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Epidemics on evolving networks with varying degrees

Epidemics on complex networks is a widely investigated topic in the last few years, mainly due to the last pandemic events. Usually, real contact networks are dynamic, hence much effort has been invested in studying epidemics on evolving networks. Here we propose and study a model for evolving networks based on varying degrees, where at each time step a node might get, with probability $r$, a new degree and new neighbors according to a given degree distribution, instead of its former neighbors. We find analytically, using the generating functions framework, the epidemic threshold and the probability for a macroscopic spread of disease depending on the rewiring rate $r$. Our analytical results are supported by numerical simulations. We find surprisingly that the impact of the rewiring rate $r$ has qualitative different trends for networks having different degree distributions. That is, in some structures, such as random regular networks the dynamics enhances the epidemic spreading while in others such as scale free the dynamics reduces the spreading. In addition, for scale-free networks, we reveal that fast dynamics of the network, $r=1$, changes the epidemic threshold to nonzero rather than zero found for $r<1$, which is similar to the known case of $r=0$, i.e., a static network. Finally, we find the epidemic threshold also for a general distribution of the recovery time.