Covid-19 – Understanding R

I was thinking about this so much yesterday, that I knew I had to write it up for today’s post. (We will return in Michael Spencer’s book – Reconsider Jesus – on Friday.)

This graph in a nutshell is what the experience of my home province of Ontario has been with Covid-19. You can click on the graph to view it full size.

I have actually merged two graphs together here and added the orange lines. The top graph if of the “Rt” value of Covid-19 in Ontario and is produced by an Ontario Statistician, Ryan Imgrund. The bottom graph is the 7 day average of reported daily new Covid-19 cases in Ontario as compiled by CTV.

Let me first explain to you what you are seeing, why it is important, and what you can do about it.

The R value of a virus is its rate of transmission. R0 is the rate that it is transmitted if no preventative measures are put in place. That is, how many other people (on average) will one person with the virus infect. With Covid-19, the R value is estimated to be somewhere between 2.5 and 3. If no preventative measures were in place each person with Covid-19 would infect on average about 2.7 people.

To illustrate how significant that number is: If you start with 1 person, after 10 rounds of transmission you end up with 20,000 people infected. After another 10 rounds of transmission you end up with 400 million people infected. This is why we saw such initial rapid growth looking like an exponential graph, because the growth was exponential.

This graph, however, is displaying the Rt number. This is the effective transmission rate of the virus. The transmission rate of the virus can be mitigated by preventative measures. Each addition of a preventative measure reduces the Rt value. Each easing or removal of a restriction increases the Rt value. The key number is the number one. If that Rt value is greater than one, the number of daily cases will continue to grow, if it is less than one the number of daily cases will eventually shrink to zero.

So in the top graph, we can see some of the preventative measures (in red) and some of the relaxing measures (in green).

School closures drastically reduced the Rt value. This was not only because the students stopped interacting, but parents started working from home. It took some pretty drastic steps to get that initial value under one. Not only did we have school closures, but all but essential businesses were closed, then the list of essential businesses was restricted greatly. Finally around April 14th the Rt value dropped below one for the first time.

Note how their is a lag between the first and the second graph. The number of cases in the 2nd graph don’t peak until 10 days later. That is because this graph is of the dates that the cases were reported. People however are reporting cases about 10 days on average after they are infected.

There is another interesting thing that you will see in the graphs, and I have tried to highlight with the orange lines. Generally speaking, every time the Rt value passes through zero you get either a peak or a valley in the number of cases. Cases rise (with a 10 day lag time) when the number is above one, and decline when the number is below one. When is passes through one it is the point at which it is no longer rising or declining and so you get a peak or valley in numbers.

You can see from the graphs that for the most part after the initial clampdowns were initiated, Ontario was able to keep its Rt number under one for most of the next four and a half months. Hence the significant decline in cases per day.

Ontario’s reopening plan has four stages. I won’t go into the details here, but you can look it up. Each stage is more permissive than the previous one. There is an interesting interplay of events happening here. You would expect that as Ontario’s businesses and events opened up, the Rt value would start to go up, but largely that didn’t happen. Why? People started wearing masks. Along with more and more people wearing masks, Ontario municipalities started implementing mask by-laws. This mitigated the impact of businesses opening up and kept the Rt number in check.

By the end of July just about everyone was wearing a mask indoors in public.

So why does the Rt value start increasing at that point, and why do we see the number of cases starting to rise?

Because once everyone is masked, there is not available benefit to be gained from additional people masking. There are no additional people masking. There is not additional downward pressure that is being placed on the Rt value.

Instead we see businesses continue to open, people eating more inside restaurants, churches being a little bolder in relaxing restrictions, friends being a little more lax about how many they socialize with, people returning to office jobs, schools reopening.

While not shown on the graph, the Rt value in Ontario has continued to climb. On August 30th it was at 1.21, the highest point at which it has been since the state of emergency was declared back in March. I am expecting it to continue to climb because of the reasons mentioned above.

Returning to school, in Ontario and elsewhere, is a giant social experiment that I believe is going to have rather severe repercussions.

So what can we do?

Here is what I would encourage each and every one of us to do. Stay a step behind the allowed activities: If you are allowed a social bubble of ten, restrict yourself to five. Want to go back to the office? Continue working from home (if possible) a little longer. If your church is not currently meeting, but is considering meeting again, ask for it to be delayed two or three more months until we start to see the impact of schools reopening. If your church is allowed 50 percent capacity, ask that it stay restricted at 30 percent. Consider home schooling (and I never thought I would hear myself say or write that!)

