SIR-PID: A Proportional–Integral–Derivative Controller for COVID-19 Outbreak Containment

Aldo Ianni,Nicola Rossi

doi:10.3390/physics3030031

Aldo Ianni, Nicola Rossi

Open Access

https://doi.org/10.3390/physics3030031

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Abstract

Ongoing social restrictions, including social distancing and lockdown, adopted by many countries to inhibit spread of the the COVID-19 epidemic, must attempt to find a trade-off between induced economic damage, healthcare system collapse, and the costs in terms of human lives. Applying and removing restrictions on a system with a given latency as represented by an epidemic outbreak (and formally comparable with mechanical inertia), may create critical instabilities, overshoots, and strong oscillations in the number of infected people around the desirable set-point, defined in a practical way as the maximum number of hospitalizations acceptable by a given healthcare system. A good understanding of the system reaction to any change of the input control variable can be reasonably achieved using a proportional–integral–derivative controller (PID), which is a widely used technique in various physics and technological applications. In this paper, this control theory to is proposed to be applied epidemiology, to understand the reaction of COVID-19 propagation to social restrictions and to reduce epidemic damages through the correct tuning of the containment policy. Regarding the synthesis of this interdisciplinary approach, the extended to the susceptible–infectious–recovered (SIR) model name “SIR-PID” is suggested.

Highlights

The diffusion of the coronavirus disease (COVID-19) outbreak, caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [1,2], is described, as for typical epidemics, by the so-called basic reproduction number, usually denoted R0 [3] and defined as the expected number of new infections from a single new case in a population where all subjects are susceptible.A sizeable fraction of people infected by COVID-19, especially those who are older and with underlying preexisting medical problems, are likely to develop a serious illness with a fatality rate well above the typical seasonal influenza [4]
To model the behaviour of the COVID-19 epidemic in the presence of social restrictions aiming at changing the basic reproduction number R0, let us consider a basic SIR compartmental model [25,26]
Values in between represent different gradations of the social distancing ruled by official orders set every 14 days, the typical time scale of the COVID-19 epidemic latency