  The global behaviour of the compact pairwise approximation of SIS epidemic
propagation on networks is studied. It is shown that the system can be reduced
to two equations enabling us to carry out a detailed study of the dynamic
properties of the solutions. It is proved that transcritical bifurcation occurs
in the system at $\tau = \tau _c = \frac{\gamma n}{\langle n^{2}\rangle-n}$,
where $\tau$ and $\gamma$ are infection and recovery rates, respectively, $n$
is the average degree of the network and $\langle n^{2}\rangle$ is the second
moment of the degree distribution. For subcritical values of $\tau$ the
disease-free steady state is stable, while for supercritical values a unique
stable endemic equilibrium appears. We also prove that for subcritical values
of $\tau$ the disease-free steady state is globally stable under certain
assumptions on the graph that cover a wide class of networks.
