Explosive Contagion in Networks

EPL (Europhysics Letters), 2015

Journal of Theoretical Biology, 2005

We present a model of contagion that unifies and generalizes existing models of the spread of social influences and microorganismal infections. Our model incorporates individual memory of exposure to a contagious entity (e.g. a rumor or disease), variable magnitudes of exposure (dose sizes), and heterogeneity in the susceptibility of individuals. Through analysis and simulation, we examine in detail the case where individuals may recover from an infection and then immediately become susceptible again (analogous to the so-called SIS model). We identify three basic classes of contagion models which we call epidemic threshold, vanishing critical mass, and critical mass classes, where each class of models corresponds to different strategies for prevention or facilitation. We find that the conditions for a particular contagion model to belong to one of the these three classes depend only on memory length and the probabilities of being infected by one and two exposures, respectively. These parameters are in principle measurable for real contagious influences or entities, thus yielding empirical implications for our model. We also study the case where individuals attain permanent immunity once recovered, finding that epidemics inevitably die out but may be surprisingly persistent when individuals possess memory. r

Diffusion of information, behavioural patterns or innovations follows diverse pathways depending on a number of conditions, including the structure of the underlying social network, the sensitivity to peer pressure and the influence of media. Here we study analytically and by simulations a general model that incorporates threshold mechanism capturing sensitivity to peer pressure, the effect of `immune' nodes who never adopt, and a perpetual flow of external information. While any constant, non-zero rate of dynamically-introduced innovators leads to global spreading, the kinetics by which the asymptotic state is approached show rich behaviour. In particular we find that, as a function of the density of immune nodes, there is a transition from fast to slow spreading governed by entirely different mechanisms. This transition happens below the percolation threshold of fragmentation of the network, and has its origin in the competition between cascading behaviour induced by innovator...

Acta Physica Polonica B

In this work, the aim is to study the spread of a contagious disease and information on a multilayer social system. The main idea is to find a criterion under which the adoption of the spreading information blocks or suppresses the epidemic spread. A two-layer network is the base of the model. The first layer describes the direct contact interactions, while the second layer is the information propagation layer. Both layers consist of the same nodes. The society consists of five different categories of individuals: susceptibles, infective, recovered, vaccinated and precautioned. Initially, only one infected individual starts transmitting the infection. Direct contact interactions spread the infection to the susceptibles. The information spreads through the second layer. The SIR model is employed for the infection spread, while the Bass equation models the adoption of information. The control parameters of the competition between the spread of information and spread of disease are the topology and the density of connectivity. The topology of the information layer is a scale-free network with increasing density of edges. In the contact layer, regular and scale-free networks with the same average degree per node are used interchangeably. The observation is that increasing complexity of the contact network reduces the role of individual awareness. If the contact layer consists of networks with limited range connections, or the edges sparser than the information network, spread of information plays a significant role in controlling the epidemics.

Physical Review E, 2014

In this paper we study the susceptible-infected-susceptible epidemic dynamics, considering a specialized setting where popular places (termed passive entities) are visited by agents (termed active entities). We consider two types of spreading dynamics: direct spreading, where the active entities infect each other while visiting the passive entities, and indirect spreading, where the passive entities act as carriers and the infection is spread via them. We investigate in particular the effect of selection strategy, i.e., the way passive entities are chosen, in the spread of epidemics. We introduce a mathematical framework to study the effect of an arbitrary selection strategy and derive formulas for prevalence, extinction probabilities, and epidemic thresholds for both indirect and direct spreading. We also obtain a very simple relationship between the extinction probability and the prevalence. We pay special attention to preferential selection and derive exact formulas. The analysis reveals that an increase in the diversity in the selection process lowers the epidemic thresholds. Comparing the direct and indirect spreading, we identify regions in the parameter space where the prevalence of the indirect spreading is higher than the direct one.

International Journal of Modeling, Simulation, and Scientific Computing

In this work, the spread of a contagious disease on a society where the individuals may take precautions is modeled. The primary assumption is that the infected individuals transmit the infection to the susceptible members of the community through direct contact interactions. In the meantime, the susceptibles gather information from the adjacent sites which may lead to taking precautions. The SIR model is used for the diffusion of infection while the Bass equation models the information diffusion. The sociological classification of the individuals indicates that a small percentage of the population takes action immediately after being informed, while the majority expects to see some real advantage of taking action. The individuals are assumed to take two different precautions. The precursory measures are getting vaccinated or trying to avoid direct contact with the neighbors. A weighted average of states of the neighbors leads to the choice of action. The fully connected and scale-f...

IEEE Conference on Decision and Control and European Control Conference, 2011

One of the popular dynamics on complex networks is the epidemic spreading. An epidemic model describes how infections spread throughout a network. Among the compartmental models used to describe epidemics, the Susceptible-Infected-Susceptible (SIS) model has been widely used. In the SIS model, each node can be susceptible, become infected with a given infection rate, and become again susceptible with a given curing rate. In this paper, we add a new compartment to the classic SIS model to account for human response to epidemic spread. Each individual can be infected, susceptible, or alert. Susceptible individuals can become alert with an alerting rate if infected individuals exist in their neighborhood. An individual in the alert state is less probable to become infected than an individual in the susceptible state; due to a newly adopted cautious behavior. The problem is formulated as a continuous-time Markov process on a general static graph and then modeled into a set of ordinary differential equations using mean field approximation method and the corresponding Kolmogorov forward equations. The model is then studied using results from algebraic graph theory and center manifold theorem. We analytically show that our model exhibits two distinct thresholds in the dynamics of epidemic spread. Below the first threshold, infection dies out exponentially. Beyond the second threshold, infection persists in the steady state. Between the two thresholds, the infection spreads at the first stage but then dies out asymptotically as the result of increased alertness in the network. Finally, simulations are provided to support our findings. Our results suggest that alertness can be considered as a strategy of controlling the epidemics which propose multiple potential areas of applications, from infectious diseases mitigations to malware impact reduction.

2010

Many behavioral phenomena have been found to spread interpersonally through social networks, in a manner similar to infectious diseases. An important difference between social contagion and traditional infectious diseases, however, is that behavioral phenomena can be acquired by non-social mechanisms as well as through social transmission.

Physical Review Letters, 2004

Models of contagion arise broadly in both the biological and the social sciences, with applications ranging from the transmission of infectious diseases to the spread of cultural fads. In this Letter, we introduce a general model of contagion which, by explicitly incorporating memory of past exposures to, for example, an infectious agent, rumor, or new product, includes the main features of existing contagion models and interpolates between them. We obtain exact solutions for a simple version of the model, finding that under general conditions only three classes of collective dynamics exist. Furthermore, we find that, for a given length of memory, the class into which a particular system falls is determined by only two parameters. Our model suggests novel measures for assessing the susceptibility of a population to large contagion events, and also a possible strategy for inhibiting or facilitating them.

2021

We study epidemic arrival times in meta-population disease models through the lens of front propagation into unstable states. We demonstrate that several features of invasion fronts in the PDE context are also relevant to the network case. We show that the susceptible-infected-recovered model on a network is linearly determined in the sense that the arrival times in the nonlinear system are approximated by the arrival times of the instability in the system linearized near the disease free state. Arrival time predictions are extended to an susceptible-exposed-infected-recovered model. We then study a recent model of social epidemics where high order interactions of individuals lead to faster invasion speeds. For these pushed fronts we compute corrections to the estimated arrival time in this case. Finally, we show how inhomogeneities in local infection rates lead to faster average arrival times.