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Efficient vaccination strategy is crucial for controlling recurrent epidemic spreading on networks. In this paper, based on the analysis of real epidemic data and simulations, it's found that the risk indicator of recurrent epidemic outbreaks could be determined by the ratio of the epidemic infection rate of the year to the average infected density of the former year. According to the risk indicator, the dynamic vaccination probability of each year can be designed to suppress the epidemic outbreaks. Our simulation results show that the dynamic vaccination strategy could effectively decrease the maximal and average infected density, and meanwhile increase the time intervals of epidemic outbreaks and individuals attacked by epidemic. In addition, our results indicate that to depress the influenza outbreaks, it is not necessary to keep the vaccination probability high every year; and adjusting the vaccination probability at right time could decrease the outbreak risks with lower costs. Our findings may present a theoretical guidance for the government and the public to control the recurrent epidemic outbreaks.

The existence of interlayer degree correlations has been disclosed by abundant multiplex network analysis. However, how they impose on the dynamics of social contagions are remain largely unknown. In this paper, we propose a non-Markovian social contagion model in multiplex networks with inter-layer degree correlations to delineate the behavior spreading, and develop an edge-based compartmental (EBC) theory to describe the model. We find that multiplex networks promote the final behavior adoption size. Remarkably, it can be observed that the growth pattern of the final behavior adoption size, versus the behavioral information transmission probability, changes from discontinuous to continuous once decreasing the behavior adoption threshold in one layer. We finally unravel that the inter-layer degree correlations play a role on the final behavior adoption size but have no effects on the growth pattern, which is coincidence with our prediction by using the suggested theory.

It is recently revealed from amounts of real data of recurrent epidemics that there is a phenomenon of hysteresis loop in the state space. To understand it, an indirect investigation from the parameter space has been given to qualitatively explain its mechanism but a more convincing study to quantitatively explain the phenomenon directly from the state space is still missing. We here study this phenomenon directly from the state space and find that there is a positive correlation between the size of outbreak and the size of hysteresis loop, implying that the hysteresis is a nature feature of epidemic outbreak in real case. Moreover, we surprisingly find a paradox on the dependence of the size of hysteresis loop on the two parameters of the infectious rate increment and the transient time, i.e. contradictory behaviors between the two spaces, when the evolutionary time of epidemics is long enough. That is, with the increase of the infectious rate increment, the size of hysteresis loop will decrease in the state space but increase in the parameter space. While with the increase of the transient time, the size of hysteresis loop will increase in the state space but decrease in the parameter space. Furthermore, we find that this paradox will disappear when the evolutionary time of epidemics is limited in a fixed period. Some theoretical analysis are presented to both the paradox and other numerical results.

Temporality is an essential characteristic of many real-world networks and dramatically affects the spreading dynamics on networks. In this paper, we propose an information spreading model on temporal networks with heterogeneous populations. Individuals are divided into activists and bigots to describe the willingness to accept the information. Through a developed discrete Markov chain approach and extensive numerical simulations, we discuss the phase diagram of the model and the effects of network temporality. From the phase diagram, we find that the outbreak phase transition is continuous when bigots are relatively rare, and a hysteresis loop emerges when there are a sufficient number of bigots. The network temporality does not qualitatively alter the phase diagram. However, we find that the network temporality affects the spreading outbreak size by either promoting or suppressing, which relies on the heterogeneities of population and of degree distribution. Specifically, in networks with homogeneous and weak heterogeneous degree distribution, the network temporality suppresses (promotes) the information spreading for small (large) values of information transmission probability. In networks with strong heterogeneous degree distribution, the network temporality always promotes the information spreading when activists dominate the population, or there are relatively fewer activists. Finally, we also find the optimal network evolution scale, under which the network information spreading is maximized.

A great deal of significant progress has been seen in the study of information spreading on populations of networked individuals. A common point in many of the past studies is that there is only one transition in the phase diagram of the final accepted size versus the transmission probability. However, whether other factors alter this phenomenology is still under debate, especially for the case of information spreading through many channels and platforms. In the present study, we adopt a two-layered network to represent the interactions of multiple channels and propose a Susceptible-Accepted-Recovered information spreading model. Interestingly, our model shows a novel double transition including a continuous transition and a following discontinuous transition in the phase diagram, which originates from two outbreaks between the two layers of the network. Furthermore, we reveal that the key factors are a weak coupling condition between the two layers, a large adoption threshold, and the difference of the degree distributions between the two layers. Moreover, we also test the model in the coupled empirical social networks and find similar results as in the synthetic networks. Then, an edge-based compartmental theory is developed which fully explains all numerical results. Our findings may be of significance for understanding the secondary outbreaks of information in real life.

