Detailed analytic study of the compact pairwise model for SIS epidemic propagation on networks

TL;DR

This paper provides a detailed mathematical analysis of a simplified model for SIS epidemic spread on networks, identifying bifurcation points and stability conditions for disease-free and endemic states.

Contribution

It introduces a reduced two-equation system for the compact pairwise model and rigorously analyzes its bifurcation and stability properties.

Findings

Transcritical bifurcation at critical infection rate 

Global stability of disease-free state under certain conditions

Existence of a stable endemic equilibrium for supercritical 

Abstract

The global behaviour of the compact pairwise approximation of SIS epidemic propagation on networks is studied. It is shown that the system can be reduced to two equations enabling us to carry out a detailed study of the dynamic properties of the solutions. It is proved that transcritical bifurcation occurs in the system at $\tau = \tau _c = \frac{\gamma n}{\langle n^{2}\rangle-n}$, where $\tau$ and $\gamma$ are infection and recovery rates, respectively, $n$ is the average degree of the network and $\langle n^{2}\rangle$ is the second moment of the degree distribution. For subcritical values of $\tau$ the disease-free steady state is stable, while for supercritical values a unique stable endemic equilibrium appears. We also prove that for subcritical values of $\tau$ the disease-free steady state is globally stable under certain assumptions on the graph that cover a wide class of networks.