Health Secretary Matt Hancock has been speaking to BBC Breakfast to talk about Thursday's launch of the test and trace system in England.
The 2,000 who tested positive on Wednesday will be approached by text, email or phone by tracers, and Hancock added he was confident they will co-operate by divulging information about who they had been in contact with.
He said: "The vast majority will say yes because if we all participate then we'll be able to, more safely, lift the lockdown measures.
"Where this works best is where the clinician and person who tested positive work together to work out who that person might have been in contact with. One of the very nasty features of this disease is that it transmits before you have symptoms - the conversation will be about what happened over the past few days before you became symptomatic.
When asked about a report by the Royal Society which stated the success of the system depended on how quickly contacts can be found, Hancock added: "We need to get test results as fast as possible - our goal is to have a turnaround of tests within 24 hours. We now have 84% of tests at drive-thru centres turned around in 24 hours."