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Suppressing Epidemics with a Limited Amount of Immunization Units
Christian M. Schneider, Tamara Mihaljev, Shlomo Havlin, Hans J. Herrmann
TL;DR
The paper addresses how to suppress spreading on real networks when immunization resources are limited. It optimizes susceptible network size starting from high-betweenness immunization sequences and tests the resulting strategy on multiple networks. The improved strategy suppresses infection more effectively, especially when only a small number of immunization units is available.
Problem
Limited immunization resources make it important to identify network immunization strategies that suppress disease spreading more effectively than existing high-betweenness methods.
Method
The method optimizes susceptible network size by iteratively modifying high-betweenness node or edge immunization sequences and retaining changes that do not increase it.
Results
Across the studied real networks, the strategy reduces average infection risk by more than 10% and achieves improvements of up to 55% for specific immunization-unit amounts.
Takeaways & Limitations
The strategy outperforms high-betweenness immunization for both node and link immunization across the studied networks, particularly with small immunization budgets.
Abstract
from arXiv · showhide
The way diseases spread through schools, epidemics through countries, and viruses through the Internet is crucial in determining their risk. Although each of these threats has its own characteristics, its underlying network determines the spreading. To restrain the spreading, a widely used approach is the fragmentation of these networks through immunization, so that epidemics cannot spread. Here we develop an immunization approach based on optimizing the susceptible size, which outperforms the best known strategy based on immunizing the highest-betweenness links or nodes. We find that the network's vulnerability can be significantly reduced, demonstrating this on three different real networks: the global flight network, a school friendship network, and the internet. In all cases, we find that not only is the average infection probability significantly suppressed, but also for the most relevant case of a small and limited number of immunization units the infection probability can be reduced by up to 55%.
I. INTRODUCTION
The paper develops an improved immunization strategy because limited resources require more effective suppression of spreading than existing targeted methods provide. Across real networks, the strategy reduces infection risk by optimizing how immunization units fragment the network.
- Motivation: Limited vaccines, screening, and manpower make selecting the most effective network immunization strategy essential.The paper motivates this need with global disease transmission through interconnected transportation and contact networks.
- Contribution: The improved strategy starts from adaptive high-betweenness immunization but reduces average infection risk by more than 10% across three real networks.The networks are the airport, school friendship, and internet networks.
- Contribution: More than 40% greater effectiveness than recalculated high-betweenness immunization is observed in the global airport SIR comparison with the same number of immunized links.The comparison uses random, high-betweenness, and improved link immunization strategies.
- Illustrative result: At q = 0.09 immunized flights, average infection probability is 84% for random, 34% for betweenness, and 20% for the improved strategy.The improved method is associated with more efficient decoupling of regions in the global airline network.
- Mechanism: The improved strategy identifies regional separations more efficiently, including decoupling East Asia and Europe from other regions at specific immunized-edge counts.The reported transitions occur at 2336 versus 2503 edges and 3169 versus 3192 edges for the compared strategies.
II. METHODS
The method evaluates immunization sequences by their effect on susceptible network size and iteratively improves a high-betweenness starting sequence. Random swaps are retained when they do not increase this measure, with additional population-based optimization.
- Performance measure: Susceptible size R is the sum of the sizes of the largest connected clusters remaining after immunizing q nodes or edges.The measure evaluates network response throughout the immunization process rather than only at disconnection.
- Initial strategy: The initial sequence uses adaptive high-betweenness centrality, repeatedly immunizing the node or edge that contributes most to shortest paths.Betweenness is calculated from shortest paths between all node pairs in the remaining network.
- Sequence optimization: The optimization randomly swaps two immunization positions and accepts the change when the susceptible size does not increase.Repeating these swaps produces an improved immunization sequence.
- Population optimization: A population-based variant starts with 1000 identical high-betweenness sequences and retains replacements that lower the population’s highest susceptible size.The final sequence is the one with the lowest susceptible size.
III. RESULTS: REAL NETWORKS
The improved immunization strategy was evaluated on airline, school friendship, and Internet networks for both link and node immunization. Across these real networks, it reduced infection probability most effectively when only a small fraction of links or nodes could be immunized.
- Real-network evaluation: The study evaluates airline, school friendship, and Internet networks, including both link and node immunization.The simulations use SIR dynamics and compare the improved strategy with high-betweenness immunization.
- Link immunization: For small numbers of immunized flights, the improved strategy is significantly more efficient than high-betweenness immunization.For the airline network, average improvement is about 15% below 20% immunized flights, with a maximum of about 55% at q ≈11.9%.
- Node immunization: The improved strategy also outperforms high-betweenness immunization for node immunization across all three real networks.Average improvements below the stated immunization fractions are about 11% for airline, 8% for school, and 12% for Internet networks.
IV. RESULTS: MODEL NETWORKS
Model-network experiments show that the improved strategy reduces susceptible network size more effectively than high-betweenness immunization. The advantage is especially pronounced when only a small number of immunization doses is available and increases with network size.
- Model-network comparisons: In Erdős-Rényi networks, the improved strategy reduces susceptible size R by about 30% overall compared with high-betweenness immunization.For small numbers of available immunization doses, the largest potentially infected component is reduced by up to a factor of 5.
- Model-network comparisons: For small immunized fractions, improved link immunization is more efficient than betweenness-based immunization across all studied infection parameters α.Figure 4 compares infection probability p against immunized fraction q for α = 0.1, 0.025, and 0.0125.
- Model-network comparisons: Node immunization comparisons across airline, school, and Internet networks show substantially greater efficiency for the improved strategy at small immunized fractions.The corresponding average savings are 18%, 7.2%, and 9.6%, respectively.
V. CONCLUSIONS
The paper introduces an immunization strategy based on susceptible size and finds that it consistently outperforms high-betweenness immunization for both nodes and links. Across the studied networks, the strategy reduces infection measures and can save immunization units, with greater efficiency in larger systems.
- Conclusions: The improved strategy outperforms high-betweenness immunization for both node and link immunization across all studied networks.The conclusion is supported by infection probability, susceptible size, and saved-immunization-unit measures.
- Conclusions: The reported improvement could result in saving human lives and resources.This is presented as a possible consequence of reducing disease spread with fewer or better-targeted immunization units.
- Conclusions: The strategy can significantly reduce disease spread in the global airline network with relatively small effort.The paper identifies the airline network as especially relevant to global disease spreading.
- Conclusions: The efficiency of the strategy increases with system size.This trend is reported across the studied network systems.