Effective epidemic containment strategy in hypergraphs

Bukyoung Jhun

doi:10.1103/physrevresearch.3.033282

Abstract

Recently, hypergraphs have attracted considerable interest from the research community as a generalization of networks capable of encoding higher-order interactions, which commonly appear in both natural and social systems. Epidemic dynamics in hypergraphs has been studied by using the simplicial susceptible-infected-susceptible ($s$-SIS) model; however, the efficient immunization strategy for epidemics in hypergraphs is not studied despite the importance of the topic in mathematical epidemiology. Here, we propose an immunization strategy that immunizes hyperedges with high simultaneous infection probability (SIP). This strategy can be implemented in general hypergraphs. We also generalize the edge epidemic importance (EI)-based immunization strategy, which is the state of the art in complex networks. However, it does not perform as well as the SIP-based method in hypergraphs despite its high computational cost. We also show that immunizing hyperedges with high H-eigenscore effectively contains the epidemics in uniform hypergraphs. A high SIP of a hyperedge suggests that the hyperedge is a ``hotspot'' of the epidemic process. Therefore, SIP can be used as a centrality measure to quantify a hyperedge's influence on higher-order dynamics in general hypergraphs. The effectiveness of the immunization strategies suggests the necessity of scientific, data-driven, systematic policy-making for epidemic containment.

Highlights

In the past two decades, extensive research has been devoted to spreading processes in complex networks [1,2,3,4,5] to model the spread of epidemic diseases [6] and innovations [7,8], opinion formation [9,10,11,12], and many other physical and social phenomena [13,14,15,16]
We show that immunizing hyperedges with the highest H-eigenscores, which is defined as the product of the elements of the H-eigenvector of the adjacency tensor with the largest H-eigenvalue of all the nodes in the hyperedge, effectively achieves herd immunity in uniform hypergraphs
III, we extend the individualbased mean-field (IBMF) and pair-based mean-field (PBMF) theories to general hypergraphs, which is required for the immunization strategies

Summary

INTRODUCTION

In the past two decades, extensive research has been devoted to spreading processes in complex networks [1,2,3,4,5] to model the spread of epidemic diseases [6] and innovations [7,8], opinion formation [9,10,11,12], and many other physical and social phenomena [13,14,15,16]. We propose an immunization strategy that targets hyperedges with high simultaneous infection probability (SIP), which is the probability that all the nodes in a hyperedge are in the infected state This probability is calculated by the individual-based mean-field (IBMF) theory [54,55]. We show that immunizing hyperedges with the highest H-eigenscores, which is defined as the product of the elements of the H-eigenvector of the adjacency tensor with the largest H-eigenvalue of all the nodes in the hyperedge, effectively achieves herd immunity in uniform hypergraphs This method generalizes the edge eigenscore in a complex network and can be implemented to contain epidemics in uniform hypergraphs.

Epidemic dynamics in networks and hypergraphs

Hypergraph static model

INDIVIDUAL- AND PAIR-BASED MEAN-FIELD THEORIES

IMMUNIZATION STRATEGIES

NUMERICAL RESULTS

CONCLUSION