Emergence of hysteresis loop in social contagions on complex networks

Dynamics of SIR model with heterogeneous response to intervention policy

In classical epidemic theory, behavior is assumed to be stationary. In recent years, epidemic models have been extended to include behaviors that transition in response to the current state of the epidemic. However, it is widely known that human behavior can exhibit strong history-dependence as a consequence of learned experiences. This history-dependence is similar to hysteresis phenomena that have been well-studied in control theory. To illustrate the importance of history-dependence for epidemic theory, we study dynamics of a variant of the SIRS model where individuals exhibit lazy-switch responses to prevalence dynamics, based on the Preisach hysteresis operator. The resulting model can possess a continuum of endemic equilibrium states characterized by different proportions of susceptible, infected and recovered populations. We consider how the limit point of the epidemic trajectory and the infection peak along this trajectory depend on the degree of heterogeneity of the response. Our approach supports the argument that public health responses during the emergence of a new disease can have fundamental long-term consequences for subsequent management efforts.

Early warning signals from the periphery: A model suggestion for the study of critical transitions

Studies on the possibility of predicting critical transitions with statistical methods known as early warning signals (EWS) are often conducted on data generated with equation-based models (EBMs). These models base on difference or differential equations, which aggregate a system's components in a mathematical term and therefore do not allow for a detailed analysis of interactions on micro-level. As an alternative, we suggest a simple, but highly flexible agent-based model (ABM), which, when applying EWS-analysis, gives reason to (a) consider social interaction, in particular negative feedback effects, as an essential trigger of critical transitions, and (b) to differentiate social interactions, for example in network representations, into a core and a periphery of agents and focus attention on the periphery. Results are tested against time series from a networked version of the Ising-model, which is often used as example for generating hysteretic critical transitions.

Dynamics of SIR model with vaccination and heterogeneous behavioral response of individuals modeled by the Preisach operator

We study global dynamics of an SIR model with vaccination, where we assume that individuals respond differently to dynamics of the epidemic. Their heterogeneous response is modeled by the Preisach hysteresis operator. We present a condition for the global stability of the infection-free equilibrium state. If this condition does not hold true, the model has a connected set of endemic equilibrium states characterized by different proportion of infected and immune individuals. In this case, we show that every trajectory converges either to an endemic equilibrium or to a periodic orbit. Under additional natural assumptions, the periodic attractor is excluded, and we guarantee the convergence of each trajectory to an endemic equilibrium state. The global stability analysis uses a family of Lyapunov functions corresponding to the family of branches of the hysteresis operator.

Coevolution spreading in complex networks

The propagations of diseases, behaviors and information in real systems are rarely independent of each other, but they are coevolving with strong interactions. To uncover the dynamical mechanisms, the evolving spatiotemporal patterns and critical phenomena of networked coevolution spreading are extremely important, which provide theoretical foundations for us to control epidemic spreading, predict collective behaviors in social systems, and so on. The coevolution spreading dynamics in complex networks has thus attracted much attention in many disciplines. In this review, we introduce recent progress in the study of coevolution spreading dynamics, emphasizing the contributions from the perspectives of statistical mechanics and network science. The theoretical methods, critical phenomena, phase transitions, interacting mechanisms, and effects of network topology for four representative types of coevolution spreading mechanisms, including the coevolution of biological contagions, social contagions, epidemic-awareness, and epidemic-resources, are presented in detail, and the challenges in this field as well as open issues for future studies are also discussed.

Critical transitions in heterogeneous networks: Loss of low-degree nodes as an early warning signal

A large number of real networks show abrupt phase transition phenomena in response to environmental changes. In this case, cascading phenomena can induce drastic and discontinuous changes in the system state and lead to collapse. Although complex network theory has been used to investigate these drastic events, we are still unable to predict them effectively. We here analyze collapse phenomena by proposing a minimal two-state dynamic on a complex network and introducing the effect of local connectivities on the evolution of network nodes. We find that a heterogeneous system of interconnected components presents a mixed response to stress and can serve as a control indicator. In particular, before the critical transition point is reached a severe loss of low-degree nodes is observed, masked by the minimal failure of higher-degree nodes. Accordingly, we suggest that a significant reduction in less connected nodes can indicate impending global failure.