Epidemic oscillations induced by social network control: the discontinuous case

Since the early days of control theory,  engineers have understood that feedback can give rise to spontaneous and sustained oscillations. Examples abound in engineering systems such as thermostats and steering devices, as well as in natural and biological systems – for example, in homeostasis or diabetes in the human metabolism. Feedback-induced oscillations have also affected recent governmental efforts to control the covid-19 pandemic crisis through measures such as social distancing, lock downs and quarantine.
This effect is also evident in epidemic models. A recent study rigorously demonstrated that compartmentalized epidemic models display oscillations in the presence of feedback between infection rate and infection states; more generally, a feedback between order and control parameters in large interacting systems subject to phase transitions triggers self-oscillations, where an Andronov-Hopf bifurcation replaces the usual phase transition. In a recent paper, LML Fellow Fabio Caccioli and Daniele De Martino study two standard epidemiological models – the SIS and SIR models – on random networks in the presence of a feedback which changes the structure of the underlying social network. They demonstrate that this feedback triggers self-oscillations and does so in a scenario elucidated by De Martino in 2018, where suitably defined connectivity properties play the role of the control parameter. In order to mimic the occurrence of lock downs, the authors focus on a simple discontinuous feedback control, deleting a certain fraction of links if the fraction of infections exceeds a given threshold