Source-linked AI summary
Fluctuating epidemics on adaptive networks
Leah B. Shaw, Ira B. Schwartz
TL;DR
The paper studies how infection-responsive rewiring changes SIRS epidemic dynamics and network structure. It combines Monte Carlo simulations with a stochastic mean field model tracking nodes and links. Rewiring produces bistability, alters network geometry, and infective fluctuations near bifurcation points show power-law scaling.
Problem
The paper addresses epidemic dynamics on adaptive networks, extending fixed-network epidemic models to include an immune recovered class and infection-responsive network changes.
Method
The study combines Monte Carlo simulations with a stochastic mean field model that tracks both node and link dynamics in an evolving SIRS network.
Results
Rewiring produces bistability between endemic and disease-free states, changes degree distributions and distances to infectives, and yields power-law infective fluctuations near bifurcation points.
Takeaways & Limitations
Adaptive rewiring affects both epidemic dynamics and network geometry, while mean field analysis can predict steady states and fluctuation scaling near appropriate bifurcations.
Abstract
from arXiv · showhide
A model for epidemics on an adaptive network is considered. Nodes follow an SIRS (susceptible-infective-recovered-susceptible) pattern. Connections are rewired to break links from non-infected nodes to infected nodes and are reformed to connect to other non-infected nodes, as the nodes that are not infected try to avoid the infection. Monte Carlo simulation and numerical solution of a mean field model are employed. The introduction of rewiring affects both the network structure and the epidemic dynamics. Degree distributions are altered, and the average distance from a node to the nearest infective increases. The rewiring leads to regions of bistability where either an endemic or a disease-free steady state can exist. Fluctuations around the endemic state and the lifetime of the endemic state are considered. The fluctuations are found to exhibit power law behavior.
I. INTRODUCTION
The paper extends epidemic modeling from fixed networks to an adaptive network whose rewiring responds to infection. It combines network-structure analysis with stochastic epidemic dynamics, including bistability and fluctuations.
- Fixed-network epidemic models do not capture stochastic phenomena such as extinction and sustained fluctuations.
- Most earlier network epidemic models used fixed architectures and commonly modeled SIS or SIR dynamics in large, isolated populations.
- Adaptive rewiring responds to susceptible-infective links by replacing them with susceptible-susceptible links, reducing infection through isolation.
- Rewiring can produce bistability between disease-free and endemic states, unlike static-network models that typically have one attracting state.
- The paper introduces an immune recovered class in an adaptive-network SIS generalization and studies network structure, bifurcations, fluctuations, and state lifetimes.
II. MODEL
The model uses SIRS node dynamics coupled to adaptive rewiring of links incident to infected nodes. Monte Carlo simulations are paired with a stochastic mean field model that tracks both nodes and links.
- SIRS dynamics comprise infection at rate pNI,nbr, recovery at rate r, and return from recovered to susceptible at rate q.
- Non-infected–infected links rewire at rate w to connect the non-infected node with another randomly selected non-infected node.
- The recovery rate is fixed at r = 0.002 because time can be rescaled while analyzing steady states.
- Monte Carlo simulations use N = 104 nodes and K = 105 links, with larger systems also considered for size dependence.
- The mean field model tracks node-state and link-state probabilities and couples rewiring to infection through the number of susceptible-infected links.
- Three-point link probabilities are closed using the approximation PABC ≈ PABPBC/PB, and the resulting ordinary differential equations are numerically integrated.
III. BIFURCATION STRUCTURE
Rewiring changes the steady-state bifurcation structure by stabilizing the disease-free state at higher transmission rates and creating bistable regimes. Mean field predictions generally agree with simulations, while the bifurcation type can differ.
- Rewiring stabilizes the disease-free state at larger transmission rates and creates a region where disease-free and endemic states coexist.
- Mean field and Monte Carlo steady-state values show fairly good agreement, although their predicted stability and bifurcation types sometimes differ.
