Decentralised adaptive-gain control for the SIS network epidemic model

This paper is concerned with the deterministic Susceptible-Infected-Susceptible (SIS) network for epidemic spreading, where each node of the network represents a population and directed edges represent transmission pathways for the disease to spread between populations. Motivated by the use of Non-Pharmaceutical Interventions (NPIs), e.g., physical distancing or mobility restrictions, to address outbreaks of novel infectious diseases, we propose a class of adaptive-gain controllers which are decentralised, being implemented independently at each node. The gains dynamically decrease over time and directly reduce the infection rates of the network at the node-level, representing an increase in NPIs for each population. We prove that for any SIS network, the adaptive-gain controllers asymptotically drive the network state to the disease-free state from any initial condition, and every gain is positive in the limit. We obtain upper bounds on the limiting gain values and the final reproduction number of the network, and conclude by proposing future research directions.