The What and the Why of Rt

We all look at our national and local COVID infection counts. But to really measure the pandemic's pulse, epidemiologists monitor a number called Rt and do their best to wrest it below 1. What is Rt, how does it work, why do we care, and what are its limitations?

What is Rt? (and a baseball analogy)

Fundamentally, Rt is the average number of people a single COVID-infected person will directly infect in turn, before that individual stops being infectious. It is called the effective reproduction ratio. It assumes a closed population, and the t refers to the fact that it varies in time.

If Rt is below 1, the infection will die out, since it is finding new victims more slowly than current victims recover (or not.) If Rt is greater than 1, the infection will spread. If Rt stays any constant number greater than 1 for some time, that spread will be exponential, the more "explosive" the greater that number is.

A special case is R0, also called the basic reproduction ratio, which is the number of people infected by one carrier at time 0, the start of a pandemic, before anyone is doing anything special and before anyone has immunity. It reflects both the infection's average "at bats" (to use a baseball analogy)--how many people an average carrier has the opportunity to infect--as well as its natural "on base percentage"--how transmissible it is, both given normal human behavioural patterns and resiliency to infection.

>> R0 for COVID seems to have been about 2.5 or higher (see below for difficulties in estimation). This is higher than typical seasonal influenzas, but much lower than traditional highly infectious diseases like chicken pox or measles.

One generally expects Rt to decrease from R0 over time, for three reasons.

1. People change their behaviour, whether of their own free will or forced by public health directives. This decreases a combination of the "at bats" and "on base percentage".
2. We get better at identifying those infected, those susceptible to infection, and at prevention or treatment. We don't allow the pandemic as many at bats, and we pitch at it in a way to induce it to harmlessly fly out when we do.
3. Finally, if there is any sort of resistance to reinfection (immunity), fewer of the contacts that would earlier have led to an infection are with people still susceptive. The baseball analogy is a bit strained here: maybe we say we keep on flooding the field with more and more outfielders until there is no gap to hit into any longer.

However, Rt can also increase, either due to peoples' behavioural changes, specifically "relaxing their guard" (frequent!), or due to the infection's dastardly mutations, and the fact that those mutations that get through our defenses better get an evolutionary advantage. We haven't conclusively seen much of the latter for COVID in particular, but it definitely happens for the seasonal flu!

Rt says nothing about how fast a carrier infects others, just how many those others are. There is a related concept, called the serial interval, which is on average how long it takes to pass on to the next "generation". Rough estimates put that at about 5 days for COVID; this is also discussed further below. Of course, the serial interval also influences how fast an infection with Rt > 1 spreads. However, pandemics can afford to be patient, so it's Rt which is most important.

The math, in simplified form

Let's consider a particularly obliging hypothetical pandemic, which starts with a single patient #1, and whose serial interval is exactly 1 day. What's more, that's not only the average time to (re)infection, but precisely the time it takes in every instance. Let's suppose R0 = 2.5. That means the 2nd day, there are 2.5 infections. The 3rd day, 2.5*2.5=6.25. Adding it together, at the end of the 3rd day, there a cumulatively 1+2.5+6.25=9.75 infected people. (The fractions are a bit artificial here; we can pretent than one patient #1 is actually 1000 to avoid them, or just ignore the problem until the numbers get bigger.)

Exponential growth would continue, except people change their behaviour and Rt begins to decrease. As long as it is > 1, the number of new infections is greater every day. And of course, the cumulative number of infections keeps on growing. The left hand side of the figure below shows what happens over time.

In the (left hand) graph panel, the orange line is the number of new infections each day, plotted on a logarithmic (each step is 10X, not plus a fixed number) scale on the left. The green line is Rt, plotted on the normal scale at right. The orange line increases while the green is above 1, and decreases when Rt < 1.

The grey bars show, also on the left hand axis, the total (cumulative) number of infections. If exponential growth continued with Rt = 2.5 throughout, the grey bars would march upwards steadily - logarithmic scales make exponential growth look like straight lines. Because Rt is driven steadily down, the march upwards is slowed down. But even in time period 20+, we're still getting 10 or so new cases a day, so the grey bars always march upwards a bit--even though by the time the cumulative cases exceed 3000, we're much less likely to pay attention to 10 cases versus in time period 4, for instance.

