  In this work we review a class of deterministic nonlinear models for the
propagation of infectious diseases over contact networks with
strongly-connected topologies. We consider network models for
susceptible-infected (SI), susceptible-infected-susceptible (SIS), and
susceptible-infected-recovered (SIR) settings. In each setting, we provide a
comprehensive nonlinear analysis of equilibria, stability properties,
convergence, monotonicity, positivity, and threshold conditions. For the
network SI setting, specific contributions include establishing its equilibria,
stability, and positivity properties. For the network SIS setting, we review a
well-known deterministic model, provide novel results on the computation and
characterization of the endemic state (when the system is above the epidemic
threshold), and present alternative proofs for some of its properties. Finally,
for the network SIR setting, we propose novel results for transient behavior,
threshold conditions, stability properties, and asymptotic convergence. These
results are analogous to those well-known for the scalar case. In addition, we
provide a novel iterative algorithm to compute the asymptotic state of the
network SIR system.
