Elsevier

Journal of Theoretical Biology

Volume 293, 21 January 2012, Pages 87-100
Journal of Theoretical Biology

Invasion threshold in structured populations with recurrent mobility patterns

https://doi.org/10.1016/j.jtbi.2011.10.010Get rights and content

Abstract

In this paper we develop a framework to analyze the behavior of contagion and spreading processes in complex subpopulation networks where individuals have memory of their subpopulation of origin. We introduce a metapopulation model in which subpopulations are connected through heterogeneous fluxes of individuals. The mobility process among communities takes into account the memory of residence of individuals and is incorporated with the classical susceptible-infectious-recovered epidemic model within each subpopulation. In order to gain analytical insight into the behavior of the system we use degree-block variables describing the heterogeneity of the subpopulation network and a time-scale separation technique for the dynamics of individuals. By considering the stochastic nature of the epidemic process we obtain the explicit expression of the global epidemic invasion threshold, below which the disease dies out before reaching a macroscopic fraction of the subpopulations. This threshold is not present in continuous deterministic diffusion models and explicitly depends on the disease parameters, the mobility rates, and the properties of the coupling matrices describing the mobility across subpopulations. The results presented here take a step further in offering insight into the fundamental mechanisms controlling the spreading of infectious diseases and other contagion processes across spatially structured communities.

Highlights

► We model the contagion spreading mediated by recurrent mobility patterns. ► We characterize analytically a phase transition between two regimes of spreading. ► We derive the threshold values in mobility rates that ensure the global spreading.

Introduction

In recent years, reaction–diffusion processes have been used as a successful modeling framework to approach a wide array of systems that, along with the usual chemical and physical phenomena (Marro and Dickman, 1999, van Kampen, 1981), includes epidemic spreading (Anderson and May, 1984, May and Anderson, 1984, Bolker and Grenfell, 1993, Bolker and Grenfell, 1995, Sattenspiel and Dietz, 1995, Lloyd and May, 1996, Keeling and Rohani, 2002, Watts et al., 2005), human mobility (Bolker and Grenfell, 1995, Sattenspiel and Dietz, 1995, Lloyd and May, 1996, Keeling and Rohani, 2002), and information and social contagion processes (Rapoport, 1953, Goffman and Newill, 1964, Goffman, 1966, Dietz, 1967, Tabah, 1999, Daley and Gani, 2000). This paradigm is extremely useful in the case of populations characterized by a highly fragmented environment in which the population is structured and localized in relatively isolated discrete patches or subpopulations connected by mobility of individuals. In this case, the spatial structure of populations is known to play a key role in the system's evolution and the reaction–diffusion dynamics is integrated in a metapopulation modeling scheme in which different subpopulations are coupled together by the mobility or migration patterns of individuals (Hanski and Gilpin, 1997, Tilman and Kareiva, 1997, Bascompte and Solé, 1998, Hanski and Gaggiotti, 2004). 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 and population dynamics, and has been successfully applied to understand the epidemic dynamics of spatially structured populations with well-defined social units (e.g., families, villages, towns, cities, regions) connected through individuals' mobility (Hethcote, 1978, May and Anderson, 1979, May and Anderson, 1984, Anderson and May, 1984, Bolker and Grenfell, 1993, Bolker and Grenfell, 1995, Sattenspiel and Dietz, 1995, Keeling and Rohani, 2002, Lloyd and May, 1996, Grenfell and Harwood, 1997, Grenfell and Bolker, 1998, Ferguson et al., 2003, Riley, 2007).The metapopulation dynamics of infectious diseases has generated a wealth of models and results that consider both mechanistic approaches that take the movement of individuals explicitly into account (Baroyan et al., 1969, Rvachev and Longini, 1985, Longini, 1988, Flahault and Valleron, 1991, Keeling and Rohani, 2002, Grais et al., 2003, Ruan et al., 2006) and effective coupling approaches wherein the diffusion process is expressed as a force of infection coupling different subpopulations (Bolker and Grenfell, 1995, Lloyd and May, 1996, Earn et al., 1998, Rohani et al., 1999, Keeling, 2000, Park et al., 2002, Vázquez, 2007). Recently, the metapopulation approach has been implemented in data-driven computational models for the large-scale analysis of the geographical spreading of infectious diseases (Grais et al., 2004, Hufnagel et al., 2004, Colizza et al., 2006a, Colizza et al., 2007a, Cooper et al., 2006, Hollingsworth et al., 2006, Riley, 2007).

