Effects of local population structure in a reaction-diffusion model of a contact process on metapopulation networks

Epidemic Spreading in Metapopulation Networks Coupled With Awareness Propagation

Understanding the feedback loop that links the spatiotemporal spread of infectious diseases and human behavior is an open problem. To study this problem, we develop a multiplex framework that couples epidemic spreading across subpopulations in a metapopulation network (i.e., physical layer) with the spreading of awareness about the epidemic in a communication network (i.e., virtual layer). We explicitly study the interactions between the mobility patterns across subpopulations and the awareness propagation among individuals. We analyze the coupled dynamics using microscopic Markov chains (MMCs) equations and validate the theoretical results via Monte Carlo (MC) simulations. We find that with the spreading of awareness, reducing human mobility becomes more effective in mitigating the large-scale epidemic. We also investigate the influence of varying topological features of the physical and virtual layers and the correlation between the connectivity and local population size per subpopulation. Overall the proposed modeling framework and findings contribute to the growing literature investigating the interplay between the spatiotemporal spread of epidemics and human behavior.

Modified diffusive epidemic process on Apollonian networks

We present an analysis of an epidemic spreading process on an Apollonian network that can describe an epidemic spreading in a non-sedentary population. We studied the modified diffusive epidemic process using the Monte Carlo method by computational analysis. Our model may be helpful for modeling systems closer to reality consisting of two classes of individuals: susceptible (A) and infected (B). The individuals can diffuse in a network according to constant diffusion rates [Formula: see text] and [Formula: see text], for the classes A and B, respectively, and obeying three diffusive regimes, i.e., [Formula: see text], [Formula: see text], and [Formula: see text]. Into the same site i, the reaction occurs according to the dynamical rule based on Gillespie's algorithm. Finite-size scaling analysis has shown that our model exhibits continuous phase transition to an absorbing state with a set of critical exponents given by [Formula: see text], [Formula: see text], and [Formula: see text] familiar to every investigated regime. In summary, the continuous phase transition, characterized by this set of critical exponents, does not have the same exponents of the mean-field universality class in both regular lattices and complex networks.

Pathogen diversity in meta-population networks

The pathogen diversity means that multiple strains coexist, and widely exist in the biology systems. The new mutation of SARS-CoV-2 leading to worldwide pathogen diversity is a typical example. What are the main factors of inducing the pathogen diversity? Previous studies indicated the pathogen mutation is the most important reason for inducing the pathogen diversity. The traffic network and gene network are crucial in shaping the dynamics of pathogen contagion, while their roles for the pathogen diversity still lacking a theoretical study. To this end, we propose a reaction-diffusion process of pathogens with mutations on meta-population networks, which includes population movement and strain mutation. We extend the Microscopic Markov Chain Approach (MMCA) to describe the model. Traffic networks make pathogen diversity more likely to occur in cities with lower infection densities. The likelihood of pathogen diversity is low in cities with short effective distances in the traffic network. Star-type gene network is more likely to lead to pathogen diversity than lattice-type and chain-type gene networks. When pathogen localization is present, infection is localized to strains that are at the endpoints of the gene network. Both the increased probability of movement and mutation promote pathogen diversity. The results also show that the population tends to move to cities with short effective distances, resulting in the infection density is high.

Infectious Diseases Spreading on an Adaptive Metapopulation Network

When an emerging acute infectious disease occurs, travel restrictions, one-way or two-way, are often taken to prevent its global spread. In order to investigate the impact of two-way travel restrictions in the global spread of infectious diseases, this paper defines a risk indicator according to the relative infection density. Based on this risk indicator and an intervention time on two-way travel restrictions, we define an adaptive metapopulation network. Then a susceptible-infectious-removed (SIR) metapopulation model on this network is proposed. The mathematical analysis shows that the basic reproduction number is independent of human mobility. Furthermore, this essay compares the effects of one-way travel restrictions and two-way travel restrictions on the global spread of infectious diseases. It is shown that the adaptive metapopulation network under two-way travel restrictions can effectively suppress the global spread of infectious diseases. We also obtain a threshold of risk indicator to prevent the global spread of infectious diseases by simulations. The earlier the intervention time on two-way travel restriction is, the better to curb the global spread of the disease. Even if two-way travel restrictions are not implemented, controlling the mobility of infectious persons would help prevent the global spread of the disease. This work will throw lights on the prevention and control of the globally spreading of an emerging infectious disease.

Infectious diseases spreading on a metapopulation network coupled with its second-neighbor network

Traditional infectious diseases models on metapopulation networks focus on direct transportations (e.g., direct flights), ignoring the effect of indirect transportations. Based on global aviation network, we turn the problem of indirect flights into a question of second neighbors, and propose a susceptible-infectious-susceptible model to study disease transmission on a connected metapopulation network coupled with its second-neighbor network (SNN). We calculate the basic reproduction number, which is independent of human mobility, and we prove the global stability of disease-free and endemic equilibria of the model. Furthermore, the study shows that the behavior that all travelers travel along the SNN may hinder the spread of disease if the SNN is not connected. However, the behavior that individuals travel along the metapopulation network coupled with its SNN contributes to the spread of disease. Thus for an emerging infectious disease, if the real network and its SNN keep the same connectivity, indirect transportations may be a potential threat and need to be controlled. Our work can be generalized to high-speed train and rail networks, which may further promote other research on metapopulation networks.

