On Control of Epidemics with Application to COVID-19

At the time of writing, the ongoing COVID-19 pandemic, caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), had already resulted in more than thirty-two million cases infected and more than one million deaths worldwide. Given the fact that the pandemic is still threatening health and safety, it is in the urgency to understand the COVID-19 contagion process and know how it might be controlled. With this motivation in mind, in this paper, we consider a version of a stochastic discrete-time Susceptible-Infected-Recovered-Death~(SIRD)-based epidemiological model with two uncertainties: The uncertain rate of infected cases which are undetected or asymptomatic, and the uncertain effectiveness rate of control. Our aim is to study the effect of an epidemic control policy on the uncertain model in a control-theoretic framework. We begin by providing the closed-form solutions of states in the modified SIRD-based model such as infected cases, susceptible cases, recovered cases, and deceased cases. Then, the corresponding expected states and the technical lower and upper bounds for those states are provided as well. Subsequently, we consider two epidemic control problems to be addressed: One is almost sure epidemic control problem and the other average epidemic control problem. Having defined the two problems, our main results are a set of sufficient conditions on a class of linear control policy which assures that the epidemic is "well-controlled"; i.e., both of the infected cases and deceased cases are upper bounded uniformly and the number of infected cases converges to zero asymptotically. Our numerical studies, using the historical COVID-19 contagion data in the United States, suggest that our appealingly simple model and control framework can provide a reasonable epidemic control performance compared to the ongoing pandemic situation.