In large-scale systems, the metapopulation approach amounts to a particle-network description in which each subpopulation populated by a certain number of individuals is connected to a set of other subpopulation by mobility flows. The particle-network framework has stimulated the broadening of reaction–diffusion models in order to deal with complex network substrates and complex mobility schemes, which has in turn allowed for the uncovering of new and interesting dynamical behaviors as well as providing a rationale for the understanding of the emerging critical points underpinning some interesting characteristics of techno-social systems (Colizza et al., 2007b, Colizza and Vespignani, 2007, Colizza and Vespignani, 2008, Vespignani, 2009). In particular, it has been shown that along the local epidemic threshold, which depends only on the disease parameters and is responsible for the epidemic outbreak within each subpopulation, structured populations may exhibit a global invasion threshold (Ball et al., 1997, Cross et al., 2005, Cross et al., 2007, Colizza and Vespignani, 2007, Colizza and Vespignani, 2008) that determines whether the metapopulation system is globally invaded by the contagion process. This novel threshold depends on the mobility rates and patterns of individuals and cannot be uncovered by continuous deterministic models as it is related to the stochastic effects of the reaction–diffusion process that describe the contagion process.

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. However, the recent accumulation of large amounts of human mobility data (Chowell et al., 2003, Barrat et al., 2004, Guimerá et al., 2005, Brockmann et al., 2006, Patuelli et al., 2007, González et al., 2008, Balcan et al., 2009) from the scale of single individuals to that of entire populations presents us with new challenges related to the high level of predictability and recurrence (Wang and González, 2009; Song et al., 2010a, Song et al., 2010b) found in the mobility and diffusion patterns in real data. For instance, commuting mobility – denoted by recurrent bidirectional flows among locations – dominates the human mobility network at the scale of census areas defined by major urban areas by one order of magnitude (Balcan et al., 2009). Highly predictable or recurrent mobility patterns do not find an easy representation in the particle-network framework as the framework is based on reaction–diffusion processes that in most cases exploit Markovian diffusion properties (Colizza and Vespignani, 2007, Colizza and Vespignani, 2008, Ni and Weng, 2009). The description of mobility processes with memory and their importance in epidemic processes have been put forward in detail in the seminal paper of Sattenspiel and Dietz (1995). The effect of recurrent and predictable mobility patterns of individuals on the onset of the global invasion behavior of contagion processes is just recently being studied both analytically and numerically (Danon et al., 2009, Keeling et al., 2010, Balcan and Vespignani, 2011, Belik et al., 2011).

