Networked SIS Epidemics with Awareness

TL;DR

The paper analyzes networked SIS epidemics where agents adjust contact patterns based on prevalence-based awareness from local and global information. It develops a mean-field approximation showing the epidemic threshold remains the same as in the benchmark model, while nontrivial fixed points are reduced under awareness. A monotone coupling framework provides a stochastic comparison that yields explicit reductions in epidemic-cost metrics and proves that awareness cannot improve the persistence threshold. Simulations on geometric, Erdős-Rényi, and scale-free graphs reveal that local (neighborhood) awareness is most effective early in an outbreak and that network structure critically shapes the spread and final size. The findings offer insights for designing information and distancing strategies in interconnected populations.

Abstract

We study an SIS epidemic process over a static contact network where the nodes have partial information about the epidemic state. They react by limiting their interactions with their neighbors when they believe the epidemic is currently prevalent. A node's awareness is weighted by the fraction of infected neighbors in their social network, and a global broadcast of the fraction of infected nodes in the entire network. The dynamics of the benchmark (no awareness) and awareness models are described by discrete-time Markov chains, from which mean-field approximations (MFA) are derived. The states of the MFA are interpreted as the nodes' probabilities of being infected. We show a sufficient condition for existence of a "metastable", or endemic, state of the awareness model coincides with that of the benchmark model. Furthermore, we use a coupling technique to give a full stochastic comparison analysis between the two chains, which serves as a probabilistic analogue to the MFA analysis. In particular, we show that adding awareness reduces the expectation of any epidemic metric on the space of sample paths, e.g. eradication time or total infections. We characterize the reduction in expectations in terms of the coupling distribution. In simulations, we evaluate the effect social distancing has on contact networks from different random graph families (geometric, Erdős-Renyi, and scale-free random networks).

Figures (5)

• Figure 1: (a) Node-level state transition diagram. (b) System-level diagram
• Figure 2: Diagram of the proof of Theorem \ref{['theorem_nontrivial_fixed_point']}. Here, $p^*$ denotes a nontrivial fixed point of $\phi$.
• Figure 3: A pair of sample paths $(h,g)$ drawn from $\Phi_\pi$.
• Figure 4: Norms of the nontrivial fixed points (solid lines) and long-run fraction of infected in stochastic simulations (diamonds) in the range of epidemic persistence, $\delta/\beta \in [0,\lambda_{\text{max}}(A_C)]$, for $n=1000$ node networks. The fixed points are computed by iterating the MFA dynamics \ref{['eq:MFA_dist']} and \ref{['eq:MFA_standard']} with an arbitrary initial condition until convergence. The stochastic long-run infected fractions are computed by averaging the levels of epidemic states in the latter half of a sample run of length 200. Vertical dashed lines indicate $\lambda_{\text{max}}(A_C)$. (a) Erdős-Renyi random network with $p_{\text{ER}}=.01$, $\lambda_{\text{max}}(A_C)=11.1$. Here, $p_{\text{ER}} > \log n/n$, the regime where the network is connected with high probability. (b) Geometric random graph with $r = .0564$, $\lambda_{\text{max}}(A_C)=16.52$. (c) Scale-free generated from the PA algorithm with $m=5$, $\lambda_{\text{max}}(A_C)=19.9$. The parameters are chosen such that all networks have the same average degree $d \approx 10$.
• Figure 5: Epidemic spreading as a function of time (same networks as Fig. \ref{['fig:FP_plots']}). Local contact information ($\alpha$ near 1, $p$ near 0) slows spread most effectively for (a),(c), and the early stages of (b), whereas global information ($\alpha$ near 0 or $\alpha=1,p$ near 1) is least effective. Note the inversion of awareness effectiveness in (b). In these simulations, $\delta = \beta = 0.2$.

Theorems & Definitions (25)

• Remark 1
• Theorem 1
• Lemma 1: Lemma 3.1, Hassibi2013
• Lemma 2
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['theorem_nontrivial_fixed_point']}:
• Corollary 1
• proof
• Definition 1