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Consider the SIS model of epidemic spreading. There is a finite graph $G(V,E)$, link infection rates $\lambda_{ij}$ and node recovery rates $\mu_i$. There are a few initial nodes which are infected at $t=0$. The infection spreads around for some time before eventually disappearing.

Under a fixed topology, rates and initial conditions, is there a way to rank order nodes by the expected time they will be in infected state? Or is there a way to rank order nodes by their expected first hitting times? Is there any work that has dealt with finding relative vulnerabilities of different nodes in a SIS model?

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Just stumble across this question.

I think the following paper provide some insights to the answers you seek.

Van Mieghem, Piet, Jasmina Omic, and Robert Kooij. "Virus spread in networks." Networking, IEEE/ACM Transactions on 17.1 (2009): 1-14. (http://www.nas.its.tudelft.nl/people/Piet/papers/IEEEToN_virusspread.pdf)

From their $N$-intertwined approach, you can compute the instantaneous time evolution of the infected state per node as well as at their steady state. So, if you have as input the adjacency matrix of the topology and the set of initial infected nodes (seed nodes), then you can actually track the probability of each node being infected over time. With this, you can rank them as you need.

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