Source-linked AI summary
Competing spreading processes on multiplex networks: awareness and epidemics
Clara Granell, Sergio Gomez, Alex Arenas
TL;DR
The paper asks how awareness and epidemic spreading compete on multiplex networks when immunization is partial and infection does not always trigger awareness. It generalizes the coupled model, analyzes its critical behavior and mass-media input, and finds that mass media eliminates the metacritical point.
Problem
The paper examines awareness-epidemic competition beyond the earlier assumptions of total immunization and immediate awareness after infection.
Method
The authors analyze coupled SIS epidemic and UAU awareness processes on multiplex networks using a generalized model and Microscopic Markov Chain Approach.
Results
Mass media makes the metacritical point of the epidemics vanish, while self-awareness does not affect the epidemic threshold and immunization degree does.
Takeaways & Limitations
The metacritical-point phenomenon is rooted in competition between spreading processes and persists across a broad set of interaction scenarios, except when mass media is present.
Takeaways & Limitations
The Microscopic Markov Chain equations assume independence of the probabilities of becoming infected or aware through different neighbors.
Abstract
from arXiv · showhide
Epidemic-like spreading processes on top of multilayered interconnected complex networks reveal a rich phase diagram of intertwined competition effects. A recent study by the authors [Granell et al. Phys. Rev. Lett. 111, 128701 (2013)] presented the analysis of the interrelation between two processes accounting for the spreading of an epidemics, and the spreading of information awareness to prevent its infection, on top of multiplex networks. The results in the case in which awareness implies total immunization to the disease, revealed the existence of a metacritical point at which the critical onset of the epidemics starts depending on the reaching of the awareness process. Here we present a full analysis of these critical properties in the more general scenario where the awareness spreading does not imply total immunization, and where infection does not imply immediate awareness of it. We find the critical relation between both competing processes for a wide spectrum of parameters representing the interaction between them. We also analyze the consequences of a massive broadcast of awareness (mass media) on the final outcome of the epidemic incidence. Importantly enough, the mass media makes the metacritical point to disappear. The results reveal that the main finding i.e. existence of a metacritical point, is rooted on the competition principle and holds for a large set of scenarios.
1. INTRODUCTION
The paper studies competing awareness and epidemic spreading on multiplex networks, extending an earlier model by relaxing total immunization and immediate-awareness assumptions. It analyzes how interaction parameters and mass media affect epidemic criticality and incidence.
- 1. INTRODUCTION: Multiplex networks use shared entities across distinct layers, enabling awareness and epidemics to propagate through different connectivity patterns.The paper highlights physical-contact and information relationships as distinct layers for studying their interplay.
- 1. INTRODUCTION: The study generalizes an earlier awareness-epidemic model by allowing partial immunization and delayed or absent awareness after infection.The generalized interaction is regulated by self-awareness probability κ and immunization parameter γ.
- 1. INTRODUCTION: Self-awareness does not affect the epidemic threshold, whereas the degree of immunization does.These conclusions are reported for the generalized model across the considered interaction scenarios.
- 1. INTRODUCTION: Mass media is modeled as an external source transmitting disease information throughout the information layer.The paper examines whether this global awareness input changes epidemic outcomes.
- 1. INTRODUCTION: Mass media makes the epidemics’ metacritical point vanish.The metacritical point is where epidemic onset begins depending on awareness incidence.
2. MODEL FOR AWARENESS AND EPIDEMIC SPREADING WITH MASS MEDIA
The model places competing SIS epidemic and UAU awareness processes on separate layers of a multiplex network. Infection can generate awareness, awareness can reduce infection, and mass media supplies global information.
- 2. MODEL FOR AWARENESS AND EPIDEMIC SPREADING WITH MASS MEDIA: The multiplex contains the same actors in both layers, but connectivity differs between physical contacts and information sharing.Information links can include online sources that are absent from the physical-contact network.
- 2. MODEL FOR AWARENESS AND EPIDEMIC SPREADING WITH MASS MEDIA: The physical-contact layer carries an SIS epidemic, while the information layer carries a UAU awareness process.SIS uses infection probability β and recovery probability µ; UAU uses awareness and unaware-recovery parameters λ and δ.
- 2. MODEL FOR AWARENESS AND EPIDEMIC SPREADING WITH MASS MEDIA: Infected nodes become aware with probability κ, while aware nodes reduce their infection probability through immunization parameter γ.κ represents self-awareness or willingness to spread information, whereas γ regulates protection against infection.
