Author: Ernie Dainow

The big question many people are asking is “When will social distancing and other measures be relaxed?” Covid-19 models provide these dates, but Health authorities are being cagey and are not revealing them.

I have actually been working on a Corona Virus model for Canada with my old friend from high school Steve. We have shared a fascination with math since grade 8. Steve is a rather brilliant theorist, although he never worked in the field of mathematics. He got a Pd. in Philosophy but spent most of his working career as an independent graphics designer.

In March, Steve summarized the basic epidemiological factors that drive an epidemic. It looked like a basic model could be built in a spreadsheet, without having to use a more complicated SIR type of model with differential equations (a video of this type of model is in the blog post “Epidemiological Growth”). So we have been working on such a model and we think it is producing some really good results. Here is a summary of the key targets in the Canada model (a 2 year projection), with the results from our model in yellow.

The Canada plan is well thought out. The aim is to keep the number of infected people low until a vaccine is available. This can be done with alternating periods of restrictions and relaxations. Here’s the chart produced by our model showing what it will look like.

This multi-wave result is similar to the Imperial College model (shown in the previous blog). We did not plan a multi-wave in our model, it is just a result of basic epidemiological  math and how it is affected by the social restrictions that are being used. After a period of relaxed restrictions, social restrictions need to be reinstated to bring the increasing virus infections down (the troughs in the chart).

Since our model matches the targets of the Canada model and also the Imperial College model, we expect that the results from our model will be similar to the information that Canada will use to help shape policies for Covid-19 going forward.

In our model current restrictions end on June 3, 2020 and will be followed by 9 weeks of social distancing, similar to what we have today, beginning on:
   Sep 13, 2020; Jan 11, 2021; May 11, 2021; Sep 14, 2021; Jan 12, 2022

There is some flexibility to adjust these dates and restrictions for each cycle. In addition, some provinces will be on different time frames. So for example, Sep 13 will not close schools. All periods will likely continue to ban large public events such as concerts and sporting events.

This does not look like a plan that many people will be happy about. That is probably why health authorities are not saying very much about it. A plan is not a blue print. We expect that health officials are hoping that the predictions of the model do not occur. There are several possible reasons why.
 
1) The thinking seems to be that by gradually relaxing restrictions in a step by step limited way, most important economic activity can resume without triggering a virus outbreak. We were able to model a case where there would be only one more period of restrictions in the next 2 years. But in order to achieve this, a very low virus transmission rate has to be maintained. The transmission rate (average number of people who get infected by someone who has the virus) is the biggest factor in determining the spread of a virus. The previous blog showed the example of how reducing the transmission rate from 3 to 2 resulted in reducing the number of additional infected people from 1000 to 100, a reduction of 90%. The transmission rate, also called the reproduction number or R0, is difficult to measure and predict. Our model needed a transmission rate of 1.15 to limit the outbreaks to one. It is going to be a trial and error process to see if if this can be achieved.

2) A big unknown is the number of asymptomatic cases (probably not in anyone’s model). There have been wide ranging estimates of the number of people who are infected with Covid-19 but do not show any symptoms. It is now believed that this may be about 50%. This makes it is more difficult to contain the virus. You cannot locate people who are infected but don’t have any symptoms without doing mass testing. Testing capabilities in Canada and the US have had numerous problems reaching target levels and it is not clear if this will improve enough to contain outbreaks as some of the social distancing measures are relaxed to “open up the economy”. https://www.theatlantic.com/ideas/archive/2020/04/were-testing-the-wrong-people/610234/

On the other hand, if there are and continue to be many more people who have had the virus than the reported case counts, it means that the percentage of the population that has had the virus and are immune is increasing a lot faster and we will reach herd immunity faster. Herd immunity for Covid-19 is predicted to be about 60%. At this level the virus is not eradicated but the increasing pandemic spread is slowed. People will still continue to get infected until closer to 100% of the population is immune.

