The impact of nodes of information dissemination on epidemic spreading in dynamic multiplex networks

TL;DR

This study analyzes epidemic spreading on a two-layer multiplex network consisting of an information diffusion layer and a disease spreading layer, incorporating a class of \Omega-nodes that never participate in information diffusion. Using the Microscopic Markov Chain Approach (MMCA), the authors derive a threshold condition \beta_c^U = \mu / \Lambda_{\max}(H) where \Lambda_{\max} is the largest eigenvalue of the matrix \mathbf{H} with elements \;h_{ij}=[1-(1-\gamma) P_i^A] b_{ji}, illustrating how awareness dynamics and network structure jointly shape contagion. Numerical simulations on BA (information) and WS (disease) networks show that high centrality in the awareness layer—via degree, betweenness, or clustering—substantially modulates epidemic spread, while low-centrality Omega-nodes have a weaker, more linear impact as their proportion grows. The results emphasize targeting influential information channels to curb outbreaks and highlight directions for extending theory to more complex networks and a broader class of Omega-nodes.

Abstract

Epidemic spreading processes on dynamic multiplex networks provide a more accurate description of natural spreading processes than those on single layered networks. To describe the influence of different individuals in the awareness layer on epidemic spreading, we propose a two-layer network-based epidemic spreading model, including some individuals who neglect the epidemic, and we explore how individuals with different properties in the awareness layer will affect the spread of epidemics. The two-layer network model is divided into an information transmission layer and a disease spreading layer. Each node in the layer represents an individual with different connections in different layers. Individuals with awareness will be infected with a lower probability compared to unaware individuals, which corresponds to the various epidemic prevention measures in real life. We adopt the micro-Markov chain approach to analytically derive the threshold for the proposed epidemic model, which demonstrates that the awareness layer affects the threshold of disease spreading. We then explore how individuals with different properties would affect the disease spreading process through extensive Monte Carlo numerical simulations. We find that individuals with high centrality in the awareness layer would significantly inhibit the transmission of infectious diseases. Additionally, we propose conjectures and explanations for the approximately linear effect of individuals with low centrality in the awareness layer on the number of infected individuals.

Figures (10)

• Figure 1: The structure of a two-layer network. The states of the nodes have been marked inside the nodes, the dark blue nodes represent $\Omega$ nodes, they do not interact with the nodes in the awareness layer and are always in the U-state. The figure demonstrates that two layer networks have different topological structures, but nodes are one-to-one corresponding.
• Figure 2: The transfer probability tree of five states. It is a visual description of Eq. (1) and Eq. (2), the state that a node is currently in is taken as the root node of this tree, and the probability that it will be in each state at the next time step can be derived from this tree. The first layer of the first tree represents whether an individual in the US-state transitions to A-state, while the second layer represents whether the individual transitions to I-state. The final probability of the transition is the product of the probabilities associated with the two arrows. The remaining trees are analogous to the first one.
• Figure 3: The proportion of final infections with the information transmission rate and disease spreading rate. The horizontal axis is the information transmission rate ($\lambda$) and the vertical axis is the disease spreading rate ($\beta$), which refers to the probability of infection of individuals in the unawareness state (U-state), and different colors in the graph mark different proportions of final infections ($\rho^R$), whose specific values are demonstrated in the bar graph on the right. It illustrates that the transmission rate of information has a significant impact on disease propagation and that there exists a clear threshold for disease transmission.
• Figure 4: The proportion of final infected individuals varies with the information transmission rate and disease spreading rate. the horizontal axis is the information transmission rate ($\lambda$), the vertical axis is the disease spreading rate ($\beta$), which refers to the probability of infection of individuals in the unawareness state (U-state), different colors in the graph mark different proportions of final infected individuals ($\rho^R$), and the experimental results are averaged over 10 simulations. The figure indicates that when $\Omega$ nodes are low-degree centrality nodes, the overall scale of disease propagation is lower than that of high-degree centrality nodes.
• Figure 5: The proportion of $\rho^A$ and $\rho^R$ with different degree centrality $\Omega$ nodes. (a) and (b) are the curves of the proportion of infected people with time, (c) and (d) are the curves of the corresponding number of people in the awareness state (A-state) with time step. The parameters are set as $N=10000$, $\mu = 0.06$, $\delta = 0.04$, $\gamma = 0.5$, and the experimental results are averaged over 10 simulations. (a) and (b) demonstrate that when $\Omega$ nodes are low-degree centrality nodes, the scale of infection and the time to reach maximum scale are lower than those of high-degree centrality nodes. (c) and (d) indicate that the difference between the two is not significant in terms of information dissemination.

Theorems & Definitions (6)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof