Decay towards the overall-healthy state in SIS epidemics on networks

TL;DR

This paper analytically computes the decay rate of SIS epidemics on complete graphs using a new algebraic method for eigenvalue calculation, providing sharp bounds on epidemic lifetime related to network size and infection rate.

Contribution

It introduces a novel algebraic approach to precisely compute the second largest eigenvalue of a stochastic matrix, advancing understanding of epidemic decay on networks.

Findings

Maximum epidemic lifetime scales exponentially with network size.

Derived tight bounds for epidemic decay time on complete graphs.

Shows the decay rate depends on network topology and infection parameters.

Abstract

The decay rate of SIS epidemics on the complete graph $K_{N}$ is computed analytically, based on a new, algebraic method to compute the second largest eigenvalue of a stochastic three-diagonal matrix up to arbitrary precision. The latter problem has been addressed around 1950, mainly via the theory of orthogonal polynomials and probability theory. The accurate determination of the second largest eigenvalue, also called the \emph{decay parameter}, has been an outstanding problem appearing in general birth-death processes and random walks. Application of our general framework to SIS epidemics shows that the maximum average lifetime of an SIS epidemics in any network with $N$ nodes is not larger (but tight for $K_{N}$) than \[ E\left[ T\right] \sim\frac{1}{\delta}\frac{\frac{\tau}{\tau_{c}}\sqrt{2\pi}% }{\left( \frac{\tau}{\tau_{c}}-1\right) ^{2}}\frac{\exp\left( N\left\{ \log\frac{\tau}{\tau_{c}}+\frac{\tau_{c}}{\tau}-1\right\} \right) }{\sqrt {N}}=O\left( e^{N\ln\frac{\tau}{\tau_{c}}}\right) \] for large $N$ and for an effective infection rate $\tau=\frac{\beta}{\delta}$ above the epidemic threshold $\tau_{c}$. Our order estimate of $E\left[ T\right] $ sharpens the order estimate $E\left[ T\right] =O\left( e^{bN^{a}}\right) $ of Draief and Massouli\'{e} \cite{Draief_Massoulie}. Combining the lower bound results of Mountford \emph{et al.} \cite{Mountford2013} and our upper bound, we conclude that for almost all graphs, the average time to absorption for $\tau>\tau_{c}$ is $E\left[ T\right] =O\left( e^{c_{G}N}\right) $, where $c_{G}>0$ depends on the topological structure of the graph $G$ and $\tau$.