The PM wrote in the Mail on Sunday that more complicated messages were needed during the next phase of the response and as restrictions changed.
But Mr Johnson said he trusted the “good sense of the British people” to observe the new rules and thanked the public for “sticking with us” so far.
It came as No 10 announced up to £93m to speed-up a new vaccine research lab.
The new fund will accelerate construction of the not-for-profit Vaccines Manufacturing and Innovation Centre in Oxfordshire so it can open a year earlier than planned, the government said.
Ministers hope the centre will be a “key component” of the UK’s coronavirus vaccine programme.
Business Secretary Alok Sharma said: “Once a breakthrough is made, we need to be ready to manufacture a vaccine by the millions.”
But Mr Johnson cautioned that, while the UK is “leading the global effort” to find a jab, “a vaccine might not come to fruition”.
In his article, Mr Johnson said changes to lockdown restrictions in England – such as unlimited exercise outdoors – were possible due to the public’s “good common sense”.
In a reference to confusion and criticism of the government’s new message urging people to “stay alert”, Mr Johnson said the government was attempting something that has “never had to be done before”.
The government changed its slogan from “stay at home” to “stay alert” in England on 10 May. The stay home message remains the same in Scotland, Wales and Northern Ireland, which have their own powers over restrictions.
Image copyrightAFP“I understand people will feel frustrated with some of the new rules,” the PM wrote.
“We are trying to do something that has never had to be done before – moving the country out of a full lockdown, in a way which is safe and does not risk sacrificing all of your hard work.
“I recognise what we are now asking is more complex than simply staying at home, but this is a complex problem and we need to trust in the good sense of the British people.”
He added that he wanted to thank the public personally “for sticking with us and – most of all – for being so patient”.
“And I want to reassure you that there is a route out of this.”
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The PM’s words came as the government sought to reassure parents in England it will be safe to send their children back to school next month.
Education Secretary Gavin Williamson told Saturday’s daily Downing Street briefing “we owe it to the children” to get pupils back in school.
Mr Williamson said he knew some parents were “very anxious” about reopening schools, but said it would be a “cautious, phased return”.
Media caption“This is a very small, tentative step, in what I believe is the right direction,” Mr Williamson tells BBC’s Branwen Jeffreys
It followed a row over the government’s plan to begin a phased reopening of primary schools in England no earlier than 1 June.
Teachers’ unions have said the date is too soon to be safe – and the British Medical Association has said it is right to be cautious.
England is the only UK nation to set a date for schools to start to reopen. Schools in Wales will not reopen on 1 June, while those in Scotland and Northern Ireland may not restart before the summer holidays.
Meanwhile, Labour leader Sir Keir Starmer has said different approaches to tackling coronavirus across the UK nations will not “help us out of this crisis”.
The number of people who have died with coronavirus in the UK across all settings increased by 468 on Saturday.
It takes the total number of UK deaths, in all settings following a positive coronavirus test, to 34,466.
There were 136,486 tests in the UK on Friday – the highest daily figure so far in the UK. Boris Johnson has set a target of 200,000 tests a day by the end of May.