Here we develop a framework based on a time-scale separation technique to analyze the behavior of contagion and spreading processes on a network of locations where individuals have memory of their location of origin. We focus on the prototypical example of the spreading of biological agents in populations characterized by bidirectional commuting patterns. We assume that individuals of a subpopulation will visit any one of the connected subpopulations with a per capita diffusion rate σ. As we aim at modeling commuting processes in which individuals have a memory of their location of origin, displaced individuals return to their original subpopulation with a per capita return rate τ. The mobility parameters σ and τ influence the probability that individuals carrying infection will export the contagion process to nearby subpopulations. If the diffusion rate approaches zero then the probability of contagion of neighboring subpopulations goes to zero as there are no occasions for the carriers of the process to visit them. On the other hand if the return rate is very high then the visiting time of individuals in neighboring populations is so short that they do not have time to spread the contagion in the visited subpopulations. This implies the presence of a transition (Ball et al., 1997, Cross et al., 2005, Cross et al., 2007, Colizza and Vespignani, 2007, Colizza and Vespignani, 2008) between a regime in which the contagion process may invade a macroscopic fraction of the network and a regime in which it is limited to a few subpopulations. The presented results extend and generalize the analysis of Colizza and Vespignani, 2007, Colizza and Vespignani, 2008, Balcan and Vespignani (2011), and Belik et al. (2011) and we include in the analytical treatment the heterogeneity of the subpopulation network and find an explicit expression for the threshold separating a regime in which the spreading phenomenon affects a macroscopic fraction of the system and a regime in which only a few locations are affected. The invasion threshold depends on the mobility parameters, providing guidance on how to control disease spreading by constraining mobility processes. The results are confirmed by mechanistic Monte Carlo simulations for the infection dynamics in synthetic metapopulation systems in which each single individual is tracked in time to account for the discreteness of the processes involved. Heterogeneous connectivity patterns among subpopulations are modeled and different values of the parameters are considered to validate the theoretical results. The theoretical approach presented in this paper extends and generalizes the results presented in Balcan and Vespignani (2011) opening the path to the inclusion of more complicated mobility or interaction schemes and at the same time provides a general framework that may be used not just as an interpretative framework. Understanding the effect of mobility and interaction patterns on the global spreading of contagion processes can then be used to enhance or suppress spreading by adjusting the basic parameters of the system in the appropriate ways.

The paper is organized as follows. Section 2 introduces the basic formalism for recurrent mobility patterns and the time-scale separation approximation that defines mixing subpopulations. Section 3 generalizes the formalism to the case of complex subpopulation networks by using a mean-field degree-block variables description equivalent to a mean-field description that includes the network heterogeneity. Section 4 incorporates the disease spreading into the mobility processes. Stochastic effects and discrete descriptions of the processes are considered with a tree-like approximation for the analysis of the invasion dynamics at the level of subpopulations. The effects of diffusion properties on the invasion dynamics are analyzed and related to the existence of an epidemic invasion threshold for the metapopulation system. In Section 5 we report extensive mechanistic Monte Carlo simulations which confirm the analytical findings of the previous sections.

Section snippets

Mobility processes with memory and commuting networks

In order to describe the mobility process induced by the commuting pattern of people among subpopulations let us consider a metapopulation system with V distinct subpopulations, each of which has a population size Ni (i=1,…,V). The subpopulations form a network in which each subpopulation i is connected to a set of other subpopulations υ(i). The edge connecting two subpopulations i and j indicates the presence of a flux of commuters. We assume that individuals in subpopulation i will visit any

Mobility processes with memory in heterogeneous networks

In order to gain analytic insight into the case of subpopulation networks with highly heterogeneous connectivity patterns we rely on the assumption of statistical equivalence of subpopulations with similar degree. This is a mean-field approximation that considers all the subpopulations with same degree as statistically equivalent, thus allowing the introduction of degree-block variables that depend only upon the subpopulation degree. While this is an obvious approximation to the system

Epidemic spreading and the invasion threshold

Here we want to consider that an infectious disease has been introduced in one or a tiny number of subpopulations. For the sake of analytical simplicity we assume the usual susceptible-infectious-recovered (SIR) model (Keeling and Rohani, 2008) for the disease. The SIR compartmental model classifies at any time t each individual by one of the disease compartments: susceptible (S); infectious (I); recovered (R). Susceptible individuals acquire infection in the case of contact with an infectious

Stochastic simulations

In the following we provide extensive numerical simulations to support the theoretical picture described above. We present in detail the mechanistic numerical simulations where each single individual is tracked in time, during both the infection dynamics and the diffusion processes, and the synthetic subpopulation networks. We report results from Monte Carlo simulations in a variety of different cases and compare them with the analytical findings.

