Metapopulation models have been a powerful tool for both theorizing and simulating epidemic dynamics. In a metapopulation model, one considers a network composed of subpopulations and their pairwise connections, and individuals are assumed to migrate from one subpopulation to another obeying a given mobility rule. While how different mobility rules affect epidemic dynamics in metapopulation models has been studied, there have been relatively few efforts on comparison of the effects of simple (i.e., unbiased) random walks and more complex mobility rules. Here we study a susceptible-infectious-susceptible (SIS) dynamics in a metapopulation model, in which individuals obey a parametric second-order random-walk mobility rule called the node2vec. We map the second-order mobility rule of the node2vec to a first-order random walk in a network whose each node is a directed edge connecting a pair of subpopulations and then derive the epidemic threshold. For various networks, we find that the epidemic threshold is large (therefore, epidemic spreading tends to be suppressed) when the individuals infrequently backtrack or infrequently visit the common neighbors of the currently visited and the last visited subpopulations than when the individuals obey the simple random walk. The amount of change in the epidemic threshold induced by the node2vec mobility is in general not as large as, but is sometimes comparable with, the one induced by the change in the diffusion rate for individuals.
Identifying influential subpopulations in metapopulation epidemic models has far-reaching potential implications for surveillance and intervention policies of a global pandemic. However, there is a lack of methods to determine influential nodes in metapopulation models based on a rigorous mathematical background. In this study, we derive the message-passing theory for metapopulation modeling and propose a method to determine influential spreaders. Based on our analysis, we identify the most dangerous city as a potential seed of a pandemic when applied to real-world data. Moreover, we particula…
It is often useful to represent the infectious dynamics of mobile agents by metapopulation models. In such a model, metapopulations form a static network, and individuals migrate from one metapopulation to another. It is known that heterogeneous degree distributions of metapopulation networks decrease the epidemic threshold above which epidemic spreads can occur. We investigate the combined effect of heterogeneous degree distributions and diffusion on epidemics in metapopulation networks. We show that for arbitrary heterogeneous networks, diffusion suppresses epidemics in the sense of an incre…
We propose a novel framework to study viral spreading processes in metapopulation models. Large subpopulations (i.e., cities) are connected via metalinks (i.e., roads) according to a metagraph structure (i.e., the traffic infrastructure). The problem of containing the propagation of an epidemic outbreak in a metapopulation model by controlling the traffic between subpopulations is considered. Controlling the spread of an epidemic outbreak can be written as a spectral condition involving the eigenvalues of a matrix that depends on the network structure and the parameters of the model. Based on …
We review and introduce a generalized reaction-diffusion approach to epidemic spreading in a metapopulation modeled as a complex network. The metapopulation consists of susceptible and infected individuals that are grouped in subpopulations symbolising cities and villages that are coupled by human travel in a transportation network. By analytic methods and numerical simulations we calculate the fraction of infected people in the metaopoluation in the long time limit, as well as the relevant parameters characterising the epidemic threshold that separates an epidemic from a non-epidemic phase. W…
Metapopulation epidemic models describe epidemic dynamics in networks of spatially distant patches connected with pathways for migration of individuals. In the present study, we deal with a susceptible-infected-recovered (SIR) metapopulation model where the epidemic process in each patch is represented by an SIR model and the mobility of individuals is assumed to be a homogeneous diffusion. Our study focuses on two types of patches including high-risk and low-risk ones, in order to evaluate intervention strategies for epidemic control. We theoretically analyze the intervention threshold, indic…
We propose a theoretical framework for the study of epidemics in structured metapopulations, with heterogeneous agents, subjected to recurrent mobility patterns. We propose to represent the heterogeneity in the composition of the metapopulations as layers in a multiplex network, where nodes would correspond to geographical areas and layers account for the mobility patterns of agents of the same class. We analyze both the classical Susceptible-Infected-Susceptible and the Susceptible-Infected-Removed epidemic models within this framework, and compare macroscopic and microscopic indicators of th…
We study the dynamics of epidemic and reaction-diffusion processes in metapopulation models with heterogeneous connectivity pattern. In SIR-like processes, along with the standard local epidemic threshold, the system exhibits a global invasion threshold. We provide an explicit expression of the threshold that sets a critical value of the diffusion/mobility rate below which the epidemic is not able to spread to a macroscopic fraction of subpopulations. The invasion threshold is found to be affected by the topological fluctuations of the metapopulation network. The presented results provide a ge…
