  This paper considers the classical Susceptible--Infected--Susceptible (SIS)
network epidemic model, which describes a disease spreading through $n$ nodes,
with the network links governing the possible transmission pathways of the
disease between nodes. We consider feedback control to eliminate the disease in
scenarios where the disease would otherwise persist in an uncontrolled network.
We propose a family of decentralised adaptive-gain control algorithms, in which
each node has a control gain that adaptively evolves according to a
differential equation, independent of the gains of other nodes. The adaptive
gain is applied multiplicatively to either decrease the infection rate or
increase the recovery rate. To begin, we assume all nodes are controlled, and
prove that both infection rate control and recovery rate control algorithms
eliminate the disease with the limiting gains being positive and finite. Then,
we consider the possibility of controlling a subset of the nodes, for both the
infection rate control and recovery rate control. We first identify a necessary
and sufficient condition for the existence of a subset of nodes, which if
controlled would result in the elimination of the disease. For a given network,
there may exist several such viable subsets, and we propose an iterative
algorithm to identify such a subset. Simulations are provided to demonstrate
the effectiveness of the various proposed controllers.
