# How much Covid-19 testing do we need?

There is a simple way to estimate how much SARS-CoV-2 PCR testing we need to start diminishing the COVID-19 pandemic. Suppose we test everyone at a rate $f$, with a PCR test with 100% sensitivity, which means we do not miss anyone who is positive but we could have false positives. The number of positives we will find is $f p$, where  $p$ is the prevalence of infectious individuals in a given population. If positive individuals are isolated from the rest of the population until they are no longer infectious with probability $q$, then the rate of reduction in prevalence is $fqp$. To reduce the pandemic, this number needs to be higher than the rate of pandemic growth, which is given by $\beta s p$, where $s$ is the fraction of the population susceptible to SARS-CoV-2 infection and $\beta$ is the rate of transmission from an infected individual to a susceptible upon contact. Thus, to reduce the pandemic, we need to test at a rate higher than $\beta s/q$.

In the initial stages of the pandemic $s$ is one and $\beta = R_0/\sigma$, where $R_0$ is the mean reproduction number, which is probably around 3.7 and $\sigma$ is the mean rate of becoming noninfectious, which is probably around 10 to 20 days. This gives an estimate of  $\beta_0$ to be somewhere around 0.3 per day. Thus, in the early stages of the pandemic, we would need to test everyone at least two or three times per week, provided positives are isolated. However, if people wear masks and avoid crowds then $\beta$ could be reduced. If we can get it smaller then we can test less frequently. Currently, the global average of $R_0$ is around one, so that would mean we need to test every two or three weeks. If positives don’t isolate with high probability, we need to test at a higher rate to compensate. This threshold rate will also go down as $s$ goes down.

In fact, you can just test randomly at rate $f$ and monitor the positive rate. If the positive rate trends downward then you are testing enough. If it is going up then test more. In any case, we may need less testing capability than we originally thought, but we do need to test the entire population and not just suspected cases.

## 3 thoughts on “How much Covid-19 testing do we need?”

1. It is curious in the UK today, or so, we are slowly leaving Lockdown – the issue mainly is testing, but on TV there is talk of airbridges to Greece & Spain, and Cyprus will pay for your trip if you get Covid – so not so bad in the Med.

One problem has been testing capability – now above 100k/day in UK but an abysmal 2kish/day in Scotland; and the latter is leaving Lockdown first. You say:
“In any case, we may need less testing capability than we originally thought, but we do need to test the entire population and not just suspected cases.” So with a population of 70 million that’s gonna take a couple of yrs ….

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2. So at a testing rate of 1 per month, the UK would need to roll out about 6 million tests per month.

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3. Ishi Crew says:

I like these kinds of equations. eg P= positives = fp

or assuming a birth and death process P =fqp . its always good to stay positive!!! (see ‘postitive psychology’. )

one could also have W=work= force times distance = fs.

i am against work whether in thermodynamics or the real world.

how much work would it take to test everyone and also follow everyone who talked to someone who tested positive?

this could be part of the ‘Green New deal’. Every unemployed or underemployed person could get a job testing people to see if they have the virus they have passed the virus onto.

i ran across people who had jobs fixing metro train seats from vandalism. after they got off work, they vandalized some more metro trains.its a self-organizing system.

its what is called work W= integral (l F. ds)

you know if you want to climb a mountain its alot of work. I think Feynman’s lecture on physics part 4 had the formula (also written by Chandeshekar) p ~ exp(-V/kT)

In economics kT = you can get your income from energy or temperature.

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in thermodynamics the only way to get work is via the partition function in statistical mechanics. . in my case if i took classical thermodynamics you get a fail.

thuis is why science education in middle schools should nto gbe abolished (i had one short job in a very upscale middle school and one in a low scale elementary school teaching science. i gave one of my students my copy of Schrodingers ‘what is ife’ and then i quit –i was getting haraased by a superior. i quit my other job after i taught Cantor’s diagonal argument to my preteen students. they got it; their parents didn’t nor did
the people who employed me. so i quit after 3 months and hitchiked to alaska. (interesting trip—i had to get past my grandmother, aunt, and then the border police station in the Yukon. and then ‘work’–change a car tire for 40\$, help people cut lumber for firewood–it could be -45F in winter, and catch salmon for food—which grizzly bears and minks or martens might take. .).
i think maybe all equations in math and physics may be written in a form like a=bc or a=bcd (abc conjecture)

https://en.wikipedia.org/wiki/Primon_gas Ramanujan’s writings had littl;e economic value in his lifetime (he lived a long happy life up until lhis 30’s). https://arxiv.org/abs/1804.02436 that discusses PCR ) Kary Mullis, astrologer, peaceful conflict resolutionist, )

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