In short, do what you can to keep your own personal number of contacts with others down. Every little bit that we do helps to bring down that Rt value.

I realize that their are people out there who are struggling, and businesses out there that are struggling. Look for ways to assist them in ways that does not have any measurable impact on the Rt value. Order takeout if you are financially able. Give an encouraging call to someone who is socially isolated. And by all means look after your own mental health. Get as much fresh air and exercise as you can. And don’t be afraid to talk to someone you trust. You are not alone in going through this.

As usual your thoughts and comments are welcome.

44 thoughts on “Covid-19 – Understanding R

  1. But Canada wasn’t obsessed with “Getting Everything Back To Normal” ASAP..
    Which causes a superspread spike, which causes another Lockdown, which…

    And Canada is not known as The Land of “YOU CAN’T MAKE ME! BFYTW!”


  2. Looking at it from a statistical quality control and acceptance sampling viewpoint, I would presume the sampling methodologies take the unreported into account. Since you cannot do 100% testing of the entire country, you are relying on the sampling (i.e. testing) to provide you data on the population. I would also do a detailed ANOVA of the samples to determine which sources of variance are the largest.


  3. I feel for you, Robert F., and you and your wife will be in my prayers. We have been fortunate so far. I have a job I can do easily from home, and my employer is allowing it, but it took a real fight to get them to do that. My wife is a teacher and is working remotely — for now. I fear the day she has to go back to in-person teaching, especially because where we live they are likely to make teachers do that before it is really safe.


  4. In addition, in our society, if you are not socioeconomically well off it usually becomes inherently more difficult to take all the necessary precautions all the time even if you have the best intentions and want to.

    My wife and my situation. It’s very frustrating, and scary.


  5. I was aware of most of this, if not the specific numbers for Ontario. As an American, I’m deeply worried that we will never get our transmission rate below 1.0. Far too many people lack the self-discipline needed, and a vocal minority cling to and promote an anti-science mindset and actions. We have prominent people, including prominent religious people, actively aggravating for, and sometimes holding, large events with many people and no preventive measures.

    In addition, in our society, if you are not socioeconomically well off it usually becomes inherently more difficult to take all the necessary precautions all the time even if you have the best intentions and want to.

    So I’m not hopeful. I think it’s going to get a lot worse before it gets better, and many more people are going to die needlessly.


  6. Any thoughts or data yet on herd immunity, Mike? I’m curious about your take on that potential glimmer of hope.


  7. The area above or below the curve is the indicator of the trend. In may, you had an extended period where it was at or above the line. Hence the little bump up. In June the area above the line was counteracted by the area below the line. Hence the plateau.


  8. But look at the Rt line between June 9 and June 23 and compare it to May 5 to May 19. They’re almost exactly the same–and if anything, the June 9-23 line is actually more severe. Each of those timeframes have stretches leading into them that are below the Rt =1 line, and each has two moments where it crosses above Rt = 1. However, the June 9-23 blips seem to have absolutely ZERO impact on the 7-day average of cases 10 days later–in fact, the June “cases” graph keeps declining despite those two June 9-23 peaks.


  9. I heard an interview on NPR with a Swedish journalist regarding the Swedish experience with coronavirus. He did say that it is typical of Swedes to comply with government guidelines and recommendations, because there is a lot of national trust for the government among Swedes, and so mandates are not necessary. But what he said is mystifying is the lowering transmission rate throughout the Summer, despite the government relaxing recommendations and guidelines and more resumption of normal activities among Swedes. In response to a request from the interviewer to compare his impression of the Swedish experience of decreasing transmission with the American one of growing transmission, he was reticent to do so, saying only, “I’m glad I’m not American.”


  10. Please pray for us in California. The fires nearest me (still 50 miles away) are 90% contained, but we still have a lot of incidents. Latest is in the Big Creek fire in the Sierras – fire started Friday and grew to 35,000 acres in 24 hours. Population is less dense, but people do have houses and cabins in the area and there are many recreation sites with only one road in/out. You can read on the news how the military had to airlift lakeside campers out of danger. Satellite photos show the massive plume. Drought years and disease have left tens of thousands of dead trees that are burning at incredible temperatures, in areas seeing record heat over the weekend and for the next few days. Only the communities right on the coastline are reasonably cool right now, and of course people are flocking to the beach (to provide a narrow thread connecting this comment to the subject of the post).