Conclusions

In this paper we have set a mathematical framework to investigate the conditions of global epidemic invasion in the case of subpopulations coupled with recurrent mobility patterns. On one hand we have extended the mathematical framework of degree-block variables (Colizza and Vespignani, 2007, Colizza and Vespignani, 2008) to gain insight into the impact of a non-Markovian mobility process on epidemic extinction/persistence, while on the other hand we have extended the time-scale separation

Acknowledgments

We would like to thank Vitaly Belik and Dirk Brockmann for sharing their results on a closely related work. We would also like to thank Chiara Poletto and Vittoria Colizza for interesting discussions during the preparation of this manuscript. This work has been partially funded by the NIH R21-DA024259 award and the DTRA-1-0910039 award to A.V. The work has also been partly sponsored by the Army Research Laboratory and was competed under Cooperative Agreement Number W911NF-09-2-0053. The views

References (95)

  • A. Vázquez

    Epidemic outbreaks on structured populations

    J. Theor. Biol.

    (2007)
  • R.M. Anderson et al.

    Spatial, temporal and genetic heterogeneity in host populations and the design of immunization programs

    IMA J. Math. Appl. Med. Biol.

    (1984)
  • N.T. Bailey

    The Mathematical Theory of Infectious Diseases

    (1975)
  • P. Bajardi et al.

    Human mobility networks, travel restrictions, and the global spread of 2009 H1N1 pandemic

    PLoS ONE

    (2011)
  • D. Balcan et al.

    Multiscale mobility networks and the spatial spreading of infectious diseases

    Proc. Natl. Acad. Sci. USA

    (2009)
  • D. Balcan et al.

    Phase transitions in contagion processes mediated by recurrent mobility patterns

    Nat. Phys.

    (2011)
  • F. Ball et al.

    Epidemics with two levels of mixing

    Ann. Appl. Probab.

    (1997)
  • O.V. Baroyan et al.

    An attempt at large-scale influenza epidemic modelling by means of a computer

    Bull. Int. Epidemiol. Assoc.

    (1969)
  • A. Barrat et al.

    The architecture of complex weighted networks

    Proc. Natl. Acad. Sci. USA

    (2004)
  • A. Barrat et al.

    Dynamical Processes on Complex Networks

    (2008)
  • J. Bascompte et al.

    Modeling Spatiotemporal Dynamics in Ecology

    (1998)
  • V. Belik et al.

    Natural human mobility patterns and spatial spread of infectious diseases

    Phys. Rev. X

    (2011)
  • B.M. Bolker et al.

    Chaos and biological complexity in measles dynamics

    Proc. R. Soc. London B

    (1993)
  • B.M. Bolker et al.

    Space persistence and dynamics of measles epidemics

    Philos. Trans. R. Soc. London B

    (1995)
  • D. Brockmann et al.

    The scaling laws of human travel

    Nature

    (2006)
  • F. Carrat et al.

    Time lines of infection and disease in human influenza: a review of volunteer challenge studies

    Am. J. Epidemiol.

    (2008)
  • M. Catanzaro et al.

    Generation of uncorrelated random scale-free networks

    Phys. Rev. E

    (2005)
  • G. Chowell et al.

    Scaling laws for the movement of people between locations in a large city

    Phys. Rev. E

    (2003)
  • V. Colizza et al.

    Invasion threshold in heterogeneous metapopulation networks

    Phys. Rev. Lett.

    (2007)
  • V. Colizza et al.

    The role of the airline transportation network in the prediction and predictability of global epidemics

    Proc. Natl. Acad. Sci. USA

    (2006)
  • V. Colizza et al.

    The modeling of global epidemics: stochastic dynamics and predictability

    Bull. Math. Biol.

    (2006)
  • V. Colizza et al.

    Modeling the worldwide spread of pandemic influenza: baseline case and containment interventions

    PLoS Med.

    (2007)
  • V. Colizza et al.

    Reaction-diffusion processes and metapopulation models in heterogeneous networks

    Nat. Phys.

    (2007)
  • B.S. Cooper et al.

    Delaying the international spread of pandemic influenza

    PLoS Med.

    (2006)
  • P. Cross et al.

