The spatial structure of populations is a key element in the understanding of the large scale spreading of epidemics. Motivated by the recent empirical evidence on the heterogeneous properties of transportation and commuting patterns among urban areas, we present a thorough analysis of the behavior of infectious diseases in metapopulation models characterized by heterogeneous connectivity and mobility patterns. We derive the basic reaction-diffusion equation describing the metapopulation system at the mechanistic level and derive an early stage dynamics approximation for the subpopulation invasion dynamics. The analytical description uses degree block variables that allows us to take into account arbitrary degree distribution of the metapopulation network. We show that along with the usual single population epidemic threshold the metapopulation network exhibits a global threshold for the subpopulation invasion. We find an explicit analytic expression for the invasion threshold that determines the minimum number of individuals traveling among subpopulations in order to have the infection of a macroscopic number of subpopulations. The invasion threshold is a function of factors such as the basic reproductive number, the infectious period and the mobility process and it is found to decrease for increasing network heterogeneity. We provide extensive mechanistic numerical Monte Carlo simulations that recover the analytical finding in a wide range of metapopulation network connectivity patterns. The results can be useful in the understanding of recent data driven computational approaches to disease spreading in large transportation networks and the effect of containment measures such as travel restrictions.
The metapopulation modeling approach is an essential theoretical framework used in population ecology, genetics and adaptive evolution to describe population dynamics whenever the spatial structure of populations is known to play a key role in the system's evolution (Hanski & Gilpin, 1997;Hanski & Gaggiotti, 2004;Tilman & Kareiva, 1997;Bascompte & Solé, 1998). Metapopulation models rely on the basic assumption that the system under study is characterized by a highly fragmented environment in which the population is structured and localized in relatively isolated discrete patches or subpopulations connected by some degree of migration. Classic metapopulation dynamics focuses on the processes of local extinction, recolonization and regional persistence (Levins, 1969(Levins, , 1970)), as the outcome of the interplay between migration processes among unstable local populations and population dynamics (e.g birth and death rates, competition and predations). This paradigm is extremely useful also in the case of infectious diseases, and can be applied to understand the epidemic dynamics of spatially structured populations with well defined social units (e.g. families, villages, city locations, towns, cities, regions) connected through individual mobility (Hethcote, 1978;May & Anderson, 1979;Anderson & May, 1984;May & Anderson, 1984;Bolker & Grenfell, 1993, 1995;Keeling & Rohani, 2002;Lloyd & May, 1996;Grenfell & Harwood, 1997;Grenfell & Bolker, 1998;Ferguson et al., 2003;Riley, 2007). The arrival of the infection in any subpopulation and its epidemic evolution are determined by the coupling generated by the mobility processes among subpopulations. The metapopulation dynamics of infectious diseases has generated a wealth of models and results considering both mechanistic approaches taking explicitly into account the movement of individuals (Baroyan et al., 1969;Rvachev & Longini, 1985;Longini, 1988;Flahault & Valleron, 1991;Sattenspiel & Dietz, 1995;Keeling & Rohani, 2002;Grais et al., 2003) and effective coupling approaches where the diffusion process is expressed as a force of infection coupling different subpopulations (Bolker & Grenfell, 1995;Lloyd & May, 1996;Earn et al., 1998;Rohani et al., 1999;Keeling, 2000;Park et al., 2002;Vázquez, 2007). Recently, the metapopulation approach is being revamped in computational approaches for the large scale forecast of infectious disease spreading (Grais et al., 2004;Hufnagel et al., 2004;Colizza et al., 2006a;Cooper et al., 2006;Colizza et al., 2007a;Hollingsworth et al., 2006;Riley, 2007).
Metapopulation epidemic models, especially at the mechanistic level, are based on the spatial structure of the environment, and the detailed knowledge of transportation infrastructures and movement patterns. The increasing computational power and informatics advances are beginning to lift the constraints limiting the collection of large spatiotemporal data on human behavior and demography, finally allowing for the formulation of realistic data driven models. On the other hand, networks which trace the activities and interactions of individuals, social patterns, transportation fluxes, and population movements on a local and global scale (Liljeros et al., 2001;Schneeberger et al., 2004;Barrat et al., 2004;Guimerá et al., 2005;Chowell et al., 2003) have been analyzed and found to exhibit complex features encoded in large scale heterogeneity, self-organization and other properties typical of complex systems (Albert & Barabási, 2002;Dorogovtsev & Mendes, 2003;Newman, 2003;Pastor-Satorras & Vespignani, 2004). In particular, a wide range of societal and technological networks exhibits very heterogeneous topologies. The airport network among cities (Barrat et al., 2004;Guimerá et al., 2005), the commuting patterns in inter and intra-urban areas (Chowell et al., 2003;Barrett et al., 2000;De Montis et al., 2007), and several info-structures (Pastor-Satorras & Vespignani, 2004) are indeed characterized by networks whose nodes, representing the elements of the system, have a wildly varying degree, i.e. the number of connections to other elements. These topological fluctuations are mathematically encoded in a heavy-tailed degree distribution P (k), defined as the probability that any given node has degree k, and have been found to have a large impact on epidemic phenomena on complex contact patterns (Anderson & May, 1992;Pastor-Satorras & Vespignani, 2001a,b;Moreno et al., 2002;Lloyd & May, 2001;Barthélemy et al., 2005). Motivated by the above findings we provide here the analysis of the behavior of epidemic models in metapopulation systems with heterogeneous connectivity patterns. In order to have a mechanistic description of the system, we derive the deterministic reaction-diffusion equations describing the evolution of the epidemic in the metapopulation systems. The heterogeneity of the network is taken explicitly into account by introducing degree block variables that provide results expresse
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