Network Inoculation: Heteroclinics and phase transitions in an epidemic model

TL;DR

This paper analyzes a complex epidemic model on adaptive, heterogeneous networks, revealing phase transitions caused by non-local bifurcations that lead to a network inoculation effect, preventing disease reinvasion.

Contribution

It provides a detailed mathematical analysis of phase transitions in an epidemic model with adaptive network rewiring, highlighting the role of non-local bifurcations in network inoculation.

Findings

Identification of two phase transitions in the model

Discovery of a non-local heteroclinic bifurcation causing network inoculation

Demonstration that exposure can lead to disease collapse without reinvasion

Abstract

In epidemiological modelling, dynamics on networks, and in particular adaptive and heterogeneous networks have recently received much interest. Here we present a detailed analysis of a previously proposed model that combines heterogeneity in the individuals with adaptive rewiring of the network structure in response to a disease. We show that in this model qualitative changes in the dynamics occur in two phase transitions. In a macroscopic description one of these corresponds to a local bifurcation whereas the other one corresponds to a non-local heteroclinic bifurcation. This model thus provides a rare example of a system where a phase transition is caused by a non-local bifurcation, while both micro- and macro-level dynamics are accessible to mathematical analysis. The bifurcation points mark the onset of a behaviour that we call network inoculation. In the respective parameter region exposure of the system to a pathogen will lead to an outbreak that collapses, but leaves the network in a configuration where the disease cannot reinvade, despite every agent returning to the susceptible class. We argue that this behaviour and the associated phase transitions can be expected to occur in a wide class of models of sufficient complexity.