Activation thresholds in epidemic spreading with motile infectious agents on scale-free networks

We investigate a fermionic susceptible-infected-susceptible model with the mobility of infected individuals on uncorrelated scale-free networks with power-law degree distributions P ( k ) ∼ k - γ of exponents 2 < γ < 3 . Two diffusive processes with diffusion rate D of an infected vertex are considered. In the standard diffusion, one of the nearest-neighbors is chosen with equal chance, while in the biased diffusion, this choice happens with probability proportional to the neighbor's degree. A non-monotonic dependence of the epidemic threshold on D with an optimum diffusion rate D ∗ , for which the epidemic spreading is more efficient, is found for standard diffusion while monotonic decays are observed in the biased case. The epidemic thresholds go to zero as the network size is increased and the form that this happens depends on the diffusion rule and the degree exponent. We analytically investigated the dynamics using quenched and heterogeneous mean-field theories. The former presents, in general, a better performance for standard and the latter for biased diffusion models, indicating different activation mechanisms of the epidemic phases that are rationalized in terms of hubs or max k -core subgraphs.

Spreading Processes in Multiplex Metapopulations Containing Different Mobility Networks

We propose a theoretical framework for the study of spreading processes in structured metapopulations, with heterogeneous agents, subjected to different 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 classical epidemic models within this framework and obtain an excellent agreement with extensive Monte Carlo simulations. This agreement allows us to derive analytical expressions of the epidemic threshold and to face the challenge of characterizing a real multiplex metapopulation, the city of Medellín in Colombia, where different recurrent mobility patterns are observed depending on the socioeconomic class of the agents. Our framework allows us to unveil the geographical location of those patches that trigger the epidemic state at the critical point. A careful exploration reveals that social mixing between classes and mobility crucially determines these critical patches and, more importantly, it can produce abrupt changes of the critical properties of the epidemic onset.

The spread of disease is strongly driven by how humans move and segregate across cities, regions, and countries. Understanding how epidemics arise from the interplay of these behaviors is crucial for implementing efficient containment and prevention policies. In this work, we propose a theoretical framework for the spread of pathogens that incorporates the real patterns of mobility, demographics, and human diversity observed in urban environments.

Our framework represents various populations in a community as layers in a multiplex network, where nodes correspond to geographical areas and layers account for movement patterns. With this framework, we analyze a real urban system, the city of Medellin in Colombia, considering the interplay between six socioeconomic classes displaying disparate demographic segregation and mobility habits. Our framework can identify those neighborhoods and classes that trigger epidemic outbreaks.

Remarkably, we observe that small changes to both mobility and social mixing among these subpopulations can trigger abrupt changes. As a consequence, containment strategies targeting a certain neighborhood can quickly change from efficient to useless. Finally, we provide a physical interpretation of the abrupt changes of those patches driving the unfolding of epidemics based on the effective number of contacts of agents.

Our framework provides a reliable, time-saving platform for analyzing the spread of pathogens among various populations, allowing researchers to identify areas critical to the unfolding of diseases. With further improvements, our formalism could be extended to accommodate more sophisticated commuting patterns and epidemic models.

Epidemic spreading on metapopulation networks including migration and demographics

Epidemic dynamics in a structured population has been widely investigated in recent years by utilizing the metapopulation framework with a reaction-diffusion approach. In this paper, we study epidemic spreading on metapopulation networks, including migration and demographics, wherein population dynamics in each node (a patch) follows the logistic model with a heterogeneous carrying capacity. The epidemic threshold is theoretically calculated at a mean-field level and is then evaluated by Monte Carlo simulations. It is shown that heterogeneity of carrying capacity drastically decreases the threshold, and conversely increasing the migration rate slightly increases the threshold. Interestingly, we observe Monte Carlo simulations showing the effect of heterogeneity of carrying capacity and migration on the epidemic prevalence above the epidemic threshold. Heterogeneity of carrying capacity enhances epidemic spreading in the initial stage, but has no impact on the final infection density. The migration rate has a pronounced impact on both temporal spreading behaviour and endemic state.

Moment closure of infectious diseases model on heterogeneous metapopulation network

The global transmission of infectious diseases poses huge threats to human. Traditional heterogeneous mean-field models on metapopulation networks ignore the heterogeneity of individuals who are in different disease states in subpopulations with the same degree, resulting in inaccuracy in predicting the spread of disease. In this paper, we take heterogeneity of susceptible and infectious individuals in subpopulations with the same degree into account, and propose a deterministic unclosed general model according to Markov process on metapopulation networks to curve the global transmission of diseases precisely. Then we make the general model closed by putting forward two common assumptions: a two-dimensional constant distribution and a two-dimensional log-normal distribution, where the former is equivalent to the heterogeneous mean-field model, and the latter is a system of weighted ordinary differential equations. Further we make a stability analysis for two closed models and illustrate the results by numerical simulations. Next, we conduct a series of numerical simulations and stochastic simulations. Results indicate that our general model extends and optimizes the mean-field model. Finally, we investigate the impacts of total mobility rate on disease transmission and find that timely and comprehensive travel restriction in the early stage is an effective prevention and control of infectious diseases.