3) Other good news is that Covid-19 may in fact be like the flu viruses and spread more slowly in the summer. Up until now, disease experts have said there is no evidence that Covid-19 would behave this way, and that the virus is spreading in Africa. A Homeland Security lab just reported significant reductions in the half-life of Covid-19 as temperature and ultraviolate (solar) light increased. 
https://metro.co.uk/2020/04/23/us-says-covid-19-lives-just-2-minutes-sunny-surfaces-21c-70f-12602532/

This map shows that in the part of the world that is hot at this time of year, between the Equator and the Tropic of Capricorn, there are very low case counts, except for Brazil, Peru and Ecuador. Note that India is not in this hot zone now – it is north of the Equator. 

The same principles in our model apply as well to the US. However, I am not aware of a national strategic plan to limit the number of infections to a particular target.States that take the recommendations of health authorities will likely follow a similar plan to what is outlined here. States that do not take these recommendations and relax restrictions too early, will see huge numbers of infected people and deaths.  

Most articles report on only a few companies doing vaccine research. This is the most comprehensive article I’ve found. https://www.nature.com/articles/d41573-020-00073-5  

“Although a number of large multinational vaccine developers (such as Janssen, Sanofi, Pfizer and GlaxoSmithKline) have engaged in COVID-19 vaccine development, many of the lead developers are small and/or inexperienced in large-scale vaccine manufacture. So, it will be important to ensure coordination of vaccine manufacturing and supply capability and capacity to meet demand.”

This last point may be a major bottleneck. If there is no deal and close co-ordination between a small Biotech and a Big Pharma who can manufacture the drug, valuable time will be lost.Big Pharma has never put public health interests before its own licensing and patent protection. 
https://www.theguardian.com/commentisfree/2020/apr/15/coronavirus-treatment-drug-companies

Estimates for development of a virus are widely said to be 12-18 months. 
https://www.theguardian.com/world/2020/apr/19/coronavirus-vaccine-when-will-we-have-one

If this is the time to get a lab proven result, the drug still needs FDA approval (even fast track takes some time), then manufacturing which needs time to source the ingredients, set up the manufacturing infrastructure and then scale it up to produce large volume. All of these steps take time, so the total time could be quite a lot longer than 12-18 months.

In addition, distribution will take time. If a target is 50% of the population to reach a safe level (somewhat less than herd immunity threshold), that’s about 4 billion people. Assuming the drug companies manipulate the supplies behind the scenes and distribute most to Europe and North America first, that’s still 50% of 1 billion.

Here’s a terrific article with interactive timelines showing the potential delays at each step until distribution. https://www.nytimes.com/interactive/2020/04/30/opinion/coronavirus-covid-vaccine.html

Most years, I have to wait for the regular flu vaccine because of distribution delays. Is anybody betting they will be able to get a vaccination in 18 months as predicted?

The BC announcement providing some details of their projections has triggered a tidal wave in Canada. Ontario, Quebec, Alberta and the Government of Canada have now all issued their own reports. Looking at these projections and lots of other stats on the virus is overwhelming, and you often can’t see the big picture. What is important is to focus on what is the really new information, and what is NOT being said.
 
As an overall reference point, Canada, the US and the UK are largely following the Imperial College analysis. A NY Times article on why their report was so influential was in the Mar 22 blog. The Imperial College graph showing a second wave when social distancing policies are used was in the Mar 19 blog (repeated here for convenience).

https://www.nytimes.com/2020/03/17/world/europe/coronavirus-imperial-college-johnson.html

What was important about the BC projection was that it is the first time that an official has acknowledged that there will likely be a second wave in the fall. Alberta and Quebec only released projections until April 30, citing uncertainty in their model and their data to go beyond that.    
The Ontario report went even further than BC and said that the effects of Covid-19 may last for 18 months to 2 years.The Government of Canada confirmed that the pandemic will continue for up to 2 years, and further that there will be a number of outbreaks after the first peak.

Here is the case from the Imperial College report that shows multiple waves over 18 months, as social distancing measures are relaxed and then reimposed when the virus starts to spread again. (This report was published March 16. It took 3 weeks for various Canadian government health authorities to release bits of information shown in this case.)

