We explore strategies of contact tracing, case isolation and quarantine of exposed contacts to control the SARS-CoV-2 epidemic using a branching process model with household structure. This structure reflects higher transmission risks among household members than among non-household members. We explore strategic implementation choices that make use of household structure, and investigate strategies including two-step tracing, backwards tracing, smartphone tracing and tracing upon symptom report rather than test results. The primary model outcome is the effect of contact tracing, in combination with different levels of physical distancing, on the growth rate of the epidemic. Furthermore, we investigate epidemic extinction times to indicate the time period over which interventions must be sustained. We consider effects of non-uptake of isolation/quarantine, non-adherence, and declining recall of contacts over time. Our results find that, compared to self-isolation of cases without contact tracing, a contact tracing strategy designed to take advantage of household structure allows for some relaxation of physical distancing measures but cannot completely control the epidemic absent of other measures. Even assuming no imported cases and sustainment of moderate physical distancing, testing and tracing efforts, the time to bring the epidemic to extinction could be in the order of months to years. This article is part of the theme issue 'Modelling that shaped the early COVID-19 pandemic response in the UK'.