Coexistence of positive and negative information in information-epidemic dynamics on multiplex networks

TL;DR

The paper addresses how coexisting positive and negative disease-related information shapes information-epidemic dynamics on multiplex networks. It develops a UA1A2U-SIS model and uses an individual-based mean-field framework to derive the epidemic threshold in terms of the largest eigenvalue of a level-dependent matrix $H$, yielding $\beta_c = \mu / \Lambda_{\max}(H)$. In the fully connected case, it provides explicit threshold expressions and fixed points for awareness densities and reveals a cusp in infection prevalence signaling a transition between coexistence and dominance of a single information type; it also shows that away from the threshold the prevalence can be monotone or non-monotone depending on $\lambda_1$ and $\lambda_2$, with a monotonicity criterion $\beta'_{\mathrm cusp}=\beta'_{\mathrm max}$. The results are validated against Gillespie Monte Carlo simulations across network structures, demonstrating robust agreement with the mean-field predictions. The work highlights how information diffusion parameters can be tuned to influence epidemic outcomes, offering insights for information-inspired epidemic control strategies.

Abstract

This paper investigates the coexistence of positive and negative information in the context of information-epidemic dynamics on multiplex networks. In accordance with the tenets of mean field theory, we present not only the analytic solution of the prevalence threshold, but also the coexistence conditions of two distinct forms of information (i.e., the two phase transition points at which a single form of information becomes extinct). In regions where multiple forms of information coexist, two completely distinct patterns emerge: monotonic and non-monotonic. The physical mechanisms that give rise to these different patterns have also been elucidated. The theoretical results are robust with regard to the network structure and show a high degree of agreement with the findings of the Monte Carlo simulation.

Figures (7)

• Figure 1: The schematic illustrations demonstrate the $\mathrm{UA_1A_2U}$-SIS model on a multiplex network in (a), along with the associated transition rates between states in (b). The information layer describes the diffusion of information that facilitates the adoption of protective measures against the disease, and nodes have three kinds of states: unaware (U), aware of positive preventive information ($\mathrm{A_1}$), and aware of negative preventive information ($\mathrm{A_2}$). The epidemic layer describes the spread of disease in populations with heterogeneous protective measures, and nodes can be either infected (I) or susceptible (S). It should be noted that UI and $\mathrm{A_2I}$ are excluded from the $\mathrm{UA_1A_2U}$-SIS model. Consequently, individuals can be in four possible states: $\mathrm{A_1S}$, $\mathrm{A_2S}$, US, and $\mathrm{A_1I}$. The quantity $k_{\mathrm{neig}}(\mathrm{X})$ represents the number of the individual's neighbors who are in the state $X$, where $\mathrm{X\in\{A_1S, A_2S, US, A_1I}\}$ and $k_{\mathrm{neig}}(\mathrm{A_1})=k_{\mathrm{neig}}(\mathrm{A_1S})+k_{\mathrm{neig}}(\mathrm{A_1I})$.
• Figure 2: The epidemic threshold as a function of awareness rate of negative information for different awareness rate of positive information on mutiplex networks. (a) Both the information layer and the disease layer are fully connected networks. The results are obtained from Eq. \ref{['eqq6']}. (b) The information layer uses the BA network with an average degree of $6$, and the epidemic layer is the ER network with an average degree of $5$. The results are obtained from Eq. \ref{['eqq4']}. Parameters: $\gamma_1=0$, $\gamma_2=0.5$; (a) $\delta_1=\delta_2=0.3$, $\mu=0.5$; (b) $\delta_1=\delta_2=1$, $\mu=1$, $N=10^4$.
• Figure 3: The epidemic prevalence $\rho^{\mathrm{I}}$ as a function of infection rate on mutiplex networks. (a) Both the information layer and the disease layer are fully connected networks. The results are obtained from Eq. \ref{['eqq5a']}-Eq. \ref{['eqq5c']}, with parameters $\delta_1=\delta_2=0.3$ and $\lambda'_1=0.5$. (b) The information layer uses the BA network with an average degree of $6$, and the epidemic layer is the ER network with an average degree of $5$. The results are obtained from Eq. \ref{['eqq1a']}-Eq. \ref{['eqq1c']}, with parameters $\delta_1=\delta_2=0.15$, $\lambda_1=0.5$, and $N=10^4$. Other parameters: $\gamma_1=0$, $\gamma_2=0.5$, $\mu=0.5$; (a1) $\lambda'_2=0.6$; (a2) $\lambda'_2=0.5$; (a3) $\lambda'_2=0.4$; (b1) $\lambda_2=0.6$; (b2) $\lambda_2=0.5$; (b3) $\lambda_2=0.4$. The diagonal shadow indicates that only $\mathrm{A_2}$ is present, without $\mathrm{A_1}$. The gray area represents the coexistence of $\mathrm{A_1}$ and $\mathrm{A_2}$ in this region. The remaining (blank) area indicates the presence of $\mathrm{A_1}$, without $\mathrm{A_2}$.
• Figure 4: The epidemic prevalence $\rho^{\mathrm{I}}$ as a function of infection rate on mutiplex fully connected networks. The lines are obtained from Eq. \ref{['eqq5a']}-Eq. \ref{['eqq5c']}. Parameters: $\mu=0.1$, $\delta_1=\delta_2=0.3$, $\lambda'_1=2.5$, $\lambda'_2=3$; (a) $\gamma_1=0$; (b) $\gamma_2=0.8$.
• Figure 5: The criteria for evaluating the monotony of the infection curve and the rationale for determining the extent of non-monotony. The difference $\beta'_{\mathrm{cusp}}-\beta'_{\mathrm{max}}$ and slope $K$ as a function of $\lambda'_1$ in panel (a), $\lambda'_2$ in panel (b), and $\mu$ in panel (c). The lines and dash lines is obtained from Eq. \ref{['eqq9']}-Eq. \ref{['eqq12']}. Parameters: $\gamma_1=0$, $\gamma_2=1$, $\delta_1=\delta_2=0.3$; (a) $\mu=0.1$, $\lambda'_2=3$; (b) $\mu=0.1$, $\lambda'_1=2.5$; (c) $\lambda'_1=2.5$, $\lambda'_2=3$. Under the grey area parameters ($\beta'_{\mathrm{cusp}}-\beta'_{\mathrm{max}}<0$), the infected curve $\rho^{\mathrm{I}}(\beta')$ shows a monotonic increase, whereas the other regions ($\beta'_{\mathrm{cusp}}-\beta'_{\mathrm{max}}>0$) exhibit non-monotonic curves.