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 inva…
Since the 21st century, the global outbreaks of infectious diseases such as SARS in 2003, H1N1 in 2009, and H7N9 in 2013, have become the critical threat to the public health and a hunting nightmare to the government. Understanding the propagation in large-scale metapopulations and predicting the future outbreaks thus become crucially important for epidemic control and prevention. In the literature, there have been a bulk of studies on modeling intra-city epidemic propagation but with the single population assumption (homogeneity). Some recent works on metapopulation propagation, however, focu…
Interevent times in temporal contact data from humans and animals typically obey heavy-tailed distributions, and this property impacts contagion and other dynamical processes on networks. We theoretically show that distributions of interevent times heavier-tailed than exponential distributions are a consequence of the most basic metapopulation model used in epidemiology and ecology, in which individuals move from a patch to another according to the simple random walk. Our results hold true irrespectively of the network structure and also for more realistic mobility rules such as high-order ran…
We propose an efficient strategy to suppress epidemic explosion in heterogeneous metapopulation networks, wherein each node represents a subpopulation with any number of individuals and is assigned a curing rate that is proportional to $k^α$ with $k$ the node degree and $α$ an adjustable parameter. We have performed stochastic simulations of the dynamical reaction-diffusion processes associated with the susceptible-infected-susceptible model in scale-free networks. We found that the epidemic threshold reaches a maximum when the exponent $α$ is tuned to be $α_{opt}\simeq 1.3$. This nontrivial p…
We investigate the effects of mobility and density on the amplitude death of coupled oscillators in metapopulation networks, wherein each node represents a subpopulation with any number of mobile individuals. We perform stochastic simulations of the dynamical reaction-diffusion processes associated with the Landau-Stuart oscillators in scale-free networks. Interestingly, we find that, with increasing the mobility rate or density, the system may undergo phase transitions from incoherent state to amplitude death, and then to frequency synchronization. Especially, there exists an extent of interm…
We investigate the effect of mobility on the response of coupled oscillators to a subthreshold external signal in metapopulation networks, wherein each node represents a subpopulation with overdamped bistable oscillators that can randomly diffuse between nodes. With increasing mobility rate, the oscillators undergo transitions from intrawell to interwell motion, demonstrating clearly mobility-enhanced signal amplification. Moreover, the response shows nonmonotonic dependence on the mobility rate, i.e., a maximal gain occurs at a moderate level of mobility. This interesting phenomenon is robust…
Infection dynamics have been studied extensively on complex networks, yielding insight into the effects of heterogeneity in contact patterns on disease spread. Somewhat separately, metapopulations have provided a paradigm for modeling systems with spatially extended and "patchy" organization. In this paper we expand on the use of multitype networks for combining these paradigms, such that simple contagion models can include complexity in the agent interactions and multiscale structure. We first present a generalization of the Volz-Miller mean-field approximation for Susceptible-Infected-Recove…
In this work we study the influence of populations mobility on the spread of a vector-borne disease. We focus on the chikungunya epidemic event that occurred in 2005-2006 on the Réunion Island, Indian Ocean, France, and validate our models with real epidemic data from the event. We propose a metapopulation model to represent both a high-resolution patch model of the island with realistic population densities and also mobility models for humans (based on real-motion data) and mosquitoes. In this metapopulation network, two models are coupled: one for the dynamics of the mosquito population and …
The analysis of contagion-diffusion processes in metapopulations is a powerful theoretical tool to study how mobility influences the spread of communicable diseases. Nevertheless, many metapopulation approaches use indistinguishable agents to alleviate analytical difficulties. Here, we address the impact that recurrent mobility patterns, and the spatial distribution of distinguishable agents, have on the unfolding of epidemics in large urban areas. We incorporate the distinguishable nature of agents regarding both, their residence, and their usual destination. The proposed model allows both a …
The complex interplay between population movements in space and non-homogeneous mixing patterns have so far hindered the fundamental understanding of the conditions for spatial invasion through a general theoretical framework. To address this issue, we present an analytical modelling approach taking into account such interplay under general conditions of mobility and interactions, in the simplifying assumption of two population classes. We describe a spatially structured population with non-homogeneous mixing and travel behaviour through a multi-host stochastic epidemic metapopulation model. D…
Human mobility, contact patterns, and their interplay are key aspects of our social behavior that shape the spread of infectious diseases across different regions. In the light of new evidence and data sets about these two elements, epidemic models should be refined to incorporate both the heterogeneity of human contacts and the complexity of mobility patterns. Here, we propose a theoretical framework that allows accommodating these two aspects in the form of a set of Markovian equations. We validate these equations with extensive mechanistic simulations and derive analytically the epidemic th…
Mathematical models represent one of the fundamental ways of studying nature. In special, epidemic models have shown to be particularly useful in the understanding of the course of diseases and in the planning effective control policies. A particular type of epidemic model considers the individuals divided into populations. When studied in graphs, it is already known that the graph topology can play an important role in the evolution of the disease. At the same time, one may want to study the effect of the presence of an underlying \emph{attraction landscape} of the vertices, apart from the re…
Multiple scales in metapopulations can give rise to paradoxical behaviour: in a conceptual model for a public goods game, the species associated with a fitness cost due to the public good production can be stabilised in the well-mixed limit due to the mere existence of these scales. The scales in this model involve a length scale corresponding to separate patches, coupled by mobility, and separate time scales for reproduction and interaction with a local environment. Contrary to the well-mixed high mobility limit, we find that for low mobilities, the interaction rate progressively stabilises t…
In this paper, we focus on susceptible-infected-susceptible dynamics on metapopulation networks, where nodes represent subpopulations, and where agents diffuse and interact. Recent studies suggest that heterogeneous network structure between elements plays an important role in determining the threshold of infection rate at the onset of epidemics, a fundamental quantity governing the epidemic dynamics. We consider the general case in which the infection rate at each node depends on its population size, as shown in recent empirical observations. We first prove that a sufficient condition for the…
The metapopulation framework is adopted in a wide array of disciplines to describe systems of well separated yet connected subpopulations. The subgroups or patches are often represented as nodes in a network whose links represent the migration routes among them. The connections have been so far mostly considered as static, but in general evolve in time. Here we address this case by investigating simple contagion processes on time-varying metapopulation networks. We focus on the SIR process and determine analytically the mobility threshold for the onset of an epidemic spreading in the framework…
In this paper, we study into the impact of the preference of an individual for public transport on the spread of infectious disease, through a quantity known as the public mobility. Our theoretical and numerical results based on a constructed model reveal that if the average public mobility of the agents is fixed, an increase in the diversity of the agents' public mobility reduces the epidemic threshold, beyond which an enhancement in the rate of infection is observed. Our findings provide an approach to improve the resistance of a society against infectious disease, while preserving the utili…
While large scale mobility data has become a popular tool to monitor the mobility patterns during the COVID-19 pandemic, the impacts of non-compulsory measures in Tokyo, Japan on human mobility patterns has been under-studied. Here, we analyze the temporal changes in human mobility behavior, social contact rates, and their correlations with the transmissibility of COVID-19, using mobility data collected from more than 200K anonymized mobile phone users in Tokyo. The analysis concludes that by April 15th (1 week into state of emergency), human mobility behavior decreased by around 50%, resultin…
Metapopulation models have been a popular tool for the study of epidemic spread over a network of highly populated nodes (cities, provinces, countries) and have been extensively used in the context of the ongoing COVID-19 pandemic. In the present work, we revisit such a model, bearing a particular case example in mind, namely that of the region of Andalusia in Spain during the period of the summer-fall of 2020 (i.e., between the first and second pandemic waves). Our aim is to consider the possibility of incorporation of mobility across the province nodes focusing on mobile-phone time dependent…
In this work, we address the connection between population density centers in urban areas, and the nature of human flows between such centers, in shaping the vulnerability to the onset of contagious diseases. A study of 163 cities, chosen from four different continents reveals a universal trend, whereby the risk induced by human mobility increases in those cities where mobility flows are predominantly between high population density centers. We apply our formalism to the spread of SARS-COV-2 in the United States, providing a plausible explanation for the observed heterogeneity in the spreading…
