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We consider the fractal Sierpi\'{n}ski gasket or carpet graph in dimension$d\geq 2,$ denoted by $G$. At time $0$, we place a Poisson point process ofparticles onto the graph and let them perform independent simple random walks,which in this setting exhibit sub-diffusive behaviour. We generalise theconcept of particle process dependent Lipschitz percolation to the (coarsegraining of the) space-time graph $G\times \mathbb{R}$, where the opened/closedstate of space-time cells is measurable with respect to the particle processinside the cell. We then provide an application of this generalised frameworkand prove the following: if particles can spread an infection when they share asite of $G$, and if they recover independently at some rate $\gamma>0$, then if$\gamma$ is sufficiently small, the infection started with a single infectedparticle survives indefinitely with positive probability.

Lipschitz cutset for fractal graphs and applications to the spread of infections