The Behavior of Epidemics under Bounded Susceptibility

TL;DR

This paper explores how a cap on infection rates, reflecting limited attention or bandwidth, affects epidemic spread in different network structures, revealing surprising impacts on spreading and extinction times.

Contribution

It introduces the concept of bounded susceptibility into epidemic models and analyzes its effects on SI and SIS dynamics across different network types.

Findings

Bounded susceptibility does not affect star-like networks in SI dynamics.

It significantly alters spreading times in clique-like networks under SI dynamics.

In SIS dynamics, bounded susceptibility impacts extinction times in star-like networks.

Abstract

We investigate the sensitivity of epidemic behavior to a bounded susceptibility constraint -- susceptible nodes are infected by their neighbors via the regular SI/SIS dynamics, but subject to a cap on the infection rate. Such a constraint is motivated by modern social networks, wherein messages are broadcast to all neighbors, but attention spans are limited. Bounded susceptibility also arises in distributed computing applications with download bandwidth constraints, and in human epidemics under quarantine policies. Network epidemics have been extensively studied in literature; prior work characterizes the graph structures required to ensure fast spreading under the SI dynamics, and long lifetime under the SIS dynamics. In particular, these conditions turn out to be meaningful for two classes of networks of practical relevance -- dense, uniform (i.e., clique-like) graphs, and sparse, structured (i.e., star-like) graphs. We show that bounded susceptibility has a surprising impact on epidemic behavior in these graph families. For the SI dynamics, bounded susceptibility has no effect on star-like networks, but dramatically alters the spreading time in clique-like networks. In contrast, for the SIS dynamics, clique-like networks are unaffected, but star-like networks exhibit a sharp change in extinction times under bounded susceptibility. Our findings are useful for the design of disease-resistant networks and infrastructure networks. More generally, they show that results for existing epidemic models are sensitive to modeling assumptions in non-intuitive ways, and suggest caution in directly using these as guidelines for real systems.