  We study the properties of the potential overlap between two networks $A,B$
sharing the same set of $N$ nodes (a two-layer network) whose respective degree
distributions $p_A(k), p_B(k)$ are given. Defining the overlap coefficient
$\alpha$ as the Jaccard index, we derive upper bounds for the minimum and
maximum overlap coefficient in terms of $p_A(k)$, $p_B(k)$ and $N$. We also
present an algorithm based on cross-rewiring of links to obtain a two-layer
network with any prescribed $\alpha$ inside the permitted range. Finally, to
illustrate the importance of the overlap for the dynamics of interacting
contagious processes, we derive a mean-field model for the spread of an SIS
epidemic with awareness against infection over a two-layer network, containing
$\alpha$ as a parameter. A simple analytical relationship between $\alpha$ and
the basic reproduction number follows. Stochastic simulations are presented to
assess the accuracy of the upper bounds of $\alpha$ and the predictions of the
mean-field epidemic model.
