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Epidemics in partially overlapped multiplex networks
C. Buono, L. G. Alvarez Zuzek, P. A. Macri, L. A. Braunstein
TL;DR
SIR epidemics have been studied in only a few network-of-networks settings, while real multiplex layers may share only some individuals. The paper uses a branching-process framework to analyze how overlap affects epidemic spread. It finds that as overlap approaches zero, the system threshold is governed by the layer with greater propagation capability, with a discontinuous change in the other layer’s isolated threshold.
Problem
Only a few studies have applied the SIR model to network-of-networks systems, despite real networks interacting and not all individuals being present in every layer.
Source: Introduction · pp. 1–2Method
A branching-process generating-function framework models SIR spread across two partially overlapped layers and derives threshold and recovered-fraction equations.
Source: Introduction · p. 3, Results · p. 5, Discussion · p. 8Results
As q→0, the epidemic threshold is dominated by the layer with greater propagation capability, and the less-propagating layer’s isolated threshold changes discontinuously.
Source: Results · p. 8, Discussion · p. 10Takeaways & Limitations
The system’s epidemic threshold depends on both layer topology and the fraction of shared nodes.
Source: Discussion · p. 8
Abstract
from arXiv · showhide
Many real networks exhibit a layered structure in which links in each layer reflect the function of nodes on different environments. These multiple types of links are usually represented by a multiplex network in which each layer has a different topology. In real-world networks, however, not all nodes are present on every layer. To generate a more realistic scenario, we use a generalized multiplex network and assume that only a fraction $q$ of the nodes are shared by the layers. We develop a theoretical framework for a branching process to describe the spread of an epidemic on these partially overlapped multiplex networks. This allows us to obtain the fraction of infected individuals as a function of the effective probability that the disease will be transmitted $T$. We also theoretically determine the dependence of the epidemic threshold on the fraction $q > 0$ of shared nodes in a system composed of two layers. We find that in the limit of $q \to 0$ the threshold is dominated by the layer with the smaller isolated threshold. Although a system of two completely isolated networks is nearly indistinguishable from a system of two networks that share just a few nodes, we find that the presence of these few shared nodes causes the epidemic threshold of the isolated network with the lower propagating capacity to change discontinuously and to acquire the threshold of the other network.
Introduction
The paper extends SIR epidemic analysis to two multiplex layers that share only a fraction of individuals, using a branching-process framework to relate transmissibility and overlap to epidemic thresholds and recovered fractions. It finds that even small overlap can make the system threshold reflect the layer with greater propagation capability.
Introduction
- Prior SIR studies of interacting networks were few, motivating analysis of nonrecurrent epidemics when individuals are not present in every layer.The paper situates this question within work on interacting networks and multiplex systems.Source: Introduction · pp. 1–2
Method
- The model has two equal-sized layers with a fraction q of shared individuals; transmissibility T is the effective probability of transmission across a link.In the SIR process, infected individuals recover after a fixed time and cannot be reinfected.Source: Introduction · pp. 2–3, Method · pp. 3–4
- A generating-function branching process tracks infection branches that remain within one layer and branches that cross layers through shared nodes.The model uses self-consistent branch probabilities and a Jacobian condition to determine the epidemic threshold.Source: Introduction · p. 3, Method · p. 5, Results · p. 5
Results
- The threshold depends on both layer topology and overlap, decreases as q increases, and is lower at q→1 than at q→0.The authors attribute the decrease to greater degree dispersion and a larger total branching factor as overlap increases.Source: Results · pp. 5–7, Figure 2 · p. 6
- Theory and simulations agree for the recovered fraction across Erdős–Rényi and cutoff power-law layers; increasing q shifts the threshold lower and makes growth in recovery more abrupt.For power-law layers, the threshold loses its q-dependence as the exponential cutoff c→∞.Source: Figure 3 · p. 7, Results · p. 8
- As q→0, the system threshold approaches that of the layer with the higher branching factor, while the less-propagating layer’s isolated threshold changes discontinuously.The joined threshold approaches the higher-branching layer’s isolated threshold quadratically; when q→1, the layers have the same recovered fraction.Source: Results · pp. 5, 8
Discussion
The analysis derives the epidemic threshold and steady-state recovered fractions for two partially overlapped SIR layers. As overlap vanishes, even a few shared nodes can shift the lower-capacity layer’s threshold discontinuously to that of the other layer.
Discussion
- The framework derives Tc and steady-state recovered fractions for a two-layer partially overlapped SIR network.The study considers two layers sharing a fraction q of nodes.Source: Discussion · p. 8
- Figure 4 plots recovered fractions RA and RB against transmissibility T; arrows mark their shared Tc(q), with isolated-layer curves shown for q = 0.The panels show layer A above and layer B below; marker shapes and colors distinguish overlap fractions.Source: Figure 4 · p. 9
- As q → 0, the system threshold is set by the layer with greater propagation capability, and a few shared nodes shift the other layer’s isolated threshold to it.The shift is discontinuous.Source: Discussion · p. 10
- As q falls, total branching falls and the threshold rises; for T > Tc, layer recovery fractions diverge, unlike their equality at q = 1.At q = 1, the threshold is at its minimum.Source: Discussion · p. 10