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23 April 2020updated 05 Aug 2021 10:26am

Why keeping the rate of transmission below one is important — and tricky

"r" is the name for the amount of new cases each infected person generates. r greater than one means each infected person more than replaces themselves as a host. 

By Tony Yates

What to do next in the battle against Covid-19? One possible route out has been briefed to TheTimes. One source says that in the British government, the “debate is now between people who think we should suppress the virus completely and those who think we should run things quite hot, use the spare capacity in the NHS and aim to keep r just below one”.

r” is the name epidemiologists settled on for the amount of new cases each infected person generates. It’s a combination of biological factors influencing infectiousness (like how easily it travels in the air, survives on surfaces, weathers heat) and our propensity to contact each other in a way that enables the virus to get from one host to another (how often we get close to one another, cough, sneeze, kiss, shake hands).  

r greater than one means each infected person, who sooner or later will either recover or die, more than replaces themselves as a host.  It is the same as the “replacement rate” in studies of population. Couples need to have at least two children to sustain the sexually transmitted disease of life and record an r equal to one. Social distancing measures enforced during lockdown lower the part of r that depends on how much we come into contact with each other.

With the exception of Sweden, in countries in poorer parts of the world that have limited state capacity to enforce a lockdown, or where people live at such close quarters that social distancing isn’t possible, the global consensus has been to adopt a three-phase strategy involving social distancing, test and trace, then hope for a vaccine.  

Phase one, strict social distancing measures, implies closing down parts of the economy, accompanied with government income replacement schemes to compensate those unable to work or whose corporate revenues have fallen. Once the flow of new cases has dropped to very low numbers, phase two begins: social distancing measures are relaxed and the authorities attempt to detect and tackle new outbreaks with large-scale and intrusive test and contact tracing, holding a further lockdown in reserve if a new outbreak gains a foothold. 

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In this second phase, the r for new outbreaks is back to its old value of about 2.5-3, but test and trace and subsequent isolation and quarantine stamps on the of localised outbreaks, reducing them to zero. Phase three begins with the arrival of a vaccine and/or effective anti-viral therapies.

One side of the Cabinet debate, if the briefing is to be taken at face value, seems in line with this global strategy. But the other does not.  

Concerned about the economic costs of forcing people not to work, and government debt piling up as a consequence of income replacement schemes, and perhaps also by worries about how long it will take to develop a test and trace infrastructure in the UK, this other faction wants to relax the lockdown somewhat, knowing that the cost is more infections, more hospitalisation, and, inevitably, more death. 

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This is a translation of “run things quite hot”: run the health system at closer to full capacity than we are now. But the hope of this second faction is that one can relax the lockdown by just enough to keep the reproductive rate of the virus below the critical threshold of one so that the flow of new cases continues to decline.

That the running-it-hot strategy is a live option should worry us all. The debate seems premised on the idea that with the current lockdown, we have managed to push r a long way below one, and that there is some looser lockdown we could move to that gives us a higher r but still below one, so that cases continue to fall. As a follow-up Times story on 21 April put it, quoting another anonymous source: “Everyone wants the rate of transmission to be below one. But the question is how much lower it should be.”  

This would be a very bold move. The number of new confirmed Covid-19 deaths has stopped increasing in the UK, and seems to have begun to fall. But countries a little further ahead with similar strategies, like Spain and Italy, have seen very gradual falls in deaths. This could be consistent with an r not much below one. The limited capacity and imperfect nature of UK tests for Covid-19 make it hard to be confident where we are in the epidemic process, and what the reproductive rate of the virus is at any time.

An added uncertainty for the “run it hot” faction is: what would happen to the degree of compliance if aspects of the lockdown were released? Would people fine-tune their activities in line with the looser lockdown? Or would they let rip after many weeks cooped up? Or ignore the advice, concerned for their health, and stay at home, (engaging in what a niche of economic epidemiologists would call “equilibrium social distancing“). TheTimes story, which indicates that the Prime Minister is siding with the “doves” (those who don’t want to relax the lockdown yet) attributes to him the observation that “relatively little is known about the effect that easing individual restrictions could have on the transmission rate”.

In these circumstances, “running it hot” is extremely risky. It is likely that there is not much room to relax distancing while keeping the reproductive rate below its critical value. If a partial relaxation leads to a new outbreak, it will take time to reveal itself and the explosive disease dynamics will mean another stringent (and economically costly) lockdown is required to regain control.

Engineers who pioneered the “control theory” that was imported into epidemiology (and into economic models) would recognise the cabinet as being like fighter pilots flying in poor visibility, with a long delay between when the pilot tweaks the joystick and new information about what the tweak has done to the plane. In these circumstances, the recommendation would be not to fly too low to the ground. Hitting the ground, which is game over for the pilot, is equivalent to “running it too hot”: the reproductive rate of coronavirus breaching one and a new epidemic breaking out.