Networks in nature rarely function in isolation but instead interact with one another with a form of a network of networks (NoN). A network of networks with interdependency between distinct networks contains instability of abrupt collapse related to the global rule of activation. As a remedy of the collapse instability, here we investigate a model of correlated NoN. We find that the collapse instability can be removed when hubs provide the majority of interconnections and interconnections are convergent between hubs. Thus, our study identifies a stable structure of correlated NoN against catastrophic failures. Our result further suggests a plausible way to enhance network robustness by manipulating connection patterns, along with other methods such as controlling the state of node based on a local rule., 5 figures

Information spreading has been studied for decades, but its underlying mechanism is still under debate, especially for those ones spreading extremely fast through Internet. By focusing on the information spreading data of six typical events on Sina Weibo, we surprisingly find that the spreading of modern information shows some new features, i.e. either extremely fast or slow, depending on the individual events. To understand its mechanism, we present a Susceptible-Accepted-Recovered (SAR) model with both information sensitivity and social reinforcement. Numerical simulations show that the model can reproduce the main spreading patterns of the six typical events. By this model we further reveal that the spreading can be speeded up by increasing either the strength of information sensitivity or social reinforcement. Depending on the transmission probability and information sensitivity, the final accepted size can change from continuous to discontinuous transition when the strength of the social reinforcement is large. Moreover, an edge-based compartmental theory is presented to explain the numerical results. These findings may be of significance on the control of information spreading in modern society., 10 pages, 8 figures

Highlights • A novel patterns made of two (or multiple) epidemic outbreak peaks from historical data is found. • The two peaks pattern may emerge from different time delays at the two layers. • The essential ingredient of two peaks pattern is a weak coupling condition between the layers themselves. • A rigorous edge-based theory are in perfect agreement with the numerical results., The study of epidemic spreading on populations of networked individuals has seen recently a great deal of significant progresses. A common point in many of past studies is, however, that there is only one peak of infected density in each single epidemic spreading episode. At variance, real data from different cities over the world suggest that, besides a major single peak trait of infected density, a finite probability exists for a pattern made of two (or multiple) peaks. We show that such a latter feature is distinctive of a multilayered network of interactions, and reveal that a two peaks pattern may emerge from different time delays at which the epidemic spreads in between the two layers. Further, we show that the essential ingredient is a weak coupling condition between the layers themselves, while different degree distributions in the two layers are also helpful. Moreover, an edge-based theory is developed which fully explains all numerical results. Our findings may therefore be of significance for protecting secondary disasters of epidemics, which are definitely undesired in real life.

Most of the studies on epidemics so far have focused on the growing phase, such as how an epidemic spreads and what are the conditions for an epidemic to break out in a variety of cases. However, we discover from real data that on a large scale, the spread of an epidemic is in fact a recurrent event with distinctive growing and recovering phases, i.e., a hysteresis loop. We show here that the hysteresis loop can be reproduced in epidemic models provided that the infectious rate is adiabatically increased or decreased before the system reaches its stationary state. Two ways to the hysteresis loop are revealed, which is helpful in understanding the mechanics of infections in real evolution. Moreover, a theoretical analysis is presented to explain the mechanism of the hysteresis loop.

In the metropolis, traffic congestion has become a very serious problem, especially in rush hours. This congestion causes people to have more chance to contact each other and thus will accelerate epidemic spreading. To explain this observation, we present a reaction–diffusion (RD) model with a periodic varying diffusion rate to represent the daily traveling behaviors of human beings and its influence to epidemic spreading. By extensive numerical simulations, we find that the epidemic spreading can be significantly influenced by traffic congestion where the amplitude, period and duration of diffusion rate are the three key parameters. Furthermore, a brief theory is presented to explain the effects of the three key parameters. These findings suggest that except the normal ways of controlling contagion in working places and long-distance traveling, controlling the contagion in daily traffic congestion may be another effective way to reduce epidemic spreading.

Epidemic spreading has been studied for a long time and is currently focused on the spreading of multiple pathogens, especially in multiplex networks. However, little attention has been paid to the case where the mutual influence between different pathogens comes from a fraction of epidemic propagators, such as bisexual people in two separated groups of heterosexual and homosexual people. We here study this topic by presenting a network model of two layers connected by impulsive links, in contrast to the persistent links in each layer. We let each layer have a distinct pathogen and their interactive infection is implemented by a fraction of propagators jumping between the corresponding pairs of nodes in the two layers. By this model we show that (i) the propagators take the key role to transmit pathogens from one layer to the other, which significantly influences the stabilized epidemics; (ii) the epidemic thresholds will be changed by the propagators; and (iii) a reverse-feeding effect can be expected when the infective rate is smaller than its threshold of isolated spreading. A theoretical analysis is presented to explain the numerical results.