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

The decay rate of SIS epidemics on the complete graph KN 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 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 KN ) than