  The contact process is a simple model for the spread of an infection in a
structured population. We investigate the case when the underlying structure
evolves dynamically as a degree-dependent dynamical percolation model. Starting
with a connected locally finite base graph we initially declare edges
independently open with a probability that is allowed to depend on the degree
of the adjacent vertices and closed otherwise. Edges are independently updated
with a rate depending on the degrees and then are again declared open and
closed with the same probabilities. We are interested in the contact process,
where infections are only allowed to spread via open edges. Our aim is to
analyze the impact of the update speed and the probability for edges to be open
on the existence of a phase transition. For a general connected locally finite
graph, our first result gives sufficient conditions for the critical value for
survival to be strictly positive. Furthermore, in the setting of
Bienaym\'e-Galton-Watson trees, we show that the process survives strongly with
positive probability for any infection rate if the offspring distribution has a
stretched exponential tail with an exponent depending on the percolation
probability and the update speed. In particular, if the offspring distribution
follows a power law and the connection probability is given by a product kernel
and the update speed exhibits polynomial behaviour, we provide a complete
characterisation of the phase transition.
