I recently posted a link to a more easily understandable display of Covid-19 infections and deaths for every country. (Incidentally its creator, a high school student, had a brief profile written about him in the latest issue of *The New Yorker*.) But one thing it lacked was the ability to visualize the differential rates of growth in each country over time to better enable us to understand how the disease is spreading and what is meant when people say that one region lags behind another by so many days, and how to know if containment methods are successful.

Part of the problem is that exponential growth is not intuitive to understand even though we have some everyday examples of it. The most common one is compound interest in a savings account. If you leave the principal untouched, it will grow slowly at first and then increasingly rapidly so that over a long time one can accumulate quite a lot. But despite that, when it comes to change, we tend to think in linear terms because many things change in close to linear fashion and it is very simple to understand and the human mind tends to grab on to simple ideas and explanations and try and force everything into that mode. But there are times, and this coronavirus pandemic is one, when the amount of change in some thing in a given period is proportional to how much of that thing exists at that moment (which is how compound interest works), and then you have exponential growth.

To better understand exponential growth and its application to the pandemic statistics and how to tell when the data shows that a country has managed to brings things under control, this video is helpful because it illustrates when the inflection point (i.e., when the rate of growth of cases starts to decrease) is reached.

The excellent time-lapse growth chart shown in the above video can be seen here and is being constantly updated. It enables us to see at once that China and South Korea seem to have been successful in curbing the pandemic’s growth. Japan and Iran seemed to show some success at one time but it looks like they have lost some ground.

The explosive nature of exponential growth is hard to fully comprehend, as can be seen in the answers given to a problem like this.

A lake has algae growing on its surface. The amount of new algae that appears in a 24 hour period is equal to the amount that existed at the beginning of that period. i.e., the amount doubles every day. If the lake starts with almost no algae but is fully covered by day 30, when was it only covering a quarter of the surface?

The answer is day 28. But some people, unless they pause and think things through, would tend to give an answer that is close to about 8 days because they are thinking linearly.

chigau (違う) says

Also this

https://en.wikipedia.org/wiki/Wheat_and_chessboard_problem

Charly says

Every gardener who plants tomatoes experiences exponential growth too. The plants seem to grow slowly fo months and then they seem to explode and each day produce a ton of leaves and branches that need to be pruned. And at first, it seems there is no fruit to be seen anywhere and then all of a sudden new ripe fruit appears faster than you manage to harvest it.

Dunc says

Combine that exponential growth with the fact that it takes quite some time to go from initial infection to showing up in the statistics, and you realise that there are probably at least 8 times as many people currently infected who don’t know it yet -- and that’s

withoutconsidering the number of mild cases that never get recorded.robert79 says

I’m not sure that exponential growth is actually the right model here. Sure, if every sick person infects two other people, you get a 1, 2, 4, 8, 16, 32, … exponential growth. However, once enough of the population has been infected, recovered, and become immune, every sick person would infect two other people — multiplied by the proportion of people that have already been infected! And so the exponential growth rate slows down.

This is called logistic growth (which is only very briefly mentioned at the end of the video) but it may mean that the drops you see on the animated graphs (china, south korea) are the countries that got infected early and have hit a saturation point where a large enough proportion of the population has gotten immune that the disease stops spreading by itself, simply because there are no other people left to infect, and not due to any government measures.

Logistic growth looks like exponential growth at first, but then slows down.

This would of course imply that there is a HUGE pool of asymptomatic (or very mild symptoms) people who have never been tested but are still infectious. This may be both a good and bad thing, since it would mean the virus is a lot less lethal than it sounds, but it also means it’s completely out of control and will continue to kill people.

(Disclaimer: I’m a mathematician, not an epidimiologist, don’t take my word for it. But neither should you take a physicists word for it, trust the experts!)

Jazzlet says

roberet79

The problem with your argument is that the indications are that at the moment a relatively small percentage of the population has been infected, so

at this pointthe increase could indeed be expotential. This is also likely because it looks like without interventions to reduce it R=3.87, have a look at this study https://www.imperial.ac.uk/media/imperial-college/medicine/sph/ide/gida-fellowships/Imperial-College-COVID19-Europe-estimates-and-NPI-impact-30-03-2020.pdfjrkrideau says

@ 4 robert79

I way be wrong but the authors may be keeping thing simple? At this stage of things it looks like an exponential? Clearly a logistic model makes more sense in the longer term.

I don’t see how this leads to a necessary conclusion of immunity. Shrinking the pool of possibles by immunization or by other means such as lock-down or aggressive tracking and testing should give the same intermediate-term results.

The immunization model assumes survivors have immunity. As I understand it, based on the behaviour of other corona viruses, this is likely but not proven with Sars-CoV-2 yet.

But I am neither a mathematician or epidemiologist.

Dunc says

If the reduction in growth rate were due to the curve shifting to logistic as the pool of possible infectees shrinks, then the time it takes for that to happen would be related to population size. However, it instead appears to be much more closely related to the point at which countries introduce strict measures to limit the spread. For example, the growth rate in China slowed at a point where the number of known infections was a much smaller proportion of the population than in Italy. In all cases, we see a slowing of the growth rate at around 2 weeks after the introduction of strict control measures, which is as expected considering the incubation period.

There have been arguments put forward in favour of the idea that there are a very large number of asymptomatic cases, but these would imply a much earlier and wider undetected spread of the virus, which is difficult to reconcile with the fact that all of the early cases can be traced to known exposures, and community transmission only becomes evident later.

(I’m not an epidemiologist either.)

mnb0 says

“Part of the problem is that exponential growth is not intuitive to understand even though we have some everyday examples of it.”

Actually the cumulative curve is going to be an S-curve, because exponential growth assumes infinity while at the very max the entire population can get infected. For the same reason the growth per day will be a Bell-curve. While I’m far from a math genius I never found this counterintuitive. In fact I took personal precautions about a week before Dutch government issued them exactly due to my intuitive understanding.

Regarding the S-curve The Netherlands have reached the stage that the graph approaches a straight line; the Bell-curve may have reached its peak. The big issue is still if there will be sufficient IC-care.