- Figure 1 compares infected fraction with transmission rate p for static and rewired networks using Monte Carlo points and stable or unstable mean field branches.
- The mean field model identifies transcritical, saddle-node, and Hopf bifurcations, including unstable periodic-orbit branches from the Hopf bifurcation.
- For q = 0.0064 and w = 0.04, endemic states become stable at a saddle-node point, producing a clear bistable interval as p increases.
- Lowering q to 0.0016 brings the Hopf and transcritical branches closer together, substantially narrowing the bistability region for sufficiently large w.
- For w = 0.04, the endemic state loses stability through a saddle-node bifurcation at q = 0.0064 but through a Hopf bifurcation at q = 0.0016.
A. Degree distributions
Rewiring significantly changes degree distributions across susceptible, recovered, and infected nodes. Mean-field calculations describe susceptible and recovered distributions, but infected-node correlations limit accurate prediction.
- Rewiring changes degree distributions from the Poissonian distributions observed for all node types in the static network.For the rewired network, distributions are reported separately for infectives, recovereds, and susceptibles.
- Mean-field analysis follows nodes as they flow from infected to recovered to susceptible to characterize recovered and susceptible degree distributions.Recovered nodes gain degree through rewiring, and susceptible nodes likewise increase degree as rewiring adds connections.
- Recovered degree distributions can be calculated from equations using the infected distribution and rewiring-related probabilities.The calculation solves the recovered distribution given dI,n and the probabilities entering the rewiring rate k.
- The susceptible distribution uses an infection-rate-per-link assumption that is approximately independent of node degree in simulations.The relevant rate is expressed using susceptible link probabilities, including PSI, PSR, and PSS.
- The approximate procedure cannot accurately predict infected-node degree distributions because correlations among neighboring infection states are important.Simulations show low-degree infected nodes have more infected neighbors than the approximation predicts, causing low-degree infectives to be over-predicted.
B. Distance from an infective
The study examines distances from nodes to their nearest infective as a measure related to how many hops infection must traverse. Rewiring changes this distance distribution by reducing direct exposure while leaving only a small disconnected fraction.
- Distance metric: Distances to the nearest infective indicate how many hops infection must make to reach an uninfected individual.This metric does not always define a disease-propagation path because recovered nodes block transmission until they become susceptible again.
- Effect of rewiring: Rewiring significantly decreases the number of nodes directly connected to an infective, while only a small fraction become fully disconnected.The comparison uses matched approximate infective numbers in rewired and non-rewired systems.
- Comparisons: Figure 6 compares nearest-infective distance distributions with and without rewiring and against random graphs.The figure marks ∞ for nodes completely disconnected from an infective.
- Random-network comparison: The distance-distribution tail in the adaptive network decays in the same way as in matched random networks.The adaptive network differs by having some nodes fully disconnected from infected components, unlike the random networks.
V. FLUCTUATIONS AND OTHER DYNAMICS
After analyzing steady states and long-time network properties, the paper turns to fluctuations and other dynamical properties of the endemic state.
- The analysis shifts from steady states and long-time network averages to fluctuations and dynamical properties of the endemic state.
- The endemic state is examined as a time-dependent dynamical regime rather than only through its average network properties.
- This section introduces the study of fluctuations around the endemic state.
A. Fluctuations near bifurcation point
The paper studies noise-driven fluctuations near the loss of endemic-state stability using Monte Carlo simulations, stochastic mean field equations, and a saddle-node approximation. Fluctuations follow power laws, with different exponents in the two full-system approaches.
- A. Fluctuations near bifurcation point: Fluctuations grow near the bifurcation point because noise can overcome weak attracting forces.The study quantifies fluctuations as the standard deviation divided by the mean of long time series.
- A. Fluctuations near bifurcation point: The analysis combines Monte Carlo simulations with stochastic mean field equations using additive noise near the endemic state.Multiplicative noise was also simulated and produced similar results to additive noise.