I've called this the "Disciplined" scenario since concerted action, voluntary or not, fairly swiftly reduces Rt steadily, and keeps it below 1. There's a bit of a relaxation at the end, but Rt stays at 0.9, not >1.

The right graph panel shows something I haven't seen used much in this context, but seems a natural fit for someone like me, trained in math and physics. It's a so-called phase diagram, which plots the trajectory of the "system", in this case the infection in terms of its "position" (infection numbers) and "momentum" (Rt). This is a well known concept in dynamical systems, even though I've abused the terminology a bit here. It's sort of a dashboard of where we stand (horizonal and vertical axes) at any point in time (the individual dots), and our trajectory (the "snake" connecting the points). In particular, in this scenario, you can clearly see how we turn the corner once Rt goes below 1 and new case numbers decrease, so the snake turns to the left.

Different scenarios relevant to COVID response

We stay with this obligingly simple hypothetical pandemic to illustrate different response scenarios through the lens of Rt. These scenarios are chosen to be relevant to COVID.

>> The Disciplined scenario, described above, is how certain, especially Asian, countries responded to COVID.

Let's now consider a "Meander" scenario, where Rt gets driven down, but rather than remaining well below 1, ends up meandering around the magic value 1. Here is the outcome, overlaid over Disciplined which is shown in thinner lines.

While the difference in green lines seems small, it causes a profound difference in the orange line. The number of new infections still decreases over time, since I have engineered the meandering Rt line to stay below 1 on average. But the infection numbers subside much more slowly (and the cumulative numbers continue to much more noticeable grow too!). In the phase diagram, the meanders show up as going around in circles, around and on average below the Rt=1 line, but moving to the left much less rapidly as a result.

>> The Meander scenario is representative of what my country, Canada, more or less did in recent months. Something between Meander and Disciplined is pretty typical for countries that have moved out of Wave 1 of the COVID pandemic.

Not all actions taken to reduce Rt are terribly painful, but they're usually constraining (a discussion will follow in another article). So public health officials would love to steer the dynamical system "just right". The Titrate scenario below (overlaid over Meander) is the holy grail.

In this scenario, the hatches are battened when the situation is not contained, but relaxed a bit when it is. Until it isn't again. I've represented this as a steeper slope down in the Rt curve (which is likely part of such a strategy, though frankly it just gives me more space on the tail end of the graph...). Then Rt is allowed to drift up to 1.15 when new daily infections are < 30, and locked down to 0.8 when infections exceed 40. These numbers are wholly artificial, but they stand in for a typical reality, influenced by, for instance, how many infections the health care system can handle.

In the phase diagram for this scenario, by design the snake is captured in a little circle. Of course it would be nice if it happened to meander more to the left, but staying in a circle is good enough.

>> It seems clear a lasting solution to COVID, e.g. an effective vaccine and/or treatment, is far off. Many areas of the world would be content with a successful Titrate strategy while waiting.

Unfortunately, whether meandering or titrating, it is easy to undershoot the mark in terms of Rt reduction since in reality Rt is known only approximately (more on that below). That leads to the Lost Courage scenario below. Unlike Titrate, the Rt reduction achieved ends up on the wrong side of 1, and the phase diagram snake slithers off to the right rather than being held to the left. The only solution when this happens is to move to a more aggressive Recover scenario, or some other "Disciplined+" battening down of the hatches. With the benefit of hindsight, this was a poorer choice than not Losing Courage in the first place!

Finally, the Lost Courage scenario above is pretty mild. The case numbers continue to grow (presumably in a fashion we don't consider sustainable), but in a bounded fashion. If courage is lost even more, we collapse to a Give Up scenario instead.

>> USA, Brazil, and a number of other regions seem to be in this scenario. [Update 07/28 - that was definitely the case a week ago when I wrote this. A week later, the U.S. as a whole may be turning the corner.]

Unsimplifying and applying to COVID

It's time to remove the simplifying mathematical assumptions in the above discussion. While the Rt levels, ranging in the graphs from 2.5 to 0.7 and oscillating in the neighbourhood of 1, are representative of COVID, its serial interval is around 5 days, not 1 day. What's more, not all infections happen precisely at (t + the serial interval). Some arrive earlier, others later. This "smears" (technical term!) the whole picture.

More importantly, data is imperfect. Only some infections are detected; I've seen numbers ranging from 80-97% are undetected, unreported, or even wholly asymptomatic. Nevertheless, as long as we assume that fraction stays constant, and our goal is to preserve a functioning health care system that deals only with the detected ones, this overall analytical approach is still useful, with limitations.

