Officials are going to lengths to stress that the easing of measures does not mean people should lower their guard.
Italian Prime Minister Giuseppe Conte said the country was still in the "full throes of the pandemic", telling La Stampa newspaper the "phase 2" of the lockdown "must not be seen as a signal that we're all free".
In the United States, even as warm weather led sunseekers to flock to green spaces in Manhattan, the epicentre of the U.S. epidemic, President Donald Trump warned the national death toll could rise to 100,000. The coronavirus has already killed almost 68,000.
Governments are reluctant to prolong draconian measures that, though often broadly backed by the public, have come at a high economic price.
Factory activity was ravaged across the world in April, business surveys showed, and the outlook looked bleak as shutdowns froze global production and slashed demand. As a result, the global economy is expected to suffer its steepest contraction on record this year.
"This past week saw the amazing coincidence of the publication of the deepest quarterly economic decline in the Western world in almost 100 years and the conclusion to the strongest monthly equity rally in more than 30 years," said Erik Nielsen, chief economist at UniCredit.
Escalating tensions between the United States and China over the origin of the pandemic drove down stock markets and oil prices on Monday as investors around the world feared a new trade war.
U.S. Secretary of State Mike Pompeo said on Sunday there was "a significant amount of evidence" that the virus emerged from a laboratory in the Chinese city of Wuhan. He did not provide evidence or dispute an earlier U.S. intelligence conclusion that the virus was not man-made.
An editorial in China's Global Times said he was "bluffing" and called on the United States to present its evidence.
