Epidemic oscillations induced by social network control: the discontinuous case

The principles of control theory assert that feedback often leads to unstable behaviour and oscillatory dynamics. Examples include engineering systems such as thermostats and steering devices, as well as natural and biological systems. Feedbacks can undermine homeostatic and metabolic stability in the human body, for example, in glycemic control and diabetes. In yet another setting, feedback-induced oscillations have emerged as governments have tried to control the spread of the covid-19 pandemic crisis with containment measures such as social distancing, lock downs and quarantine.

Recent research has demonstrated that compartimentalised epidemic models display oscillations in the presence of feedbacks stimulated by the infection rate and infection states – that is, if individuals tend to modify their contact rates with others, and therefore the chance of infection spreading, based on what they observe about the current level of infection in the population. In a new paper inspired by this recent work, LML External Fellow Fabio Caccioli and Daniele De Martino of the Biofisika Institute and Ikerbasque Basque Foundation for Science now study the SIS and SIR models in a full microscopic settings on random networks in presence of a feedback that changes the structure of the underlying social network. They show that such feedback do trigger self-oscillations.

These results, they note, support a recent general theory of feedback-induced oscillations in large systems of interacting elements, where certain connectivity properties play the role of the control parameter. In order to mimic the occurrence of lock downs, Caccioli and De Martino focus on a simple discontinuous feedback control, where a certain fraction of links is deleted if the fraction of infections exceeds a given threshold 𝐼1. The same links are then be reinstated once the fraction of infections has been reduced below a second threshold value 𝐼1 < 𝐼2. This class of models is described by time-independent equations and parameters, yet gives rise to emergent self-oscillations.

The paper is available at https://arxiv.org/abs/2012.02552

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