- 2. MODEL FOR AWARENESS AND EPIDEMIC SPREADING WITH MASS MEDIA: Mass media is represented by one node connected to every information-layer node, making individuals aware with probability m.It models regularly broadcast information from sources such as television, radio, or newspapers.
3. MICROSCOPIC MARKOV CHAIN APPROACH
The MMCA represents awareness and epidemic competition by tracking four joint node states through four sequential phases. Transition probability trees yield dynamical equations that can be iterated for time evolution and analyzed at stationarity.
- State representation: Nodes occupy four joint states: US, UI, AS, or AI, combining awareness status with susceptibility or infection.These states are defined for every node in the multiplex model.
- Transition process: Transition trees encode every possible change from a node’s state at time t to its state at time t + 1, with branch probabilities that may depend on node and time.The trees provide the construction basis for the MMCA equations.
- MMCA formulation: The MMCA equations combine persistence, recovery, and neighbor-driven infection or awareness contributions to update each node-state probability.For the SIS example, infection arises either from remaining infected or from a susceptible node being infected by a neighbor.
- Transition process: Each time step applies awareness spreading, mass-media broadcast, epidemic spreading, and self-awareness of infection in sequence.The transition trees organize state changes across these four phases.
- MMCA formulation: The coupled equations assume independence of infection and awareness acquisition from different neighbors, while normalization holds at every time step.The system can then be iterated from any initial condition and analyzed for its stationary epidemic onset.
4. THE ONSET OF THE EPIDEMICS IN THE PRESENCE OF LOCAL AND GLOBAL AWARENESS
The epidemic onset is obtained from the stationary MMCA system by linearizing near vanishing infection and reducing the threshold condition to an eigenvalue problem. The resulting threshold depends on the stationary awareness state and the model parameters, including local and mass-media awareness.
- Threshold formulation: The epidemic threshold is defined through the stationary order parameter ρI, the fraction of infected nodes in the system.The stationary state satisfies pi(t + 1) = pi(t) for every node and state.
- Threshold formulation: Near epidemic onset, infected-node probabilities are approximated as pI_i = ε_i ≪ 1, allowing higher-order infection terms to be removed.This linearization is applied to the stationary equations governing infection and awareness.
- Awareness background: The threshold calculation uses the stationary awareness equations, including the decoupled UAU process with mass media, to determine the background awareness state.The relevant awareness probabilities are solved iteratively.
- Eigenvalue criterion: Non-trivial solutions of the linearized epidemic equation are eigenvectors of matrix H, with eigenvalues equal to μ/βU.The epidemic onset is determined by the largest eigenvalue of H.
5. RESULTS
The results show that immunization degree and mass media shift epidemic thresholds, whereas self-awareness has negligible effects. Mass media also removes the metacritical point linking epidemic onset to awareness incidence.
- Experimental setup: The multiplex setup varies immunization γ, self-awareness κ, and mass media m across a 1000-node power-law contact network and an information layer with 400 additional links.The epidemic layer uses exponent 2.5, while β denotes βU.
- Self-awareness: Varying self-awareness κ does not affect the epidemic onset and produces no significant change in final incidence across the tested scenarios.This holds with or without mass media and under total or partial immunization.
- Immunization: Lower γ, corresponding to higher immunity, lowers final epidemic incidence and shifts the critical point to larger β for all tested m and κ combinations.The comparison includes the non-existent-coupling case γ = 1.
- Mass media: Higher mass-media intensity m shifts epidemic onset to larger β and lowers final incidence, with the strongest effect at low γ.At γ = 1, mass media has no effect because the epidemic layer is disconnected from the information layer.
- Critical threshold: The critical threshold follows the empirical fit βc ∼(a + bx)^−1 for x = γ or x = m.The fitted constants a and b depend on the respective parameter.
- Metacritical behavior: With m = 0, epidemic onset is independent of λ below a metacritical point; any m > 0 makes that metacritical point disappear.Mass media maintains a finite pool of aware individuals that reduces epidemic incidence.
6. CONCLUSIONS
The paper presents an extended analysis of a generalized model of competing spreading processes on multiplex networks.
- The study generalizes a model of competing awareness and disease spreading processes on multiplex networks.