Without contact tracing and isolation of positive cases a lockdown won’t work. Since no system like this is in place I reckon it will take a lot longer than 2 months.
Think about it this way. The Ro spread factor is somewhere between 2.2 and 3.0 by most reports. People are contagious for 7 to 14 days before they know it. Therefore every infected person spreads a new case only once every 2 to 5 days on average. The problem isn't that the virus is explosively contagious per-interaction (measles is higher), it's that the incubation period is so long across so many interactions.
Cut that recognition time from that 7-14 days to 2, cut off those interactions, and that's how you get Ro below 1 to stop it. That's what South Korea did, by way of both rapid testing and contact tracing to test them too. Reports are that their rapid test gives false positives of up to 10%, but that's okay, a false positive is much less consequential than a false negative.
What jurisdictions in the Western world have the means and will to implement that remains to be seen.
Though I guess whenever you are producing enough virus to be contagious you would probably test positive at that point too.
At any rate, that's an approximation. The key point is for the average testing and isolation interval to be shorter than the average time to pass on a new case, whatever that interval actually is. If that happens, then it's stoppable.
Why? In a total lockdown situation, the positive cases might infect their close people the live with, but not really anyone else. And in a half lockdown, with still going out/shopping ... the infection rate is much slowed, if a high enough percentage of the infected isolate enough. That is all that we want, I thought.
That works if there is a relatively low number of cases, or you have enough testing capability to carpet bomb everyone with tests.
Once the number of cases gets too high, contact tracing has lost, because you have too many unknown cases in the community.
If you can’t test a large percentage of your population, you can’t detect pre-symptomatic or asymptomatic people that are spreading the virus.