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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 thedisease between nodes. We consider feedback control to eliminate the disease inscenarios where the disease would otherwise persist in an uncontrolled network.We propose a family of decentralised adaptive-gain control algorithms, in whicheach node has a control gain that adaptively evolves according to adifferential equation, independent of the gains of other nodes. The adaptivegain is applied multiplicatively to either decrease the infection rate orincrease the recovery rate. To begin, we assume all nodes are controlled, andprove that both infection rate control and recovery rate control algorithmseliminate the disease with the limiting gains being positive and finite. Then,we consider the possibility of controlling a subset of the nodes, for both theinfection rate control and recovery rate control. We first identify a necessaryand sufficient condition for the existence of a subset of nodes, which ifcontrolled would result in the elimination of the disease. For a given network,there may exist several such viable subsets, and we propose an iterativealgorithm to identify such a subset. Simulations are provided to demonstratethe effectiveness of the various proposed controllers.

Decentralised adaptive-gain control for eliminating epidemic spreading on networks