Abstract
Testing remains a key tool for managing health care and making health policy during the coronavirus pandemic, and it will probably be important in future pandemics. Because of false negative and false positive tests, the observed fraction of positive tests-the surface positivity-is generally different from the fraction of infected individuals (the incidence rate of the disease). In this paper a previous method for translating surface positivity to a point estimate for incidence rate, then to an appropriate range of values for the incidence rate consistent with the model and data (the test range), and finally to the risk (the probability of including one infected individual) associated with groups of different sizes is illustrated. The method is then extended to include asymptomatic infections. To do so, the process of testing is modeled using both analysis and Monte Carlo simulation. Doing so shows that it is possible to determine point estimates for the fraction of infected and symptomatic individuals, the fraction of uninfected and symptomatic individuals, and the ratio of infected asymptomatic individuals to infected symptomatic individuals. Inclusion of symptom status generalizes the test range from an interval to a region in the plane determined by the incidence rate and the ratio of asymptomatic to symptomatic infections; likelihood methods can be used to determine the contour of the rest region. Points on this contour can be used to compute the risk (defined as the probability of including one asymptomatic infected individual) in groups of different sizes. These results have operational implications that include: positivity rate is not incidence rate; symptom status at testing can provide valuable information about asymptomatic infections; collecting information on time since putative virus exposure at testing is valuable for determining point estimates and test ranges; risk is a graded (rather than binary) function of group size; and because the information provided by testing becomes more accurate with more tests but at a decreasing rate, it is possible to over-test fixed spatial regions. The paper concludes with limitations of the method and directions for future work.