    Duelling timescales of host movement and disease recovery determine invasion of disease in structured populations

    Ecol. Lett.

    (2005)
  • P. Cross et al.

    Utility of R0 as a predictor of disease invasion in structured populations

    J. R. Soc. Interface

    (2007)
  • D.J. Daley et al.

    Epidemic Modeling: An Introduction

    (2000)
  • J. de Dios Ortúzar et al.

    Modelling Transport

    (2001)
  • K. Dietz

    Epidemics and rumours: a survey

    J. R. Stat. Soc. A

    (1967)
  • D.J.D. Earn et al.

    Persistence chaos and synchrony in ecology and epidemiology

    Proc. R. Soc. London B

    (1998)
  • P. Erdős et al.

    On random graphs

    Publ. Math.

    (1959)
  • S. Erlander et al.

    The Gravity Model in Transportation Analysis

    (1990)
  • N.M. Ferguson et al.

    Planning for smallpox outbreaks

    Nature

    (2003)
  • A. Flahault et al.

    A method for assessing the global spread of HIV-1 infection based on air-travel

    Math. Popul. Stud.

    (1991)
  • J. Gani et al.

    Markov chain methods in chain binomial epidemic models

    Biometrics

    (1971)
  • W. Goffman

    Mathematical approach to the spread of scientific ideas—the history of mast cell research

    Nature

    (1966)
  • W. Goffman et al.

    Generalization of epidemic theory: an application to the transmission of ideas

    Nature

    (1964)
  • Cited by (58)

    • Social physics

      2022, Physics Reports
      Citation Excerpt :

      The threshold was found to decrease as the network becomes more node-degree heterogeneous. Refs. [952,953] extend this line of work to account for recurrent mobility patterns. Although several seminal studies [957–959] have explored the potential for developing a simple summary statistics to approximate the EAT, a general analytical framework leading to a closed-form expression for the probability distribution of the EAT has remained elusive.

    • Epidemic dynamics on metapopulation networks with node2vec mobility

      2022, Journal of Theoretical Biology
      Citation Excerpt :

      For example, the propensity to move may depend on the degree of the subpopulation that the individual currently visits and the number of the individuals in the subpopulation (Colizza and Vespignani, 2008). Furthermore, empirical mobility patterns may be better approximated by non-simple random walks (Belik et al., 2011; Balcan and Vespignani, 2011; Balcan and Vespignani, 2012; Poletto et al., 2013; Rosvall et al., 2014; Scholtes et al., 2014; Matamalas et al., 2016) or recurrent mobility patterns (Balcan and Vespignani, 2011, 2012; Gómez-Gardeñes et al., 2018; Granell and Mucha, 2018; Soriano-Paños et al., 2018, 2020; Feng et al., 2020). Other extensions of metapopulation network models include multilayer ones, in which individuals having different mobility patterns are assigned to different network layers (Xuan et al., 2013; Wang et al., 2014; Soriano-Paños et al., 2018).

    • Association of built environment attributes with the spread of COVID-19 at its initial stage in China

      2021, Sustainable Cities and Society
      Citation Excerpt :

      Activity centers are important nodes in urban networks and usually attract a large number of people. Some studies have found that the travel time by public transport from residential areas to activity destinations (e.g., the city center) has a major impact on the spread of epidemics within cities (Yashima & Sasaki, 2014; Balcan & Vespignani, 2011; Balcan & Vespignani, 2012). Eventually, we identified a total number of 976 activity centers in 255 cities across the whole of China.

    • The spreading of infectious diseases with recurrent mobility of community population

      2020, Physica A: Statistical Mechanics and its Applications
      Citation Excerpt :

      As a result, individuals move from one community to other communities in a short-term or recurrent pattern so as to form temporal communities. The study on spreading process of infectious diseases during recurrent mobility of community population helps to better understand the spreading of diseases and adopt a more effective immunization strategy [20–28]. The uncertainty and randomness of community population mobility propose a challenge to the study of mobility dynamics.

    View all citing articles on Scopus
    View full text