"If we do not take immediate action now, our NHS could be overwhelmed and more people will die", he added.
Prime Minister Boris Johnson has contributed by issuing his individual impassioned plea to the public.
Dr Adam Kucharski, another scientist advising the government and an associate professor of infectious disease epidemiology at the London School of Hygiene and Tropical Medicine, said that because the new variant was more transmissible "each interaction we have has become riskier than it was before".
In a statement yesterday, Khan said: "The situation in London is now critical with the spread of the virus out of control".
Britain on Friday reported a record 1,325 deaths over a 24 period from people testing positive for coronavirus, bringing the total number of deaths to 79,833.
But the current lockdown has been branded "too lax" by a member of the Scientific Advisory Group for Emergencies (Sage).More news: Coronavirus symptoms persist for 6 months after recovery, - Wuhan scientists
"With over 1,000 people dying yesterday it's more important than ever everyone sticks to rules", a source told the BBC.
While the infection rate for the city as a whole is one in 30, it is as high as one in 20 in some localised areas.
"We have the winter season and the virus survives longer in the cold, plus people spend more time indoors and we know aerosol transmission, which happens indoors, is a very big source of transmission for this virus".
The Office for National Statistics estimated that 1.1 million people in England had the coronavirus in the week to January 2, the equivalent of one person in 50.More news: NFL Wild Card Weekend, 1/9/21 Bucs vs. Washington Betting Predictions
Seriously ill patients are having to wait up to 30 hours for a hospital bed in London, with a major incident being declared in the capital as the NHS buckles under the strain of rising Covid infections.
"Most hospitals have expanded their intensive care capacity to somewhere in the region of three times their normal capacity".
In London, though, Khan said Covid cases were now "out of control" running at more than 1,000 per 100,000 people, putting pressure on the state-run health services and emergency services.
According to the leaked NHS England cancer resilience plan, London needs to treat more than 500 cancer patients a week to stay on top of demand, but only 122 cancer cases were treated in the city's NHS hospitals this week, with another 101 being seen in private hospitals.
NHS England Chief Executive Simon Stevens said on Thursday that the pressures facing hospitals in London and the southeast of England are so acute that the Nightingale hospital at the ExCel London will be opened next week.More news: Sony to discontinue all but one PlayStation 4 model in Japan