Despite the advanced stage of epidemic modeling, there is a major demand for methods to incorporate behavioral responses to the spread of a disease such as social distancing and adoption of prevention methods. Mobility plays an important role in epidemic dynamics and is also affected by behavioral changes, but there are many situations in which real mobility data is incomplete or inaccessible. We present a model for epidemic spreading in temporal networks of mobile agents that incorporates local behavioral responses. Susceptible agents are allowed to move towards the opposite direction of infe…
Human mobility plays a key role on the transformation of local disease outbreaks into global pandemics. Thus, the inclusion of human movements into epidemic models has become mandatory for understanding current epidemic episodes and to design efficient prevention policies. Following this challenge, here we develop a Markovian framework which enables to address the impact of recurrent mobility patterns on the epidemic onset at different temporal scales. This formalism is validated by comparing their predictions with results from mechanistic simulations. The fair agreement between both theory an…
Recently developed techniques to acquire high-quality human mobility data allow large-scale simulations of the spread of infectious diseases with high spatial and temporal resolution.Analysis of such data has revealed the oversimplification of existing theoretical frameworks to infer the final epidemic size or influential nodes from the network topology. Here we propose a spectral decomposition-based framework for the quantitative analysis of epidemic processes on realistic networks of human proximity derived from urban mobility data. Common wisdom suggests that modes with larger eigenvalues c…
Infectious diseases spread through human networks. Susceptible-Infected-Removed (SIR) model is one of the epidemic models to describe infection dynamics on a complex network connecting individuals. In the metapopulation SIR model, each node represents a population (group) which has many individuals. In this paper, we propose a modified metapopulation SIR model in which a latent period is taken into account. We call it SIIR model. We divide the infection period into two stages: an infected stage, which is the same as the previous model, and a seriously ill stage, in which individuals are infect…
Recently, many empirical studies uncovered that animal foraging, migration and human traveling obey Levy flights with an exponent around -2. Inspired by the deluge of H1N1 this year, in this paper, the effects of Levy flights' mobility pattern on epidemic spreading is studied from a network perspective. We construct a spatial weighted network which possesses Levy flight spatial property under a restriction of total energy. The energy restriction is represented by the limitation of total travel distance within a certain time period of an individual. We find that the exponent -2 is the epidemic …
Vector-borne epidemics are the result of the combination of different factors such as the crossed contagions between humans and vectors, their demographic distribution and human mobility among others. The current availability of information about the former ingredients demands their incorporation to current mathematical models for vector-borne disease transmission. Here, relying on metapopulation dynamics, we propose a framework whose results are in fair agreement with those obtained from mechanistic simulations. This framework allows us to derive an expression of the epidemic threshold captur…
Contagious diseases can spread quickly in human populations, either through airborne transmission or if some other spreading vectors are abundantly accessible. They can be particularly devastating if the impact on individuals' health has severe consequences on the number of hospitalizations or even deaths. Common countermeasures to contain the epidemic spread include introducing restrictions on human interactions or their mobility in general which are often associated with an economic and social cost. In this paper, we present a targeted model of optimal social distancing on metapopulation net…
Synthetic populations for epidemic, demographic and mobility modeling
ISI Foundation（ISI基金会）；Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB)（跨学科物理与复杂系统研究所 IFISC (CSIC-UIB)）；Sorbonne Université, INSERM, Sorbonne Public Health Institute（索邦大学、INSERM、索邦公共卫生学院）；Instituto Mediterráneo de Estudios Avanzados IMEDEA (CSIC-UIB)（地中海高级研究所 IMEDEA (CSIC-UIB)）；Universitat Pompeu Fabra（庞培法布拉大学）
Epidemics are emergent phenomena depending on the epidemiological characteristics of pathogens and the interaction and movement of people. Public transit systems have provided much important information about the movement of people, but there are also other means of transportation (e.g., bicycle and private car), that are invisible to public transit data. This discrepancy can induce a bias in disease models that leads to mispredictions of epidemic growth (e.g., peak prevalence and time). In our study, we aim to advance and compare the epidemic spreading dynamics using public transit trips, in …
The spreading of epidemics is very much determined by the structure of the contact network, which may be impacted by the mobility dynamics of the individuals themselves. In confined scenarios where a small, closed population spends most of its time in localized environments and has easily identifiable mobility patterns --such as workplaces, university campuses or schools-- it is of critical importance to identify the factors controlling the rate of disease spread. Here we present a discrete-time, metapopulation-based model to describe the transmission of SIS-like diseases that take place in co…