Epidemic spreading between two coupled subpopulations with inner structures

The structure of underlying contact network and the mobility of agents are two decisive factors for epidemic spreading in reality. Here, we study a model consisting of two coupled subpopulations with intra-structures that emphasizes both the contact structure and the recurrent mobility pattern of individuals simultaneously. We show that the coupling of the two subpopulations (via interconnections between them and round trips of individuals) makes the epidemic threshold in each subnetwork to be the same. Moreover, we find that the interconnection probability between two subpopulations and the travel rate are important factors for spreading dynamics. In particular, as a function of interconnection probability, the epidemic threshold in each subpopulation decreases monotonously, which enhances the risks of an epidemic. While the epidemic threshold displays a non-monotonic variation as travel rate increases. Moreover, the asymptotic infected density as a function of travel rate in each subpopulation behaves differently depending on the interconnection probability.

The Scaling of Human Contacts and Epidemic Processes in Metapopulation Networks

We study the dynamics of reaction-diffusion processes on heterogeneous metapopulation networks where interaction rates scale with subpopulation sizes. We first present new empirical evidence, based on the analysis of the interactions of 13 million users on Twitter, that supports the scaling of human interactions with population size with an exponent γ ranging between 1.11 and 1.21, as observed in recent studies based on mobile phone data. We then integrate such observations into a reaction- diffusion metapopulation framework. We provide an explicit analytical expression for the global invasion threshold which sets a critical value of the diffusion rate below which a contagion process is not able to spread to a macroscopic fraction of the system. In particular, we consider the Susceptible-Infectious-Recovered epidemic model. Interestingly, the scaling of human contacts is found to facilitate the spreading dynamics. This behavior is enhanced by increasing heterogeneities in the mobility flows coupling the subpopulations. Our results show that the scaling properties of human interactions can significantly affect dynamical processes mediated by human contacts such as the spread of diseases, ideas and behaviors.

Sufficient conditions of endemic threshold on metapopulation networks

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 endemic threshold (i.e., its upper bound), previously derived based on a mean-field approximation of network structure, also holds true for arbitrary networks. We also derive an improved condition showing that networks with the rich-club property (i.e., high connectivity between nodes with a large number of links) are more prone to disease spreading. The dependency of infection rate on population size introduces a considerable difference between this upper bound and estimates based on mean-field approximations, even when degree-degree correlations are considered. We verify the theoretical results with numerical simulations.

Hybrid epidemics--a case study on computer worm conficker

Conficker is a computer worm that erupted on the Internet in 2008. It is unique in combining three different spreading strategies: local probing, neighbourhood probing, and global probing. We propose a mathematical model that combines three modes of spreading: local, neighbourhood, and global, to capture the worm's spreading behaviour. The parameters of the model are inferred directly from network data obtained during the first day of the Conficker epidemic. The model is then used to explore the tradeoff between spreading modes in determining the worm's effectiveness. Our results show that the Conficker epidemic is an example of a critically hybrid epidemic, in which the different modes of spreading in isolation do not lead to successful epidemics. Such hybrid spreading strategies may be used beneficially to provide the most effective strategies for promulgating information across a large population. When used maliciously, however, they can present a dangerous challenge to current internet security protocols.

Optimizing hybrid spreading in metapopulations

Epidemic spreading phenomena are ubiquitous in nature and society. Examples include the spreading of diseases, information, and computer viruses. Epidemics can spread by local spreading, where infected nodes can only infect a limited set of direct target nodes and global spreading, where an infected node can infect every other node. In reality, many epidemics spread using a hybrid mixture of both types of spreading. In this study we develop a theoretical framework for studying hybrid epidemics, and examine the optimum balance between spreading mechanisms in terms of achieving the maximum outbreak size. We show the existence of critically hybrid epidemics where neither spreading mechanism alone can cause a noticeable spread but a combination of the two spreading mechanisms would produce an enormous outbreak. Our results provide new strategies for maximising beneficial epidemics and estimating the worst outcome of damaging hybrid epidemics.

Epidemic spreading in metapopulation networks with heterogeneous infection rates

In this paper, we study epidemic spreading in metapopulation networks wherein each node represents a subpopulation symbolizing a city or an urban area and links connecting nodes correspond to the human traveling routes among cities. Differently from previous studies, we introduce a heterogeneous infection rate to characterize the effect of nodes' local properties, such as population density, individual health habits, and social conditions, on epidemic infectivity. By means of a mean-field approach and Monte Carlo simulations, we explore how the heterogeneity of the infection rate affects the epidemic dynamics, and find that large fluctuations of the infection rate have a profound impact on the epidemic threshold as well as the temporal behavior of the prevalence above the epidemic threshold. This work can refine our understanding of epidemic spreading in metapopulation networks with the effect of nodes' local properties.