- A. Fluctuations near bifurcation point: The mean field fluctuation study uses q = 0.0064 because its saddle-node structure corresponds best to the Monte Carlo scaling observed.For q = 0.0016, the mean field endemic state loses stability through a Hopf bifurcation; for q = 0.0064, it does so through a saddle-node bifurcation.
- A. Fluctuations near bifurcation point: Both Monte Carlo and mean field fluctuations exhibit power law scaling as infection rate p approaches the critical point pc.The critical point is exact for mean field and approximated for Monte Carlo by selecting the value producing the most linear log-log plot.
- A. Fluctuations near bifurcation point: −0.59 is the Monte Carlo exponent, compared with −0.27 for mean field fluctuations.
- A. Fluctuations near bifurcation point: The saddle-node approximation assumes additive noise sufficiently small to avoid shifting the parameter value where the saddle-node disappears.It also treats the dynamics near the attracting branch and uses a stationary probability density near steady state.
B. Delayed outbreaks
The paper examines phase relationships between infectives and their infective neighbors. In the rewired system, infective fluctuations lag behind neighbor fluctuations, and larger rewiring rates produce longer delays.
- B. Delayed outbreaks: Infective counts lag behind fluctuations in the number of non-infected nodes neighboring infectives.This effect appears in the rewired system and is observed in both mean field and Monte Carlo simulations.
- B. Delayed outbreaks: Increasing rewiring rates produces increasing time lags and delayed outbreaks.The same trend occurs in the mean field model with multiplicative noise.
- B. Delayed outbreaks: Cross-correlation identifies the lag that maximizes correspondence between infective and infective-neighbor time series.
C. Lifetime of endemic steady state
Near the saddle-node bifurcation, the endemic state's finite lifetime is governed by stochastic escape from weak stability. Simulations examine how its average lifetime depends on infection rate and compare the behavior with saddle-node scaling.
- Figure 9 compares Monte Carlo and mean-field descriptions of delayed outbreaks, with infectives lagging behind their infective neighbors.
- Figure 10 plots endemic-state average lifetime against infection rate p using Monte Carlo points and a best-fit line.
- Near the bifurcation point, the endemic state is weakly stable, making disease die-out easier to observe.
- For a generic one-dimensional saddle-node bifurcation, ln T is expected to vary linearly with (p − p0)^3/2.
- A system with 4 × 10^4 nodes and 4 × 10^5 links enabled longer simulations closer to the bifurcation point.
VI. CONCLUSIONS AND DISCUSSION
Adaptive rewiring produces bistability between endemic and disease-free states while changing network structure and epidemic dynamics. Mean-field analysis captures steady states and some fluctuations, but several network properties require more specialized treatment.
- Epidemic dynamics: Rewiring produces bistability between endemic and disease-free steady states, with the bistability region adjustable through the recovered-class dynamics.The SIRS model’s re-susceptibility rate changes the locations of bifurcation points and therefore the width of the bistability region.
- Epidemic dynamics: Power-law scaling appears in infective fluctuations near the bifurcation point, consistent with mean-field results and saddle-node scaling expectations.
- Network structure: Rewiring alters degree distributions; mean-field arguments predict susceptible and recovered distributions, but correlations are needed to fully explain them.
- Network structure: Distances from non-infected nodes to the nearest infective depend mainly on node and link dynamics and can be predicted from random graphs.This distance distribution is identified as potentially relevant to disease spreading and control.
- Epidemic dynamics: Delayed outbreaks occur because infective-fraction peaks lag behind peaks in the number of nodes neighboring infectives.For the studied parameters, the lag is about 10% of the mean infectious period, although the parameters were not chosen for a specific disease.
- Epidemic dynamics: Near the saddle-node bifurcation, endemic-state extinction requires escape from weak attractive forces, and the escape rate is analytically characterized by the local topology.