The English SARS-CoV-2 epidemic has been affected by the emergence of new viral variants such as B.1.177, Alpha and Delta, and changing restrictions. We used statistical models and the agent-based model Covasim, in June 2021, to estimate B.1.177 to be 20% more transmissible than the wild type, Alpha to be 50-80% more transmissible than B.1.177 and Delta to be 65-90% more transmissible than Alpha. Using these estimates in Covasim (calibrated 1 September 2020 to 20 June 2021), in June 2021, we found that due to the high transmissibility of Delta, resurgence in infections driven by the Delta variant would not be prevented, but would be strongly reduced by delaying the relaxation of restrictions by one month and with continued vaccination. This article is part of the theme issue 'Technical challenges of modelling real-life epidemics and examples of overcoming these'.