Boris Johnson warned against complacency as lockdown measures continued to ease across the country.
According to 10 Downing Street, the number of patients admitted to hospitals is still falling, and now stands at just over 100 each day.
In April there were more than 3,000 coronavirus patients in mechanical ventilation beds, but now the latest figure is 87.
Addressing the nation, the Prime Minister said he “wouldn’t stand by and allow the virus to cause more pain and heartache in this country.”
He said: “The number of deaths continues to fall. That is obviously encouraging.
“But I have also consistently warned that this virus could come back and that we would not hesitate to take swift and decisive action as required.
“I am afraid that in parts of Asia and Latin America the virus is now gathering pace. And our European friends are also struggling to keep the virus under control.
“As we see these rises around the world, we cannot fool ourselves that we are exempt. We must be willing to react to the first signs of trouble.”
Recent figures from the Office for National Statistics show that the prevalence of the virus in the community in England is rising for the first time since May.
Around 1 in 1,500 now have the virus, compared to 1 in 1,800 on 15 July and 1 in 2,000 on 2 July.
The ONS also estimates there are now 4,900 new infections every day, up from around 3,000 per day on 14 July and 2,000 per day at the end of June.
Mr Johnson said: “We can’t afford to ignore this evidence.
“It’s vital to stress that we are in a far better position to keep the virus under control now than we were at the start of the pandemic – because we know so much more about the virus and have so many more tools at our disposal to deal with it.
“Our testing capacity has increased 100-fold.
“We have a contact tracing system up and running which has led to over 184,000 people isolating who may otherwise have spread the virus and is capable of tracing thousands of contacts every day.
“We have secured supplies of billions of items of PPE to withstand new demands on hospitals and care homes.
“And, of course, we have new treatments, like dexamethasone and remdesivir, to shorten recovery times and reduce mortality rates.
“But as I say, we cannot be complacent. I cannot – I won’t stand by and allow the virus to cause more pain and heartache in this country.”