In the US, very little is being revealed about their model projections. The focus there is on whether they are reaching the peak and flattening the curve. The best way to detect that is by looking at the change in the daily number of new cases. When this number starts to fall, the curve is flattening, when there are no new cases the curve is at the peak and when new cases start to decline you are going down the other side of the curve. Here is a typical picture showing flattening the curve .

Sites that have daily updates of the daily number of new cases are:

U.S.  
https://datausa.io/coronavirus     
scroll to GROWTH RATE. This chart is very crowded, click on the map to select the states you want to see.

Canada 
https://newsinteractives.cbc.ca/coronavirustracker/     
scroll to Daily new cases. Click Canada off and click one province at a time to see the new case count in most detail.

So now here is what is not being said — Flattening the curve is NOT reducing the number of people who will get infected, it is just spreading the cases over a longer period, in an attempt to keep the hospitals from being overwhelmed. And coming down the curve on the far side, is NOT the end of the pandemic. There is likely a second wave and a third wave and so on, as shown on the Imperial College graph above.

So how does a pandemic actually end?

A pandemic occurs when someone who is infected passes on the virus to many other people. If you infect 3 people and they in turn infect 3 more people, that is a total of 3+(3 x 3)=12. If the average time to pass on the infection is 5 days, then at the end of 1 month you have passed on the virus to about 1000 people. 

Each pandemic has different transmission characteristics. The numbers used in this example (3 and 5) are thought to be good estimates for Covid-19 (measured by calculating averages from many cases). As the virus spreads, more of the people you encounter have already had the virus and have recovered. They are now immune, so you now may only be infecting on average 2 people instead of 3. This is 100 people per month, down from 1000. This slows down the spread of the virus. When on average you are infecting less than 1 person, the pandemic will slow down and end.
 
General estimates are that 33% to 66% of the population needs to be immune before the pandemic ends.
https://www.sciencenews.org/article/covid-19-when-will-coronavirus-pandemic-social-distancing-end

A vaccine can rapidly increase the immunity in a population but this generally considered to be at least 12 months away. Social distancing restrictions can slow the spread of Covid-19, but it does NOT change immunity. The virus will continue to spread until there is enough immunity in the population to slow it down so that it is not a pandemic.

This post has a lot of math/graph information. Even if you are not usually interested in this, it might be worth looking at some of it.

The Covid-19 growth curve is often characterized as exponential growth. Here is an amazing video that shows exponential growth. You don’t need to follow all the math to be impressed by the animation. I think it deserves an Oscar nomination for that category.
https://www.youtube.com/watch?time_continue=14&v=54XLXg4fYsc&feature=emb_logo

In fact, epidemics are not exponential growth curves. They are Logistic growth curves. But the very early part can be approximated by an exponential curve, as shown in this picture.

Here is a more sophisticated video that explains Logistic growth (be patient, not until about 5:50).
https://www.youtube.com/watch?v=Kas0tIxDvrg

A separate but related topic is “What are these Covid-19 models?” These are mathematical models that try to predict the progress of the virus. A widely used model is the SIR model. It divides the population into 3 groups and uses a series of equations to show the change over time to S the number of susceptible people, I the number of people infected, and R the number of people who have recovered (or died).Here is a picture of the output of this model. 

By modelling different policies such as social distancing, you can see if the Infected curve flattens or not.

This next video has a lot more math than the previous ones but you don’t really have to understand differential equations to follow it. You can also skip the math definition part and advance to 13:15 to see an animation of the model.
https://www.youtube.com/watch?time_continue=1337&v=k6nLfCbAzgo&feature=emb_logo

BC comes even cleaner than the US and the rest of Canada — it expects the novel coronavirus pandemic will continue to impact daily life until the summer, followed by a potential second wave of the virus in the fall.
https://nationalpost.com/pmn/news-pmn/canada-news-pmn/covid-19-likely-part-of-b-c-life-until-summer-says-dr-henry-five-more-deaths 

I don’t think you will see a similar admission from US health authorities, until just before their April 30 deadline, when they will announce another extension.