However, in particular issues with detection, testing, and reporting easily bias our estimate of the serial interval and Rt. For instance, sporadic infections are likely to be detected less often and later in their lifecycle than the ones in a sequence of close-contact infections. People get tested, and testing numbers are reported, with differing levels of enthusiasm and accuracy on different days of the week. All this affects observed serial interval and Rt, unless corrected for, which is in itself guesswork. What's more, Rt is not itself an observable variable. While it is (hopefully) the result of our actions, it has to be statistically fitted to infection numbers. There's a number of methodologies out there for this, from the domain of advanced analytics, but they in turn make their own hidden assumptions. They all in some way try to answer the question, "bearing in mind we don't know anything 100% accurately, what plausible Rt curve would best explain what we're measuring?" (This process is called "calibration of the model".)

The upshot is that the best we can do is estimate Rt (and serial interval too, by the way) with error-bars, or more accurately confidence intervals. The site rt.live shows up to date estimates, and trends over time (like our green curve above) for all U.S. states. At time of writing (07/22), the situation looks like this, the bars representing confidence intervals (Pet peeve: it's not easy to find exactly how statistically the C.I's are defined...).

In my home province of Ontario, Canada, a biostatistician publishes Rt updates (actually by city/region as well) daily on Twitter, though with different methodology (especially the error bars). Example:

Interpreting COVID Rt's

So, what to do with the data, especially in the context of the error bars around any sensible estimates of Rt? The answer is to think in the context of the phase diagram snake (and relevant health care system capacity), whether or not you actually draw the phase diagram itself or just think it through mentally.

Let's take a look at the situation in Florida at rt.live/us/FL. Rt estimated as of July 21 is shown in the thumbnail at right, but better to see it at the site, larger size, and with reference to the positive tests and implied infections curves also shown there. Those correspond to the horizontal axis of the phase diagram.

As we all know, Florida continues to have horrifically high infection rates. But best statistical estimates would currently indicate they are most likely (after removing noise and adjusting for timing issues) stabilizing, reflected in Rt having trended towards 0. The confidence intervals are still pretty broad; Rt may be > or < 1. If it stabilizes <1, they will be in the Meander scenario or better. If it stabilizes >1, then at best the Give Up may have transitioned to Lost Courage; still not good at all.

Contrast with Montana at rt.live/us/MT (Wyoming is similar). It turns out their infection numbers are 5-10X lower than Florida, adjusted for population. However, it's pretty clear from the Rt graph that there is no path in sight to turn away from the Give Up scenario at the present time. I am equally, if not more worried about them than Florida in terms of the medium-term future.

[Update 07/28. Interestingly, the overall rt.live picture has not changed much. Florida has stayed close to 1, so infection numbers have stabilized. Montana has stayed near 1.23.]

Finally, let's look at Ontario, more specifically current [07/22] regional numbers, also from Ryan Imgrund (condensed from 2 pages to highlight only a few regions).

First, as important context, the weekly infection counts per 100k population are far lower than the U.S. (currently ~150 overall), never mind Florida (363). Statistical calibration of Rt with low case counts, both due to fortunately low infection levels as well as just by focusing on smaller local areas, can be very unstable. So local Rts can have very large confidence intervals and be very unstable in time (that's fancy speak for: "they'll calibrate to a different number tomorrow when we have just one day more of data").

Even with this in mind, my home town Ottawa's Rt, clearly currently well > 1, is cause for concern. Especially since new infections seem to be coming from a spike of younger people not taking social distancing seriously (but doubtless with more vulnerable friends and relatives...) and with a further, controlled relaxation of restrictions on bars and restaurants announced for this Friday! That is not a recipe for easily dragging Rt lower quickly. But the situation is not worthy of a panic, yet, since overall infection levels are low and have decreased significantly from their peak, Rt has spiked this badly only recently, and hospitals are well under capacity.

>> Meta-observation: In the last paragraph, I effectively drew out the phase diagram mentally/verbally, including grounding the horizontal axis (infection numbers) in constraints. The real-time calculation of Rt at this level of granularity for Ontario cities is fairly new; I lack the data to actually draw the phase diagram accurately. But it is clear a Meander scenario is in some danger of tipping into Lost Courage rather than Titrate. However, with the small number of infections there is still time to get it under control.

