The decay rate of SIS epidemics on the complete graph KNβ 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 KNβ) 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 Ο=δββ above the epidemic
threshold Οcβ. Our order estimate of E[T] sharpens the
order estimate E[T]=O(ebNa) 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 Ο>Οcβ
is E[T]=O(ecGβN), where cGβ>0 depends on
the topological structure of the graph G and Ο