  We study the stability properties of a susceptible-infected-susceptible (SIS)
diffusion model, so-called the $n$-intertwined Markov model, over arbitrary
directed network topologies. As in the majority of the work on infection spread
dynamics, this model exhibits a threshold phenomenon. When the curing rates in
the network are high, the disease-free state is the unique equilibrium over the
network. Otherwise, an endemic equilibrium state emerges, where some infection
remains within the network. Using notions from positive systems theory, {we
provide novel proofs for the global asymptotic stability of the equilibrium
points in both cases over strongly connected networks based on the value of the
basic reproduction number, a fundamental quantity in the study of epidemics.}
When the network topology is weakly connected, we provide conditions for the
existence, uniqueness, and global asymptotic stability of an endemic state, and
we study the stability of the disease-free state. Finally, we demonstrate that
the $n$-intertwined Markov model can be viewed as a best-response dynamical
system of a concave game among the nodes. This characterization allows us to
cast new infection spread dynamics; additionally, we provide a sufficient
condition for the global convergence to the disease-free state, which can be
checked in a distributed fashion. Several simulations demonstrate our results.