Note that though Rt is estimated from historical observed data, it is a forward-looking indicator, predicting future behaviour. Ottawa's current (small) spike in cases, and elevated Rt, is no surprise: the estimated Rt was about 1.2 a week ago, though with a fairly wide confidence interval. Therefore I continue to look with concern at Windsor, where a series of known-cause outbreaks seem to refuse to die out and Rt remains stubbornly high. Finally, Toronto and its neighbour Peel would look less concerning than Ottawa from this snapshot in time, but it is important to recognize their apparent Rt has languished around 1 for some time, and so therefore also their infection rates. If we want those to go down, easy on the relaxation! Ontario is actually implementing a version of the Titrate scenario, with different Phases of relaxation of restrictions. Toronto is being kept back for another week or so (hopefully not more). Once again, these dynamics are easily interpretable by drawing a mental phase diagram.

What about Peterborough, the other Ontario town in the last example above? The statistical point estimate of its current [07/22] Rt is 2.79, however the confidence interval is very wide. Essentially, the estimation algorithm is throwing up its hands and saying, "we don't know". That is not surprising, given there have been no cases in the past 7 days! Actually, even where there are some cases, when the number is small, the confidence in the estimate is always low. In fact, in such cases the estimates are also "unstable", i.e. they are very sensitive to small changes in the inputs, whether that's corrections to observed infection numbers or random variations over time.

This sensitivity or instability has led some people to call Rt unhelpful, especially when case counts are low. I'd prefer to say Rt and case count trajectories are jointly helpful, with Rt less relevant when case counts are and remain very low.

Interventions and Rt

This article is more a primer on the math, and how to read Rt. However, let's briefly (re)consider how interventions affect Rt, and therefore also how Rt can guide interventions.

In a simplified, hands-off system, Rt would start at its initial value, R0. Over time, people would naturally adjust their habits, hopefully reducing Rt. Assuming it would still be >0, the infection would grow exponentially until it would find its success replicating being increasingly limited by its "at-bats" (returning to baseball) being duds, since more and more of the people encountered by the current victim would already be immune (assuming there is immunity conferred; or dead/nonexistent, if not). Therefore Rt would then decrease further, down to 1. This is herd immunity, and it actually occurs when the fraction of the population that has built up immunity is 1-1/R0*. I've added a weird * there, since it would only be 1-1/R0 (without the *) if peoples' habits failed to reduce R0. If they do, it is "that part of Rt which comes from behaviours, not the part from emerging immunity".

That's one of the ways we get out of this mess. We find a vaccine, it is effective, and enough people get vaccinated for herd immunity to kick in, driving down Rt and therefore infection numbers. But R0 seems to be about 2.5, and we have to be trying pretty hard to keep Rt < 1.5. That means that immunity needs to be present in somewhere around 1/2 of the population for this to vanquish the virus (that comes from doing the calculation 1-1/1.5=33% and 1-1/2.5=60%). So the vaccine needs to have a lot of uptake! Or we find reliable ways to treat COVID with much lower death rates or lasting damage, so that we're comfortable with 50% or more of the population getting it.

Finally, of course public health measures can be used to drive down Rt. Our whole Titrate scenario was driven by this premise. Importantly, over the long term, hopefully interventions can be found which are not broadly constraining on our economy and everyone's personal freedom, but are much more finely targeted at preventing those infected (or more likely to be infected) from passing the virus on. That's the role of quarantining travellers, for instance, or those awaiting the results of lab tests. Back to baseball: we learn to adjust the outfield based on the batter. Hence the sort of circling in our Titrate scenario will become more tolerable, not just by resetting our baseline, but by better targeting.

* * *

All of this will of course change in time, but hopefully this (too long!) article will at least have explained what Rt is and what role it plays. And perhaps actual graphed phase diagrams, or mental/verbal qualitative descriptions of them where data is not available, will help in interpreting what Rt means about COVID dynamics in different localities, and how it can help guide decisions.

The author, Martin Pergler, has 20 years experience in consulting in risk management and uncertainty, as well in teaching risk-informed thinking and effective risk communication. This article reflects also his work on COVID infection rates for several localities earlier in the pandemic.

Text and scenario diagrams (c) Martin Pergler and Balanced Risk Strategies, 2020. No copyright asserted for visuals from other sources. Thanks to Rafael Gupta (you really need to set up a LinkedIn profile!!!) for helping with the model and visuals.