(4/1/20) I’m a little tardy with my weekly update for a number of reasons: (i) I was hoping to see some rolling over of the near exponential growth in the death rate so far still evident in almost every country and U.S. state last week, except China and S. Korea, and was waiting to report that, and (ii) I have a day job!
Unfortunately, the news remains grim for nearly all of the countries that we follow.
The Model for China
The Figure below shows the pattern of daily and total deaths for China. This pattern is what we hope to see in all countries. If social distancing is initiated soon enough then one should see a peaking in daily deaths about 3 weeks later (the time from infection to death). This is a big if and failing in many countries.
Our Death Rate Model
The Plots below represent a model for death rate and accumulated death. For our model we assume a Gaussian (Bell-curve) distribution for the pattern of increasing followed by decreasing deaths per day and the total deaths (which for your math nerds is the integral of the death rate). Other distributions can be used and some may be more appropriate, but a Gaussian starts rising exponentially followed by a near linear region and then rolls over to a peak, which is behavior we expect from an epidemic. My model is primarily for tutorial purposes, but might also have some predictive capability. This should look familiar to some as it resembles the so-called “flattening the curve” discussions you have read about. Frankly, I’m not sure I subscribe to the currently trending version of the flattening of the curve theory as that says if you exercise reduced human exposure you peak later but at a lower death rate. I think you must peak earlier not later if social distancing is working. Regardless, both theories do predict a much greater peak death rate by delaying social distancing.
According to our model and the discussion above, if social distancing is working then we should see the peak in the death rate about 3 weeks later. Sadly, once a population reaches the peak in death rate, they are only at the halfway point for total deaths as there will be just as many deaths on the way down the curve as going up it. (assuming the behavior is symmetric). You will notice that our plots resemble what was observed in China (Figure above).
One reality that needs to be stressed is that if a population delays social distancing by just a couple of weeks it can have a profound effect on the total deaths. In our model we show two cases: social distancing starting on 3/1/20 vs. 3/15/20. We allow for the growth rate to be similar in these two cases, but the former case will show a “flattening” of the curve sooner than the latter case, which can be seen occurring about a month after 3/1/20. Assuming this model has some validity, it predicts that by delaying social distancing by just two weeks will lead to a death rate peaking two weeks later and at a 5-fold higher value. The accumulated deaths are plotted on the right plot. For these two cases the total deaths are 50,000 and 250,000, respectively. [Note: This model assumes a width at half max of 6 weeks, when in fact historical trends suggest a narrower distribution. This would make the severity of a two-week delay even greater than a 5-fold factor.]
So where does the U.S., who waited until 3/16/20 to implement social distancing, fall on these plots. Well over the past week the death rate was 3,280/week and on 4/1/20 the number of deaths exceeded 1,000/day, which is now the highest in the world, which projects to at least 7,000/week by this coming week. Sadly, since the blue curve peaks at 7,000 deaths/week, and the U.S. death rate trend (Plot below) is nowhere near a peak, we are probably more closely following the orange curves. This would then project a total death count as falling between the 50,000 and 250,000 levels but much closer to the latter. We are in grave danger. If there is any solace in this conjecture it is that we will reach a peak in early May and pronounce a recovery by June/July.
I solicit critiques on this model and/or referrals to other models by the presumed many experts out there.
Weekly Statistics Around the World
OK now for more grim statistics. My blog of 3/23/20 (Blog #6) made forecasts for deaths and new cases using not an exponential, but a binomial, function, which I believed would be more reasonable for allowing for some slowing from social distancing. I was very wrong and most countries continued on their exponential death growth rate. New cases may already be tailing off, but we won’t know that for sure until 2-3 weeks after from the recorded deaths. Given that we are about 2-3 weeks into social distancing in most countries (at least ours started 3/16) I had expected some tailing and maybe peaking by 4/1/20. This is not generally seen, certainly not yet in the U.S., but there are some encouraging signs in other countries. Below are plots of growth rates for the most seriously affected countries.
Key observations are:
- The U.S. death rate continues to grow at an alarming rate. Other countries not showing a reduction in the growth rate include France and the U.K.
- Spain may be showing signs of a rate decrease, but it is too soon to tell.
- Iran particularly and to a lesser extent Italy are showing signs of rolling over on the death rate curve.
- S. Korea and China are of course models for us all in being well past their death rate peaks. We should be keeping an eye on whether they have a death rebound at some point so we can learn from that.
In the Bar Chart below we look at growth rate statistics in a visual way by showing how countries are doing in reducing their death rate by plotting the percentage change for one week over the next. We also place a qualitative symbol for countries in control (green), starting to control (yellow), and still no evidence of control (red).
Late breaking: Here is a better way to visualize trends in death rates. (I owe this to a great video that you should watch: https://www.youtube.com/watch?v=54XLXg4fYsc, though it plots cases, and not deaths, which is a less precise measure). By plotting death rate vs. cumulative deaths on a log-log plot one can immediately see deviations from exponential growth representing the start to recovery.
Key observations are:
- Similar conclusions are reached compared to the bar chart above.
- China and S. Korea have significantly decelerated their death rates. S. Korea as noted in earlier blogs never really reached exponential growth having gotten well ahead of the epidemics usual pattern of growth.
- Iran seems clearly on the path to recovery (also evident in the daily death rate plots above) and there is a hint now over a week that Italy may be starting to roll over on the death rate curve.
- The U.S., Spain, France, and U.K. are still showing approximately exponential growth.