Metastability of the contact process on slowly evolving scale-free networks

We investigate the contact process on scale-free networks evolving by a stationary dynamics whereby each vertex independently updates its connections with a rate depending on its power. This rate can be slowed down or speeded up by virtue of decreasing or increasing a parameter $\eta$, with $\eta\downarrow-\infty$ approaching the static and $\eta\uparrow\infty$ the mean-field case. We identify the regimes of slow, fast and ultra-fast extinction of the contact process. Slow extinction occurs in the form of metastability, when the contact process maintains a certain density of infected states for a time exponential in the network size. In our main result we identify the metastability exponents, which describe the decay of metastable densities as the infection rate goes to zero, in dependence on $\eta$ and the power-law exponent $\tau$. While the fast evolution cases have been treated in a companion paper, Jacob, Linker, M\"orters (2019), the present paper looks at the significantly more difficult cases of slow network evolution. We describe various effects, like degradation, regeneration and depletion, which lead to a rich picture featuring numerous first-order phase transitions for the metastable exponents. To capture these effects in our upper bounds we develop a new martingale based proof technique combining a local and global analysis of the process.