Scalable Distributed Reproduction Numbers of Network Epidemics With Differential Privacy

Reproduction numbers are widely used to analyze epidemic spreading processes over networks. However,conventional reproduction numbers of an overall network, which require spreading information from the entire network, do not indicate where an epidemic is spreading. To address this limitation, we first propose a novel class of local distributed reproduction numbers that capture spreading behaviors at the level of individual nodes. We demonstrate how to compute these values in a distributed way and use them to derive new threshold conditions for network spreading analysis. Due to the fact that epidemic data are often collected at multiple geographic or administrative scales, we then define a class of cluster distributed reproduction numbers to describe the spread between groups of nodes such as communities, cities, or states. We further show that the local distributed reproduction numbers can be aggregated to form the cluster distributed reproduction numbers. Unlike conventional network-level reproduction numbers, these distributed measures reveal fine-grained interaction patterns that may raise privacy concerns by exposing the frequency or intensity of interactions across regions. To address this issue, we propose a privacy-enhanced distributed reproduction number framework that implements differential privacy. This framework enables scalable and privacy-preserving analysis of epidemic spreading processes in networked populations through the calculation of privacy-preserving distributed reproduction numbers. Numerical experiments show that while maintaining differential privacy, the private distributed reproduction numbers yield accurate estimates of epidemic spread while also offering more insights than conventional reproduction numbers.