The problem with lockdown in the cities only, is that people flee to country - spreading it further, and putting pressure on local NHS/services. That exodus is happening already, and it really needs to be stopped.
Lockdown would mean you can go out to buy food - there have been no food shortages in Italy and Spain that I've heard of.
To learn from Italy, we would lock down all at once. Essential work (incl care for vulnerable people), food shopping and pharmacy only.