But the national estimate for R, which now sits between 0.7 and 0.9, is an average. The figure hides hotspots including Leicester, Bradford, Blackburn and Herefordshire where case numbers have jumped again.

Dr Kucharski suggests the concept of "transmission budgets", where the value of R is broken down into its constituent parts, is a useful one for policymakers looking forward.

"Countries essentially have a 'transmission budget' when it comes to Covid-19," he Tweeted as the Prime Minister pondered a further relaxation of lockdown this week.

"Global patterns suggest there are only so many things that can go back to normal before flare-ups start happening".

One problem in the UK is that we don't know with any precision how different measures, such as school or office reopenings, contribute to the overall budget.

However, there is some evidence from overseas. A study conducted in Hong Kong when schools were closed and track and trace was operating was able to isolate and measure the effectof other interventions as they were introduced and then relaxed.

"One of the most consequential physical distancing measures appears to be the work at home policy for civil servants, which was mirrored by many other institutions and private employers", say the authors.

"We estimated that the effectiveness of implementation of civil servants working from home was 67 per cent and the effectiveness of implementation of additional physical distancing measures including closure of high-risk places/facilities [such as bars and restaurants] was 58 per cent."

A separate study looked at nine interventions in 41 countries between January and April 2020 and found eight of them to be effective.

Closing schools reduced R by an average of 50 per cent. Other things that worked included: closing non-essential businesses (34 per cent); closing high-risk businesses (26 per cent); limiting gatherings to 10people or less (28 per cent); and issuing stay-at-home orders (14 per cent).

The "surprisingly large role for schools" the study found will worry policymakers and underlines how tight the UK's transmission budget is. It may partly explain why the Prime Minister did not order a full return to office work this week.

"Working out these budgets is incredibly difficult," says Prof Hunter, who like many other experts urges caution in the absence of firm evidence. "Wedon'tknow with certainty what the real numbers are for each activity in the UK."

Preventing a resurgence in Covid cases is one thing, stopping a separate winter wave another perhaps. Five out of 10of the last big respiratory outbreaks have featured significant subsequent waves, four of them after a summer trough.

Already there is evidence that winter in the southern hemisphere is having an effect, the cold weather forcing people inside and the Three Cs starting to overlap. The strong second peak of cases in Victoria, Australia provides the strongest hint of possible trouble to come in the UK once winter sets in.

"As soon as the bad weather comes and we start living indoors again, thats likely to be a big boost to the R value," says Prof Hunter.

"I suspect that unless by then we have an effective test, track and trace system in place which to be honest isn't looking promising at the moment then we, almost certainly, will go into a second wave."

The "reasonable worst case scenario"for the UK winter was laid out in a Government commissioned report by the Academy of Medical Sciences last week.

The modelling suggested a further 120,000 hospital patients would die in a peak occurring early in the new year if R rose to 1.7 in the UK over winter.

A less severe second wave might lead to 1,300 or 75,000 deaths between September 2020 and June 2021 if the R value rises to 1.1 or 1.5 respectively, it said.

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