Models of epidemic spreading on complex networks have attracted great attention among researchers in physics, mathematics, and epidemiology due to their success in predicting and controlling scenarios of epidemic spreading in real-world scenarios. To understand the interplay between epidemic spreading and the topology of a contact network, several outstanding theoretical approaches have been developed. An accurate theoretical approach describing the spreading dynamics must take both the network topology and dynamical correlations into consideration at the expense of increasing the complexity of the equations. In this short survey we unify the most widely used theoretical approaches for epidemic spreading on complex networks in terms of increasing complexity, including the mean-field, the heterogeneous mean-field, the quench mean-field, dynamical messagepassing, link percolation, and pairwise approximation. We build connections among these approaches to provide new insights into developing an accurate theoretical approach to spreading dynamics on complex networks.

The spread of disease through a physical-contact network and the spread of information about the disease on a communication network are two intimately related dynamical processes. We investigate the asymmetrical interplay between the two types of spreading dynamics, each occurring on its own layer, by focusing on the two fundamental quantities underlying any spreading process: epidemic threshold and the final infection ratio. We find that an epidemic outbreak on the contact layer can induce an outbreak on the communication layer, and information spreading can effectively raise the epidemic threshold. When structural correlation exists between the two layers, the information threshold remains unchanged but the epidemic threshold can be enhanced, making the contact layer more resilient to epidemic outbreak. We develop a physical theory to understand the intricate interplay between the two types of spreading dynamics.

A key ingredient in social contagion dynamics is reinforcement, as adopting a certain social behavior requires verification of its credibility and legitimacy. Memory of nonredundant information plays an important role in reinforcement, which so far has eluded theoretical analysis. We first propose a general social contagion model with reinforcement derived from nonredundant information memory. Then, we develop a unified edge-based compartmental theory to analyze this model, and a remarkable agreement with numerics is obtained on some specific models. We use a spreading threshold model as a specific example to understand the memory effect, in which each individual adopts a social behavior only when the cumulative pieces of information that the individual received from his or her neighbors exceeds an adoption threshold. Through analysis and numerical simulations, we find that the memory characteristic markedly affects the dynamics as quantified by the final adoption size. Strikingly, we uncover a transition phenomenon in which the dependence of the final adoption size on some key parameters, such as the transmission probability, can change from being discontinuous to being continuous. The transition can be triggered by proper parameters and structural perturbations to the system, such as decreasing individuals' adoption threshold, increasing initial seed size, or enhancing the network heterogeneity.

Epidemic spreading has been well studied in the past decade, where the main concentration is focused on the influence of network topology but little attention is paid to the individual's crisis awareness. We here study how the crisis awareness, i.e., personal self-protection, influences the epidemic spreading by presenting a susceptible-infected-recovered model with information-driven vaccination. We introduce two parameters to quantitatively characterize the crisis awareness. One is the information creation rate λ and the other is the information sensitivity η. We find that the epidemic spreading can be significantly suppressed in both the homogeneous and heterogeneous networks when both λ and η are relatively large. More interesting is that the needed vaccine will be significantly reduced when the information is well spread, which is a good news for the poor countries and regions with limited resources.

Identifying the most influential spreaders is an important issue in understanding and controlling spreading processes on complex networks. Recent studies showed that nodes located in the core of a network as identified by the k-shell decomposition are the most influential spreaders. However, through a great deal of numerical simulations, we observe that not in all real networks do nodes in high shells are very influential: in some networks the core nodes are the most influential which we call true core, while in others nodes in high shells, even the innermost core, are not good spreaders which we call core-like group. By analyzing the k-core structure of the networks, we find that the true core of a network links diversely to the shells of the network, while the core-like group links very locally within the group. For nodes in the core-like group, the k-shell index cannot reflect their location importance in the network. We further introduce a measure based on the link diversity of shells to effectively distinguish the true core and core-like group, and identify core-like groups throughout the networks. Our findings help to better understand the structural features of real networks and influential nodes.

Although there is always an interplay between the dynamics of information diffusion and disease spreading, the empirical research on the systemic coevolution mechanisms connecting these two spreading dynamics is still lacking. Here we investigate the coevolution mechanisms and dynamics between information and disease spreading by utilizing real data and a proposed spreading model on multiplex network. Our empirical analysis finds asymmetrical interactions between the information and disease spreading dynamics. Our results obtained from both the theoretical framework and extensive stochastic numerical simulations suggest that an information outbreak can be triggered in a communication network by its own spreading dynamics or by a disease outbreak on a contact network, but that the disease threshold is not affected by information spreading. Our key finding is that there is an optimal information transmission rate that markedly suppresses the disease spreading. We find that the time evolution of the dynamics in the proposed model qualitatively agrees with the real-world spreading processes at the optimal information transmission rate.

The spread of disease on complex networks has attracted widely attention in the physics community. Recent works have demonstrated that heterogeneous degree and weight distributions have a significant influence on the epidemic dynamics. In this study, a novel edge-weight based compartmental approach is developed to estimate the epidemic threshold and epidemic size (final infected density) on networks with general degree and weight distributions, and a remarkable agreement with numerics is obtained. Even in complex network with the strong heterogeneous degree and weight distributions, this approach is worked. We then propose an edge-weight based removal strategy with different biases, and find that such a strategy can effectively control the spread of epidemic when the highly weighted edges are preferentially removed, especially when the weight distribution of a network is extremely heterogenous. The theoretical results from the suggested method can accurately predict the above removal effectiveness.

The interplay between individual behaviors and epidemic dynamics in complex networks is a topic of recent interest. In particular, individuals can obtain different types of information about the disease and respond by altering their behaviors, and this can affect the spreading dynamics, possibly in a significant way. We propose a model where individuals' behavioral response is based on a generic type of local information, i.e., the number of neighbors that has been infected with the disease. Mathematically, the response can be characterized by a reduction in the transmission rate by a factor that depends on the number of infected neighbors. Utilizing the standard susceptible-infected-susceptible and susceptible-infected-recovery dynamical models for epidemic spreading, we derive a theoretical formula for the epidemic threshold and provide numerical verification. Our analysis lays on a solid quantitative footing the intuition that individual behavioral response can in general suppress epidemic spreading. Furthermore, we find that the hub nodes play the role of “double-edged sword” in that they can either suppress or promote outbreak, depending on their responses to the epidemic, providing additional support for the idea that these nodes are key to controlling epidemic spreading in complex networks.