Dr Fauci mentioned “a glimmer of hope” based a decline in the number of new cases reported recently. It would be nice to see that on a graph. It’s a real simple graph but I couldn’t find any. I uploaded a spreadsheet of all the case data into a google spreadsheet and then generated this chart. To be able to compare the US and Canada, I plotted % change in new cases rather than actual number of cases.

This shows US new cases declining by 51% to 15% (March 20 to March 31), but with a bump back up to 25% from March 26-28. You have to get to 0% to see flattening of the curve.This shows what is wrong with reading too much into this data – changes in the number of new cases is erratic. You can especially see this in the spike in Canadian new cases on March 26. This was due to some labs clearing a backlog of Covid-19 tests, which naturally led to more cases being reported than on other days.

Another graph that I have not been able to find is cases by % of total population. All case graphs I have seen plot actual number of cases, which makes it hard to compare countries with very different populations. Here’s the chart, generated from the same spreadsheet.

This shows several interesting things:
– The US is on a similar trajectory as Italy. Canada is on a lower trajectory.
– US, Canada are about 20 days behind Italy, 10 days behind Spain.
– It doesn’t look like any of these countries are close to flattening the curve yet.

In the US and Canada, health authorities expect the peak in mid April.

US finally comes clean on virus predictions.
– Fatalities could hit more than 2 million without any measures.
– Best case scenario is 100,000 to 200,000 American deaths.  
https://www.nbcnews.com/news/us-news/dr-deborah-birx-predicts-200-000-deaths-if-we-do-n1171876

Dr. Deborah Birx is finally being allowed to speak. The only person in the Trump administration with any experience dealing with a pandemic. Dr. Anthony Fauci said she is a “superstar”.
https://www.cnn.com/videos/politics/2020/03/20/dr-deborah-birx-profile-coronavirus-taskforce-marquardt-dnt-newday-vpx.cnn

Better predictions should be coming. https://thehill.com/policy/healthcare/489774-birx-cautions-against-inaccurate-models-predicting-signficant-coronavirus

The view from here is that if you simply look at the various graphs posted earlier, you can see that the new US deadline April 30 will not reach a peak and a turnaround that would warrant an end to current social distancing measures.

If you want to follow the hockey stick up the curve and watch for any flattening, these are the best updating graphs I have found.

Canada  
https://www.canada.ca/en/public-health/services/diseases/2019-novel-coronavirus-infection/health-professionals/epidemiological-summary-covid-19-cases.html#a2 

U.S.
https://www.statista.com/statistics/1103185/cumulative-coronavirus-covid19-cases-number-us-by-day/

This page has an incredible amount of info, graphs and stats https://ourworldindata.org/coronavirus#confirmed-covid-19-deaths-by-country

There is a really compact table showing doubling time of deaths by country. You have to scroll down about 12 pages to get to it.
Warning: this is pretty scary.

Here is a very interesting report on the influential study that showed the graph with the second wave (previous blog). It also explains why the UK suddenly changed its policy from one that would develop “herd immunity” to suppression. 
https://www.nytimes.com/2020/03/17/world/europe/coronavirus-imperial-college-johnson.html

The number of cases in the US have skyrocketed in just a few days, and is now higher than Spain. This trend will probably accelerate.

Canada at 1,328 is also on the exponential “hockey stick”, but the lower number of cases are more manageable for our health care systems, so far.

Highlights of the best info I have found. Not really a lot to read because the graphs tell it all. This is for people who are not afraid of math, statistics and graphs.

I think these graphs really show the whole story which is generally not being reported in the news. People are clinging to the hope that the recent closures (schools, restaurants, …) will flatten the curve and will end after the 3 week period. It’s clear from most countries that started on the Covid-19 path earlier this is pretty unrealistic for Canada.

World map, by country shows how Canada has been on a somewhat lower track than many other countries.

The US is on higher track, approaching Spain and Iran.  

But a detailed Canada graph of total cases by day shows how we have entered real exponential growth. Cases from Mar 15 to 18 have more than doubled, 250 to 569. This sudden change rattled Canadian Health authorities and led to a whole string of closures and other support measures being announced since last Friday. We probably will not stay on the lower track in the previous graph and will follow a path more like Europe.