Survival time of the susceptible-infected-susceptible infection process on a graph

Abstract

The survival time T is the longest time that a virus, a meme, or a failure can propagate in a network. Using the hitting time of the absorbing state in an uniformized embedded Markov chain of the continuous-time susceptible-infected-susceptible (SIS) Markov process, we derive an exact expression for the average survival time E[T ] of a virus in the complete graph K N and the star graph K 1,N−1 . By using the survival time, instead of the average fraction of infected nodes, we propose a new method to approximate the SIS epidemic threshold τ c that, at least for K N and K 1,N−1 , correctly scales with the number of nodes N and that is superior to the epidemic threshold τ . However, when the average fraction of infected nodes is used as a basis for comparison, the virus will survive in the star graph longer than in any other graph, making the star graph the worst-case graph instead of the complete graph. Finally, in non-Markovian SIS, the distribution of the spreading attempts over the infectious period of a node influences the survival time, even if the expected number of spreading attempts during an infectious period (the non-Markovian equivalent of the effective infection rate) is kept constant. Both early and late infection attempts lead to shorter survival times. Interestingly, just as in Markovian SIS, the survival times appear to be exponentially distributed, regardless of the infection and curing time distributions.

Epidemic extinction paths in complex networks

“…Since the sums in Eqs. (27)(28) are independent of i, the limit of largeβλη i N η , gives: o i / o j ∼ η j /η i and µ in i /µ in j ∼ 1. The latter can be seen in Fig.(9).…”

Accuracy criterion for the mean-field approximation in susceptible-infected-susceptible epidemics on networks

“…Moreover, a study [33] on the average extinction (or virus survival) time seems to hint that for Markovian SIS epidemics, K N and K 1,N−1 are two extremes among all graphs.…”

Rare events in networks with internal and external noise

“…We point out that the known pair-approximation equations for k-regular networks are found by setting the momenta equal to zero in Eqs. (16)- (17), p = m = 0 [28]. In fact, this is a general feature of our approach: network mean-field theories are zero-momentum invariant manifolds of Hamilton's equations of motion [24].…”