The infectious disease spread between households and communities depends on two factors: first, the way in which the disease is transmitted between people; and second, the dynamics that connect people to each other.
In theory, a strong generalised lockdown successfully implemented early enough could close both social spaces, restricting the further spread of the epidemic to that within the households.
Where a generalised lockdown successfully closed the third-level social space, but left the second open, the spread would be affected by the number of communities with at least one household infected.
These measures are unlikely to eliminate spread, but will contain opportunities for super-spreading and transmissions between communities and households.
Redirecting efforts to managing the high-risk social spaces between communities and households may go a long way towards lowering the maximum infection peaks and reducing the risk to society.