    Here is a prayer my pastor posted in our weekly bulletin:

    O Gracious God, save Your people from nature’s fury of raging fire. Save those whose homes and families are in the path of the wild fires consuming our land, and watch over the lives of those who, with courage, fight the fires.

    Comfort the families who labored for their homes. Do not allow the forces of nature to be a penance for us who have sinned. Save us, O Lord, from the fires’ destructive power. Fallen nature may rage, but You, O Lord, are merciful. Grant us the humility to repent of our negligence in tending to the order of Creation, and saves our souls, O Good One. Amen.




  11. There is a lot of misinformation about Sweden out there.

    Here are some facts though:

    1. They had infection rates at 10 times that of the neighboring countries.
    2. Although they did not have a lockdown, per se, changing behavior made it look a lot like a lockdown.

    Swedes in general have changed their behavior to a great extent during the pandemic and the practice of social distancing as well as physical distancing in public places and at work has been widespread.

    During the months of March to early June, all shops were practically empty, people stopped dining with friends, and families stopped seeing even their closest relatives… A lock-down could not have been more effective. Handwashing, excessive use of hand sanitizers, and staying home at the first sign of a cold became the new normal very quickly.
    – Maria Furberg, MD, PhD, an infectious diseases expert at Umea University Hospital in northeastern Sweden.
    as quoted in


  12. Is it possible for herd immunity to be achieved at a lower percentage of community infection. If so, it seems to me that would be a great thing.


  13. You are reading it wrong Rick.

    When it momentarily goes above one, the curve flattens. You don’t have a high enough spike above one for cases to increase, and so when it drops below one again, cases continue to decline. I could have drawn those lines on the graph but the effect wasn’t as obvious as it was in other places.

    Note also that there is a confidence interval that has a bit of latitude, it doesn’t negate the general principle.

    Also, as noted further up the thread, the R value is somewhat predictive. That is why he put in a 95% confidence interval as his expectation is that 19 times out of 20 you will see a correlation matching the confidence interval.


  14. Agreed. Though Canada’s data in this regard is severely lacking and is currently good for a retrospective analysis.


  15. Yes, the top graph is calculated from the data in the bottom graph, using a 7 day rolling average. You can read how he does the calculations here:

    But the R value is also predictive in two ways.

    His R value lags the data by 7 days, but determines the future for another three. (7 + 3 = 10).

    Plus on top of that a high or low R value will help you to determine if the number of daily cases is going to continue to climb or decline. A very high R value March 1st resulted in cases continuing to climb for another two months. Even though the R value started coming down, it took drastic measures and six weeks for it to pass through the one level.


  16. I’m not sure that the data is good enough in most places to really justify the results as anything but a ballpark.

    You will note that the closer you get to the present, the wider the degree of error from the 95% confidence interval.

    I will make some comments on how the data was gathered below.


  17. In Ontario there was a period of about six weeks where non-essential businesses (read most small retail and restaurants) were forced to close. Then, as Rivikah says, rules were put in place, and businesses had to operate within those rules.


  18. As I posted in reply to another of your comments, it could also mean that herd immunity is achieved at a lower percentage of community infection than originally thought. I mean, even New York city seems to have achieved some sort of lesser infection/hospitalization/death rate to the point they’ve loosened things up.


  19. I wonder if this kind of apparent outcome is indicative that the herd immunity percentage might indeed be lower than originally thought.

    Bottom line…. This is such a strange virus.


  20. What? You think you have a life to lead other than managing moderated comments on a blog?

    Gheez. Some people.

    [sarcasm off] I’m sure without someone would miss it.


  21. Hi all, I am busy this morning but will be responding to comments at about 2:30. Be sure to check back later this afternoon as I clear up a few mistaken assumptions made in the comments and provide some further clarification.


  22. In other words, there are a number of times where the Rt reaches or breaks through 1.0 (going one way or the other) with no direct correlation 10 days later on the 7-day moving average.


  23. Another quibble: The author of this piece also conveniently leaves out the spots where the correlation “analysis” breaks down. For instance, between June 2 and June 20 there was a gradual increase in effective Rt, but the 10-day lag in the 7-day moving average continued to drop.


  24. But even if all those differences played a factor in Sweden’s much lower death and hospitalization rate,

    Lower than the US? Yes.

