Linear Models

The linear model continues to preoccupy my mind. Today confirmed for me that many of these countries and states are now behaving as a linear model, by successfully predicting the numbers for the selected regions. Comparing against eight different models, the linear growth was consistently the best predictor. See the comparison numbers in the charts above.

The good news out of these numbers is that Switzerland is continuing to slightly reduce their active cases of COVID-19, as people are recovering faster than it is spreading. This marks three days in a row, establishing a clear pattern of slow recovery. Also providing hope, the Mideast seems to be improving, as all three tracked countries seem to be switching over to a linear model now.

I am still puzzling over how a region can maintain a linear growth rate, regardless of the size of the region or the number of infected. This means that a roughly constant number of new people are confirmed each day. In looking deeper, it turns out this is not strictly constant after all. All across the world, there is a week long cycle of ebb and flow within that linear growth, so that growth peaks in the latter half of the week around Thursday, and reaches a consistent low around Sunday/Monday. This is most easily seen in South Korea, which has been holding a roughly steady rate of new cases for almost three weeks now. This could be a reporting anomaly, the result of testing or results availability, but it was interesting to see anyway.

Day-to-day changes can make it easy to lose the significance of this model change within the background noise. To see how different a linear model is from the exponential, consider projections of active cases in the UK. In the first graph, you can see that the country switched over from exponential growth to linear growth on March 29th (five days after the stay-at-home order was issued, which happens to be median time from infection to symptoms). The difference between the two models over the period of March 29th to April 3rd is distinct. However, continuing this trend out just one more week can make a huge difference, as the next chart shows. You can imagine what happens after that.



This has important implications on how the pandemic plays out. On one hand this is great news, as it means the pandemic can be contained and the curve can be flattened much more effectively than we had hoped. However, if the linear model continues indefinitely, as South Korea suggests might be the case, then the lockdowns may need to stay in place much longer than anyone is anticipating. Returning to life as it was before the spread of COVID-19 would restart the exponential growth again. Until something changes on the healthcare side, such as a vaccine or a realistic implementation of an aggressive test-track-isolate strategy, the regions willing to enforce stay-at-home orders may be in a bit of a stalemate.

In other news stateside, Alabama has issued a stay-at-home order starting on April 4 and Missouri has decided to start one April 6. South Carolinians may be feeling a little left out at this point. Fear not, though, for neighboring Georgia has effectively weakened their stay-at-home order and reopened the beaches, as Governor Kemp took inspiration from Governor DeSantis (FL) and decreed that state law overrides any local regulations that might be more restrictive.

The Models are Changing

Switzerland is doing particularly well today, with no reported growth in active cases for two days in a row. The rest of the countries are holding steady, with the exception of France. France's numbers today have a big asterisk, as they reported a whole new host of 'updated' numbers based on a difference between how they are tracking these numbers versus the international standard. For instance, deaths of COVID-19 patients in nursing homes were not being counted. It will take me a little while to integrate that new information into the reports.

In the States, MS, OK, TN, and TX have become the latest states to enact state-wide stay-at-home orders. The governor of Oklahoma actually issued their order yesterday, but I had missed it at the end of the day. That leaves five states with partial coverage of their stay-at-home orders, and four more with no orders at all. At this rate, Missouri and South Carolina are due to change their mind in the day or two. For the Dakotas and Wyoming, it may take awhile before they feel threatened enough to act.

Also of note today, Florida's governor quietly issued an edict stating that no local governments could enact tougher rules on shutdowns or stay-at-home orders than the order he had (belatedly) issued for the state yesterday. This is notable, since throughout March, Florida only kept its numbers down through a series of mayors and county commissions closing down many of the most populous cities, including the hot spots in South Florida. For instance, this has the effect of lifting the curfew that was in place in Leon County (which happens to contain Tallahassee, the state capital). The fact that most of these locations trend Democratic is just one of those odd coincidences. Their display of initiative was not taken well and such leadership will not be allowed to continue in the future. It will be interesting to see if Governor DeSantis is challenged on this order and how it is enforced.

Louisiana is seeing a major jump in active cases over the past few days, rising to 43% growth today. The governor is claiming a "log jam" of testing is the reason behind the sudden increase, though the Mardi Gras celebration is definitely on everyone's mind. Given that the stay-at-home order was enacted 10 days ago, this surge is definitely unexpected.

For the most part, however, the major infection sites are maintaining steady growth numbers, and typically coming in under my model expectations. Which brings me to the main topic of today's post: the models are changing.

Previously, most regions were following an exponential growth model. This makes sense, as each individual has a chance to infect N people around them every day. Once infected, those individuals can infect others around them, too. This leads to more people, each of which can spread the infect to N new people, on top of the ones that were already infected and spreading the disease. This model matches the data very well in regions that have not taken steps to protect themselves. For instance, look at Tennessee or South Carolina, which only enacted a stay-at-home order today. The blue line is the actual cases over the past ten days, while the red line is the predictions of the exponential model. For reference, the gold line is the best-fit linear model.

