Empirical work described in this paper explains the daily evolution of the reproduction rate, R, and mobility for a large sample of countries, in terms of containment and public health policies. This is with a view to providing insight into the appropriate policy stance as countries prepare for a potentially protracted period characterised by new infection waves. While a comprehensive package of containment measures may be necessary when the virus is widespread and can have a large effect on reducing R, they also have effect on mobility and, by extension, economic activity. A wide-ranging package of public health policies – with an emphasis on comprehensive testing, tracing and isolation, but also including mask-wearing and policies directed at vulnerable groups, especially those in care homes – offer the best approach to avoiding a full lockdown while containing the spread of the virus. Such policies may, however, need to be complemented by selective containment measures (such as restricting large public events and international travel or localised lockdowns) both to contain local outbreaks and because implementing some of the recommended public health policies may be difficult to achieve or have unacceptable social costs.