Infectious diseases outbreaks are often characterized by a spatial component induced by hosts' distribution, mobility, and interactions. Spatial models that incorporate hosts' movements are being used to describe these processes, to investigate the conditions for propagation, and to predict the spatial spread. Several assumptions are being considered to model hosts' movements, ranging from permanent movements to daily commuting, where the time spent at destination is either infinite or assumes a homogeneous fixed value, respectively. Prompted by empirical evidence, here we introduce a general …
Initially emerged in the Chinese city Wuhan and subsequently spread almost worldwide causing a pandemic, the SARS-CoV-2 virus follows reasonably well the SIR (Susceptible-Infectious-Recovered) epidemic model on contact networks in the Chinese case. In this paper, we investigate the prediction accuracy of the SIR model on networks also for Italy. Specifically, the Italian regions are a metapopulation represented by network nodes and the network links are the interactions between those regions. Then, we modify the network-based SIR model in order to take into account the different lockdown measu…
Human mobility is a key component of large-scale spatial-transmission models of infectious diseases. Correctly modeling and quantifying human mobility is critical for improving epidemic control policies, but may be hindered by incomplete data in some regions of the world. Here we explore the opportunity of using proxy data or models for individual mobility to describe commuting movements and predict the diffusion of infectious disease. We consider three European countries and the corresponding commuting networks at different resolution scales obtained from official census surveys, from proxy d…
The networked structure of contacts shapes the spreading of epidemic processes. Recent advances on network theory have improved our understanding of the epidemic processes at large scale. The relevance of several considerations still needs to be evaluated in the study of epidemic spreading. One of them is that of accounting for the influence of origin and destination patterns in the flow of the carriers of an epidemic. Here we compute origin-destination patterns compatible with empirical data of coarse grained flows in the air transportation network. We study the incidence of epidemic processe…
We study epidemic processes using a metapopulation approach on the line featuring random transport rates between arbitrarily distant sites. An average transport network is found using a recently developed variant of the effective medium approximation (EMA) that is capable of dealing with these long-range connections. Using a Feynman-Kac argument in the effective medium, we derive an estimate on the size of the infected domain, and reproduce the known result of its exponential growth in time. We hereby demonstrate the applicability of long-range EMA to dynamical processes on networks more intri…
We study how international flights can facilitate the spread of an epidemic to a worldwide scale. We combine an infrastructure network of flight connections with a population density dataset to derive the mobility network, and then we define an epidemic framework to model the spread of the disease. Our approach combines a compartmental SEIRS model with a graph diffusion model to capture the clusteredness of the distribution of the population. The resulting model is characterised by the dynamics of a metapopulation SEIRS, with amplification or reduction of the infection rate which is determined…
In response to the COVID-19 pandemic, National governments have applied lockdown restrictions to reduce the infection rate. We perform a massive analysis on near real-time Italian data provided by Facebook to investigate how lockdown strategies affect economic conditions of individuals and local governments. We model the change in mobility as an exogenous shock similar to a natural disaster. We identify two ways through which mobility restrictions affect Italian citizens. First, we find that the impact of lockdown is stronger in municipalities with higher fiscal capacity. Second, we find a seg…
Understanding dynamic human mobility changes and spatial interaction patterns at different geographic scales is crucial for assessing the impacts of non-pharmaceutical interventions (such as stay-at-home orders) during the COVID-19 pandemic. In this data descriptor, we introduce a regularly-updated multiscale dynamic human mobility flow dataset across the United States, with data starting from March 1st, 2020. By analyzing millions of anonymous mobile phone users' visits to various places provided by SafeGraph, the daily and weekly dynamic origin-to-destination (O-D) population flows are compu…
Understanding and predicting outbreaks of contagious diseases are crucial to the development of society and public health, especially for underdeveloped countries. However, challenging problems are encountered because of complex epidemic spreading dynamics influenced by spatial structure and human dynamics (including both human mobility and human interaction intensity). We propose a systematical model to depict nationwide epidemic spreading in Côte d'Ivoire, which integrates multiple factors, such as human mobility, human interaction intensity, and demographic features. We provide insights to …
