Revisiting SIR in the Age of COVID-19: Explicit Solutions and Control Problems

Abstract

The nonpopulation conserving SIR (SIR-NC) model to describe the spread of infections in a community is studied. Unlike the standard SIR model, this does not assume population conservation. Although similar in form to the standard SIR, SIR-NC admits a closed form solution while allowing us to model mortality and also provides a different, and arguably a more realistic, interpretation of model parameters. Numerical comparisons of this SIR-NC model with the standard, population conserving, SIR model are provided. Extensions to include imported infections, interacting communities, and models that include births and deaths are presented and analyzed. Several numerical examples are also presented to illustrate these models. A discrete time control problem for the SIR-NC epidemic model is presented in which the cost function depends on variables that correspond to the levels of lockdown, the level of testing and quarantine, and the number of infections. We include a switching cost for moving between lockdown levels. Numerical experiments are presented.