Now look at a region like NY, which enacted statewide protections on March 22, with additional measures taken even before then. The data is almost a perfect linear model. The best exponential model is still shown in red.

Finally, to round out the point, examine a state like Florida, which has seen a federation of local protections with a lack of any statewide protections. The growth sits perfectly between the two models.

The same is true in other countries as well. So what does this mean? What is causing the residual linear model? What drives the slope of this linear growth? These are questions that I would really like to answer, as they may help reveal when this will peak, and what measures need to be taken to finally reverse course. Stay tuned.

Stay-at-Home Orders

Western Europe is maintaining a steady daily growth rate, with only France standing out as having a higher, uneven rate. The UK and Canada are both starting to establish a pattern of percentage growth in the low teens. The US seems to fall into the same category, but as we will see in a moment, the country is anything but homogeneous.

In addition to the Western European countries, I am also adding the Mideast to my watch list. Unfortunately, I don't have good historical numbers for many of those countries yet, so for now I will just present these three. The latest reported numbers are not shocking, but Turkey has real cause for worry around 19% growth. However, these numbers are volatile, and not completely trustworthy.

In the United States, a wave of states including AZ, FL, GA, ME, MS, NV, and PA have all issued a statewide stay-at-home order, although FL, GA, and MS won't go into effect for another two days. Repeatedly, Trump's "changing demeanor" was cited as reasons for governors changing their minds. Hopefully this will lead to greater compliance and cooperation across all states, so that we have fewer attempts to create "Woodstock-like gatherings" in defiance of social distancing. However, there are still nine states with only partial coverage of stay-at-home orders, and five states with no orders at all. In a remarkable coincidence, all 14 of these states have Republican governors. Apparently safe practices during a pandemic have become a pawn in the culture wars.

Tuesday Update

Submitted tonight without comment. Sorry, it's been a busy day.

Flattening the Curve

Western Europe is mostly holding steady. Germany posted a phenomenal 0.1% growth in active cases today, but I suspect that is a blip of noise, like France's 3% yesterday or Switzerland's -3% a few days ago. It's a great sign when it can get that low, however, even from a random undercount. Looking to some of the other countries of concern that we raised, Turkey's reported numbers are stabilizing somewhat, suggesting that the sudden surge was a result of under-reporting catching up. Iran hasn't changed.

The story in the US is looking a little better today. The national growth rate dropped to 14% today, marking three straight days under 20%. For a change, it isn't just NY and a few states driving the improvement- across the nation, growth rates were better than they were yesterday, showing that the message is starting to get through. To underscore that, KS, MD, NC, and VA all enacted statewide stay-at-home orders today. That makes a total of 22 states that have enacted stay-at-home orders in the past 6 days, which is the earliest we would expect to see results. While we are still a long way from a peak, Americans are getting smarter and buying some time.

Sunday Update

Numbers across Europe look good today. No country saw reduced numbers of infected today, but the growth rates were all smaller than yesterday. the difference between a 5% daily growth and 15% is incredible. Even Canada and the UK are seeing serious improvements from their shutdowns.

In the US, the daily growth rate is lower than it has been since they hit 100 confirmed cases. Again, this national number is mostly driven by NY, but other states are looking good today as well. In fact, WA posted a reduction in active cases today.

One caveat is that there seem to be several regions that are late reporting their numbers today. Many of these lower growth rates could be turned around with updated numbers tomorrow.

Europe Improves, Focus on Iran and Turkey

As a whole, western Europe continues to see a slow, steady decrease in the rate of infection. Italy and Spain, two of the most active countries, saw significant decreases to 5% and 7%, respectively. Switzerland jumped back up to 10% growth in active cases, but that gives them a 7% growth over the past to days, which is very admirable.

There are two more countries to keep an eye on as well, that may be flying under the radar right now. First, Turkey has seen incredible growth in confirmed cases over the past three days, exceeding 50% daily growth. This may be an artifact of reporting, since Erdogan's government keeps a tight grip on the flow of information. However, this is a very scary trend for even a short time. Given the country's critical strategic location, could have dramatic implications for many other countries in the region.

The other country of concern is Iran, another country with dubious reporting quality. Early on in the pandemic, Iran was front and center as their case numbers jumped way ahead of others. After awhile, the Iranian government seemed to get things under some semblance of control, reaching daily rates under 5%. However, reports out of Iran are painting a dire picture more recently, with reports of panic and chaos. Rumored remedies have driven a hysteria for strong alcohol, in a country that outlaws drinking. The alcohol that has hit the black markets is often tainted with toxic ethanol and bleach or other chemicals, leading to horrible cases of blindness and death. Some regions even claim that they are seeing more deaths from alcohol related poisoning than from COVID-19. Against the backdrop of this breakdown, new cases are surging again, and the exponential curve has re-established at 14% per day and growing.

The United States continues to see the rate of active cases drop to 18%, largely due to NY and a few other highly active states like CA, WA, and IL. NJ still hasn't been able to bring its new cases under control, and unless their shelter-in-place rules can start yielding results soon, they will have more cases than NY in 10 days. Over the next four days, we can expect MA to pass MI and CA.