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Modelling the spread of infectious diseases in complex metapopulations. (English) Zbl 1204.92062

Summary: Two main approaches have been considered for modelling the dynamics of SIS models 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. 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.

MSC:

92D30 Epidemiology
92C42 Systems biology, networks
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
15A18 Eigenvalues, singular values, and eigenvectors
65C20 Probabilistic models, generic numerical methods in probability and statistics
92D40 Ecology
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