Network epidemics is a ubiquitous model that can represent different
phenomena and finds applications in various domains. Among its various
characteristics, a fundamental question concerns the time when an epidemic
stops propagating. We investigate this characteristic on a SIS epidemic induced
by agents that move according to independent continuous time random walks on a
finite graph: Agents can either be infected (I) or susceptible (S), and
infection occurs when two agents with different epidemic states meet in a node.
After a random recovery time, an infected agent returns to state S and can be
infected again. The End of Epidemic (EoE) denotes the first time where all
agents are in state S, since after this moment no further infections can occur
and the epidemic stops.
For the case of two agents on edge-transitive graphs, we characterize EoE as
a function of the network structure by relating the Laplace transform of EoE to
the Laplace transform of the meeting time of two random walks. Interestingly,
this analysis shows a separation between the effect of network structure and
epidemic dynamics. We then study the asymptotic behavior of EoE (asymptotically
in the size of the graph) under different parameter scalings, identifying
regimes where EoE converges in distribution to a proper random variable or to
infinity. We also highlight the impact of different graph structures on EoE,
characterizing it under complete graphs, complete bipartite graphs, and rings