    But from what I understand the highest in Europe. But this is based on comments (by people I trust) who have looked hard at the data. I haven’t doe that look so I can’t argue the point based on facts. Doubt I have time to track the data down today.

    Anyone else with such data?


  25. With increased chatter about false-positive tests I anticipate we will hear less about Rt and more about 2020 total deaths from all causes compared to each country/community’s expected 2020 deaths from all causes.

    In addition to capturing how lethal Covid-19 is, this comparison addresses the false 6% argument we heard last week in the US, accounts for people that were going to die in 2020 but died early due to Covid-19, accounts for the discussion that some people died in 2020 because they overdosed on drugs, died from suicide, or didn’t seek medical treatment due to Covid-19, and considers things like lower auto death rates because less people are driving in 2020. Finally, it’s a way to measure the ultimate effectiveness of the measures Mike mentions above.


  26. One minor quibble: the top graph is almost certainly calculated directly from the data in the bottom graph. So the correlation between the two (the orange lines) is not too surprising. In particular, it’s probably true for other reasons that infections take about 10 days to get detected, but the lag time between the two graphs isn’t proof of that because the model-makers built that 10-day lag time into their model as an assumption.

    It’s better to think of the graph of Rt as just a different and more helpful way of viewing the data.


  27. As an addition to my initial long comment in moderation: But even if all those differences played a factor in Sweden’s much lower death and hospitalization rate, after the initial period in the Spring, it doesn’t account for the precipitously and steadily dropping coronavirus transmission rate in a country that had few mandated mitigation measures, and has through the Summer relaxed even those as well as the voluntary ones.


  28. I have a long comment in moderation, as usual. Move over on the moderated bench, Mule, I want to sit down.


  29. How do you relate Sweden’s coronavirus experience with this? No lock down, few businesses closures, no mask mandate, restaurants open, etc. In general, Sweden had very few restrictions or mandates compared to other countries, and yet, although admittedly having a lot of deaths in April, since then transmission has been low and getting lower every week, deaths and hospitalizations have decreased steadily as well, even through the Summer weeks and months when there was more socializing and group activity and laxer voluntary distancing by the populace, and even though Sweden hasn’t gotten close to anything like herd immunity. Sweden is doing much better in terms of all this than other countries that have done much more.

    What’s going on here? Did the virus kill off most of the vulnerable targets? Were, and are, Swedes just generally healthier and fitter, with fewer people in the vulnerable categories, even among the elderly? I know they are much thinner, and obesity is one of the two greatest risk factors, along with diabetes, for COVID-19 complications and death. Americans are very unfit, unhealthy, and overweight (and I’m in all three categories), that is for certain. And there is far less poverty in Sweden and Europe in general, and better access to healthcare for everyone than in America, so has that played an important factor as well, after the initial couple of months when most of the vulnerable suffered and died in great numbers, perhaps leaving few vulnerable to develop complications and high death rate?


  30. “Was this something the US or Canada or other countries did, or was it just us?”

    We never did anything even close to this here in the states, at least on a nationwide level. And our infection numbers prove it.


  31. The Canadian response hasn’t been entirely uniform across the country, but I don’t think that anywhere in the country saw the kind of lockdown or shelter in place orders that other parts of the world have used. Ontario explicitly closed a few businesses for a while, here in BC I don’t think anything was ever required to close, there were just occupancy rules imposed and businesses had to decide for themselves how (or if) they could operate within those rules.


  32. My PhD involved mathematical models of infectious disease. Lots of R values and…personally, I’d want to see the details of the methodology before putting too much stock in Rt calculations. They usually require making a certain number of assumptions and I’m not sure that the data is good enough in most places to really justify the results as anything but a ballpark.

    Biggest question in my mind: How are undiagnosed/unreported cases being handled?

    But maybe the ballpark estimate is good enough for the point you’re trying to make. My kid is going back to school this week, but I don’t really expect that will last very long.


  33. It’s interesting that a drop in infection rate due to school closure may be partly attributed to parents working from home: until the last few weeks here in the UK, even before all non-essential retail business were shut down it was a requirement that all staff who could work from home were obliged to, even before the schools were shut; you had to have a letter from your employer explaining why you were not working from home to travel to work during lockdown, even if your business was not closed. Was this something the US or Canada or other countries did, or was it just us?
    (I am still working from home although the kids are back to school and hoping to stay that way: I have no particular desire to risk Covid-19 in order to save city centre sandwich shops, however much our government apparently thinks this is our patriotic duty.)


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