Coupled dynamics of endemic disease transmission and gradual awareness diffusion in multiplex networks

TL;DR

This paper introduces the UWAU-SIS model, a coupled disease-awareness framework on multiplex networks that captures gradual, stage-based diffusion of disease-related information through a separate communication layer and endemic disease transmission through a contact layer. By combining the Microscopic Markov Chain Approach with the law of total probability, it derives discrete-time and continuous-time epidemic thresholds and proves exactness for unclustered networks, outperforming a probability-tree baseline. The key finding is that merely informing the unaware is insufficient; strong, timely awareness and protective behavior (governed by the transition rate $\alpha$ and forgetting rate $\delta$) are required to elevate the epidemic threshold, with intricate interactions among behavioral parameters and network overlap. These results highlight the nontrivial coupling between information diffusion and contagion dynamics and suggest public health strategies should account for stage-based awareness progression and its time scales. The work also provides a continuous-time formulation and demonstrates robustness across phase-space explorations, offering a general methodology for analyzing coupled contagion processes in multilayer networks.

Abstract

Understanding the interplay between human behavioral phenomena and infectious disease dynamics has been one of the central challenges of mathematical epidemiology. However, socio-cognitive processes critical for the initiation of desired behavioral responses during an outbreak have often been neglected or oversimplified in earlier models. Combining the microscopic Markov chain approach with the law of total probability, we herein institute a mathematical model describing the dynamic interplay between stage-based progression of awareness diffusion and endemic disease transmission in multiplex networks. We analytically derived the epidemic thresholds for both discrete-time and continuous-time versions of our model, and we numerically demonstrated the accuracy of our analytic arguments in capturing the time course and the steady-state of the coupled disease-awareness dynamics. We found that our model is exact for arbitrary unclustered multiplex networks, outperforming a widely adopted probability-tree-based method, both in the prediction of the time-evolution of a contagion and in the final epidemic size. Our findings show that informing the unaware individuals about the circulating disease will not be sufficient for the prevention of an outbreak unless the distributed information triggers strong awareness of infection risks with adequate protective measures, and that the immunity of highly-aware individuals can elevate the epidemic threshold, but only if the rate of transition from weak to strong awareness is sufficiently high. Our study thus reveals that awareness diffusion and other behavioral parameters can nontrivially interact when producing their effects on epidemiological dynamics of an infectious disease, suggesting that future public health measures should not ignore this complex behavioral interplay and its influence on contagion transmission in multilayered networked systems.

Figures (6)

• Figure 1: (Color online) The time-evolution of infection density $\rho^I(t)$ in coupled disease-awareness spreading models at a fixed rate of transition from weakly-aware to strongly-aware nodes $\alpha$ and at a fixed infectivity rate $\beta^U$. The red curves in all panels represent the ensemble averages of 100 independent stochastic simulation runs (depicted by green lines) that are further contrasted against the numerical outcomes of two theoretical models, including our new UWAU-SIS model (\ref{['eqnn1']}) predictions (blue line) and the predictions obtained with the corresponding probability tree based model (\ref{['eqn1']}) (dashed black line) in a scale-free multiplex network with the parameter values $q=0.2$, $k_{0} = 3$, $\Delta = 2.5$, and $N = 1000$. The model parameter values employed in our stochastic simulations were $\gamma=0$, $\delta=0.6$, $\lambda=0.15$, $\mu=0.4$, and in the initial condition there were always $20$ infected/aware nodes. Panels (a)-(d) summarize the results for different combinations of values of $\alpha,\beta^U$ parameters.
• Figure 2: (Color online) Full phase diagrams $\lambda-\beta^U$ for the same multiplex as described in Figure 1, contrasting Monte Carlo simulations and the MMCA approach for the fraction $\rho^I$ of infected individual agents in the stationary state (colors represent the fractions of infected individuals). Panel (a) shows the stochastic simulation results and panel (b) represents the numerical simulations of the MMCA model. In all simulations, our selected model parameter values were $\delta=0.6$, $\mu=0.4$, $\gamma=0$,$\alpha=0.5$ and $q=0.2$.
• Figure 3: (Color online) The density of infectious nodes $\rho^I$ in the stable state as a function of $\beta^U$ for two different values of $\alpha$ in a multiplex network with $q=0.2,\Delta= 2.5$, $k_{0}= 3$, and $N=1000$. Our other model parameter values were: $\lambda=0.15,\delta=0.6,\gamma=0,\mu=0.4.$ The displayed simulation results represent the performance of the MMCA model (\ref{['eqnn1']}), and the epidemic threshold is computed with the formula (\ref{['threshold']}).
• Figure 4: (Color online) The epidemic threshold as a function of the model parameters $\alpha$ (a), $\lambda$ (b), $\gamma$ (c) and $\delta$ (d). In panel (a), the results are shown for two different, fixed values of $\lambda$; in panels (b)-(d), the results are shown for three distinct values of $\alpha$. In numerical simulations, unless otherwise indicated and if not varied, the remaining parameter values were always fixed to $\delta=0.6$, $\mu=0.4$, $\gamma=0$, and $\lambda=0.2$.
• Figure 5: (Color online) The final epidemic size as a function of the infection rate $\beta^U$ in scale-free multiplex networks with different levels of the network links overlap $q$. Panel (a) shows the results of the stochastic simulations and panel (b) corresponds to the outcomes of the numerical simulations of the MMCA model. In all simulations, our other parameter values were selected as $\delta=0.6$, $\mu=0.4$, $\gamma=0$, $\lambda=0.5$, and $\alpha=0.5$.