Chancellor Rishi Sunak tweeted: "Whilst the test and trace pilot is fairly restrictive, allowing only essential government business, I recognise that even the sense that the rules aren't the same for everyone is wrong.
"To that end I'll be self-isolating as normal and not taking part in the pilot.
Covid herd immunity could be reached if the next wave of infections peaks without a change of course like lockdown measures, Prof Neil Ferguson said.
He told the BBC's Andrew Marr Show: "Fundamentally, this will be a different wave from the previous two.
"The previous two peaked because we introduced lockdown measures, and that's the only reason.
"This time if we don't have to change course, then it will peak because herd immunity is being reached, and then it overshoots and you still get infections beyond that point, but the epidemic will be in decline."
Prof Ferguson said the precise level of immunity in the population was not known, with data on vaccines being "not perfect".
Asked if high levels of people with sufficient antibodies could be reached without vaccinating children, Prof Ferguson replied: "In the absence of vaccinating teenagers ... we're already seeing very high numbers of cases in teenagers - and we won't be able to reach herd immunity without significant immunity in basically people under 18."