This paper argues for the regular testing of people in groups that are more likely to be exposed to SARS-CoV-2, to reduce the spread of COVID-19 and resume economic activity. We call this 'stratified periodic testing'. It is 'stratified' as it is based on at-risk groups, and 'periodic' as everyone in the group is tested at regular intervals. We argue that this is a better use of scarce testing resources than 'universal random testing', as has been recently discussed globally. We find that, under reasonable assumptions and allowing for false negative results 30 per cent of the time, 17 per cent of a subgroup would need to be tested each day to lower the effective reproduction number R from 2.5 to 0.75, under stratified periodic testing. Using the same assumptions the universal random testing rate would need to be 27 per cent (as opposed to 7 per cent as argued by Romer (2020b)). We obtain this rate of testing using a corrected method for calculating the impact of an infectious person on others, and allowing for asymptomatic cases. We also find that the effect of one day's delay between testing positive and self-isolating is similar to having a test that is 30 per cent less accurate.