Epidemic paradox induced by awareness driven network dynamics

TL;DR

While the epidemic size in the susceptible-aware and the all-aware models scales linearly with the network size, the scaling becomes sublinear in the infected-aware model; hence, fewer aware nodes may reduce the epidemic size more effectively; a phenomenon reminiscent of Braess's paradox.

Abstract

We study stationary epidemic processes in scale-free networks with local awareness behavior adopted by only susceptible, only infected, or all nodes. We find that while the epidemic size in the susceptible-aware and the all-aware models scales linearly with the network size, the scaling becomes sublinear in the infected-aware model. Hence, fewer aware nodes may reduce the epidemic size more effectively; a phenomenon reminiscent of Braess's paradox. We present numerical and theoretical analysis, and highlight the role of influential nodes and their disassortativity to raise epidemic awareness.

Figures (8)

• Figure 1: (a) 2D histogram of local awareness and concerns of respondents about their own health in case of an infection in the MASZK survey. The two most frequent answers are framed and written in white. (b) The age distribution of the two most frequent answers from subfigure (a). Among the most aware respondents, those who had medium health concerns (3) were middle-aged (red) and those who had the highest health concerns (5) were older (yellow).
• Figure 2: (a) Illustration of the set of nodes (in shaded areas) counted in the exponents of the awareness functions in the (S) S-aware $M_{\text{S}}$, (I) I-aware $M_{\text{I}}$ and (SI) SI-aware $M_{\text{SI}}$ models. (b) The fractal dimension $d_f$ measured as the slope of the metastable epidemic size $I_{\infty}$ as a function of the network size $n$ in Chung-Lu networks with $\gamma = 2.3$ and $\kappa = 10$. (c) For degree exponent $\gamma < 3$, the fractal dimension of the $M_{\text{I}}$ is smaller than 1, whereas the fractal dimensions of the $M_{\text{S}}$ and the $M_{\text{SI}}$ remain 1. Paradoxically, the infection becomes smaller in the $M_{\text{I}}$ despite that more nodes are aware in the $M_{\text{SI}}$. (d) The density of infection in the metastable state $\varrho(k)$ as function of the node degree $k$ both in simulations (solid) and in mean-field numerical approximations (dashed). Infection is concentrated on low-degree nodes in the $M_{\text{S}}$ and $M_{\text{SI}}$ models, while for $M_{\text{I}}$, the high degree nodes dominate the infection. Inset (left) verifies that the metastable epidemic size is the lowest for $M_{\text{I}}$. Inset (right) shows no paradox in the perceived infection density $\theta$, defined as the probability that a randomly chosen edge has an infected node on its end.
• Figure 3: The fractal dimension $d_f$ of the infection size as a function of the degree distribution exponent $\gamma$ and the average degree exponent $\delta$ in the I-aware and the SI-aware models is in agreement in (a) stochastic simulations, (b) mean-field numerical results and (c) analytical results (Eq. \ref{['eq:d_asymp']}). For $\gamma\in(2,3)$, the fractal dimension in the I-aware model is strictly smaller than in the SI-aware model in all three approaches.
• Figure 4: The separation between paradox-exhibiting (red) and non-exhibiting (blue) real networks. The main plots shows $I_{\text{rat}}$, defined in Eq. \ref{['eq:Irat']}, as a function of the average degree $\langle k \rangle$ and the degree assortativity $\xi$. We compute $\xi$ by fitting the exponent of $k_{\text{nn}}(k) = ck^{\xi}$, where $k_{\text{nn}}(k)$ is the average neighbor degree of nodes with degree $k$correlationPastor. If $\xi > 0$, the network is assortative; otherwise, it is disassortative. The continuous surface is fitted on the data points via linear interpolation. The upper inset shows a similar behavior on synthetically generated Chung-Lu networks with tunable assortativiy (see in Supplementary Section 8).
• Figure A.1: Infection density over time for different models: (a) Stochastic simulations show that the infection stabilizes rapidly, presenting the paradox, as the $M_{\text{I}}$ model reaches the lowest infection density. The inset provides a closer view of the initial spreading phase, where the $M_{\text{I}}$ model starts with the highest density, followed by $M_{\text{SI}}$, and $M_{\text{S}}$. After a few steps, the order reverses. (b) The numerical solution stabilizes around $t=1000$. All presented populations consist of $10^4$ nodes.

Theorems & Definitions (24)

• Lemma A.1
• proof
• Lemma A.2
• proof
• Lemma A.3
• proof
• Lemma A.4
• proof
• Lemma A.5
• proof