Claudius Gros, Roser Valenti, Kilian Valenti, Daniel Gros
In response to the rapid spread of the Coronavirus (COVID-19), with ten thousands of deaths and intensive-care hospitalizations, a large number of regions and countries have been put under lockdown by their respective governments. Policy makers are confronted in this situation with the problem of balancing public health considerations, with the economic costs of a persistent lockdown. We introduce a modified epidemic model, the controlled-SIR model, in which the disease reproduction rates evolve dynamically in response to political and societal reactions. Social distancing measures are triggered by the number of infections, providing a dynamic feedback-loop which slows the spread of the virus. We estimate the total cost of several distinct containment policies incurring over the entire path of the endemic. Costs comprise direct medical cost for intensive care, the economic cost of social distancing, as well as the economic value of lives saved. Under plausible parameters, the total costs are highest at a medium level of reactivity when value of life costs are omitted. Very strict measures fare best, with a hands-off policy coming second. Our key findings are independent of the specific parameter estimates, which are to be adjusted with the COVID-19 research status. In addition to numerical simulations, an explicit analytical solution for the controlled continuous-time SIR model is presented. For an uncontrolled outbreak and a reproduction factor of three, an additional 28% of the population is infected beyond the herd immunity point, reached at an infection level of 66%, which adds up to a total of 94%.