Distributed feedback control on the SIS network model: An impossibility result

This paper considers the deterministic Susceptible-Infected-Susceptible (SIS) epidemic network model, over strongly connected networks. It is well known that there exists an endemic equilibrium (the disease persists in all nodes of the network) if and only if the effective reproduction number of the network is greater than 1. In fact, the endemic equilibrium is unique and is asymptotically stable for all feasible nonzero initial conditions. We consider the recovery rate of each node as a control input. Using results from differential topology and monotone systems, we establish that it is impossible for a large class of distributed feedback controllers to drive the network to the healthy equilibrium (where every node is disease free) if the uncontrolled network has a reproduction number greater than 1. In fact, a unique endemic equilibrium exists in the controlled network, and it is exponentially stable for all feasible nonzero initial conditions. We illustrate our impossibility result using simulations, and discuss the implications on the problem of control over epidemic networks.