7] V. Colizza, A. Vespignani. Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations J. theor. Biol. 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 2007 315 324

[9] A. Fall, A. Iggidr, G. Sallet, J.J. Tewa Epidemiological models and Lyapunov functions Math. Model. Nat. Phenom. 2007 62 83

[10] L. Hufnagel, D. Brockmann, T. Geisel. Forecast and control of epidemics in a globalized world PNAS 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. E 2009

[12] M. J. Keeling, P. Rohani. Modeling infectious diseases in humans and animals. Princeton University Press, 2008.

[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

[14] L. S. Liebovitch, I. B. Schwartz. Migration induced epidemics: dynamics of flux-based multipatch models Phys. Lett. A 2004 256 267

[15] M. E. J. Newman, S. H. Strogatz, D. J. Watts Random graphs with arbitrary degree distributions and their applications Phys. Rev. E 2001

[16] M. E. J. Newman Mixing patterns in networks Phys. Rev. E 2003

[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. A 2008 5017 5025

[18] L.A. Rvachev, I.M. Longini. A mathematical model for the global spread of influenza Math. Biosci. 1985 3 22

[19] J. Saldaña Continuous-time formulation of reaction-diffusion processes on heterogeneous metapopulations Phys. Rev. E 2008

[20] W. Wang, X.-Q. Zhao. An epidemic model in a patchy environment Math. Biosci. 2004 97 112