Long-Term Regulation of Prolonged Epidemic Outbreaks in Large Populations via Adaptive Control: A Singular Perturbation Approach

In order to control highly-contagious and prolonged outbreaks, public health authorities intervene to institute social distancing, lock-down policies, and other Non-Pharmaceutical Interventions (NPIs). Given the high social, educational, psychological, and economic costs of NPIs, authorities tune them, alternatively tightening up or relaxing rules, with the result that, in effect, a relatively flat infection rate results. For example, during the summer of 2020 in parts of the United States, daily COVID-19 infection numbers dropped to a plateau. This letter approaches NPI tuning as a control-theoretic problem, starting from a simple dynamic model for social distancing based on the classical SIR epidemics model. Using a singular-perturbation approach, the plateau becomes a Quasi-Steady-State (QSS) of a reduced two-dimensional SIR model regulated by adaptive dynamic feedback. It is shown that the QSS can be assigned and it is globally asymptotically stable. Interestingly, the dynamic model for social distancing can be interpreted as a nonlinear integral controller. Problems of data fitting and parameter identifiability are also studied for this model. This letter also discusses how this simple model allows for a meaningful study of the effect of population size, vaccinations, and the emergence of second waves.