Dear Florida

Germany on the Mend

Good news out of Germany, seeing two consecutive days of significant improvement in the number of active cases. Additionally, the number of new cases is trending downward as well (after accounting for the regular weekly ebb and flow). Switzerland also continues a slow and steady improvement. In addition, Spain and Italy are very close to halting their growth. Overall, very good news in Europe. Canada, UK, and US numbers are all continuing on a 7-9% growth rate. In the Mideast, Iran's numbers are encouraging, but we must take their reports with a grain of salt. Conflicting numbers continue to surface, particularly around the reported number of deaths.

Unfortunately, there are no new stay-at-home orders in the US. In fact, Kansas Republican lawmakers weakened the order in their state, deciding that their governor's order does not apply to religious gatherings. I'm pretty sure that viruses are functionally atheists, however. Religious gatherings in South Korea were one of the biggest transmission vectors and the hardest to get under control.

Today I am adding a couple of new columns to show the linear and quadratic model predictions for each state. This will help show which states are breaking their trend lines, and hopefully help identify new trends and improvements. Today, Georgia, Texas, Maryland, and Rhode Island are the outlier performers, coming high on both the linear and quadratic models. Montana and Wyoming has shocking reversals in their active cases, seeing significant drops. Maybe the sparse, rural nature of these states enforces a de facto social distancing. It's also possible that their testing is uneven and we will see an equal jump in cases tomorrow.

One thing that I am noticing is that the regular weekly variation is affecting the growth models. This is enough to make linear models seem accurate for early in the week, while quadratic models are better later in the week. In looking into this phenomenon with more long-term models and regularization of the variation over the week, it seems that the weak quadratic model is a better overall fit for many of the states right now. This is interesting, since it would be more consistent with Kiskowski and Chowell's local saturation model. More data over the next few days will help validate or refute some of my suspicions.

Predictions

Some of our readers are understandably having a hard time understanding the numbers and where the analysis is coming from. I thought I would try something a little different by making a few predictions - well, extrapolations really - and showing where they are coming from. I will focus for the moment on the United States, since it's a little more like comparing apples to apples than, say, showing models for Italy and Turkey side-by-side.
First up, New York (State). New York was hit early and hit hard. It still has 1/3 of the active cases in the US right now. Also, a highly cited model keeps predicting that NY is about to peak in the next few days. Let's take a look at the past two weeks, and see how things are looking right now. The blue line is the active confirmed cases in NY over the past, while the purple and red lines extrapolate (predict) those numbers into the next few days. The purple line is a straight line estimate (linear growth). This estimate has been pretty good at predictions over the past week, though it tends to be a little conservative. The red line is an N-squared model, assuming some curvature to the data. This has more flexibility, so it can easily overfit the data, and in fact does a slightly worse job of prediction over the past week, typically overshooting the real numbers. The reality will probably be somewhere between these two lines, but hugging much closer to the linear (purple) line. That means by the end of the day today, we expect the number of active cases to be between 127k-131k. By April 12, the expectation is between 155k-167k.

The next chart shows the behavior of some of the states with the most active infections. Each of these is transitioning from exponential growth to linear, and the N-squared model has been a good fit so far. Extrapolating out, we see that each of these are behaving very similarly, with only Michigan break from the pack and growing slightly faster than the rest.

Finally, a few of the other states, including both states with linear growth and transitional models. These states are chosen specifically because they are stable over the past ten days, so states with new events with a sudden impact on the model, or very noisy data are not show, as they aren't good examples for our explanations. However, the same principles apply.

Wuhan Lockdown Lifted

China has lifted the lockdown in Wuhan province today after 11 weeks. It could be a great symbol of how pandemics can be controlled by strong measures, or it could be too early and lead to a new flare-up. I am very curious to see what happens there.

In Europe, Germany posted their first drop in active cases, and Switzerland continues to improve. There are early signs in the numbers that we may begin to see sublinear growth in Europe in the near future, but the data is not conclusive yet. We may be seeing noise, or simply a realignment to a lower linear growth rate.

In the US, Wisconsin held primary elections today despite strong protests from the Democrats. Results will not be reported until the 13th. It remains to be seen how this affected turnout. In South Carolina, Governor McMaster finally made an official stay at home order yesterday, after many in the state had already begun. The state had already shifted to a linear growth model over the past week, but it is still good to have the leadership on board.

Most States Shifting to Linear Growth

Sorry I didn't post yesterday, but I needed a day off. Across the board, case counts came in lower than would be otherwise expected, but we're learning to expect that on Sundays.

Today, Spain's cases are looking especially good, with only about 3/4 of the new cases that my models predict, even accounting for Mondays being lower on average. Also, Spain, Italy, and Germany are all looking a little better than expected. Yesterday, Switzerland saw the total number of active cases grow by about 4%, but that's not enough to counteract Saturday's amazing drop of -5%. Today, they posted another encouraging drop of 8%, due to a slightly lower than average set of new cases combined with a large number of reported recoveries. France, and the Netherlands are all very well in line with the linear model. The UK, US, and Canada are all still not on the linear growth model yet, but appear to be slowly heading in that direction, as they come in under the exponential growth curve. If they continue at this rate, I expect Canada and UK to be linear within a week, and the US shortly after, but there are a lot of heterogeneous components to that actually happening. In the Mideast, Iran has fewer confirmed active cases, but that all comes from an increase in resolved cases, as new cases remains constant. Israel is doing slightly better today, and Turkey is still on track.

In the United States, most of the states with a large number of infections are switching over to the linear growth model. Maryland is still exponential, while Pennsylvania and Texas are still in a transition model. Louisiana and Tennessee are unpredictable right now, so no telling what will happen there, yet. Arizona and Alabama look like they are continuing with exponential growth. Connecticut had a bad day, with 1214 new confirmed cases (22% growth), but that looks like an outlier. The states with under 2000 cases are mostly unpredictable right now, with especially erratic behavior out of the Midwest. Many of these areas have limited resources and logistics for testing and reporting, contributing to the noise. Taken as a group, however, these states are showing distinct exponential growth.

I am still searching for an explanation to the linear growth effect we are seeing. While there are plenty of epidemiology papers describing late-stage linear (or at least sub-exponential) growth, these are all saturation effects that do not accurately describe the scale or duration of what we are observing. Admittedly, the stay-at-home behavior we are using to fight this epidemic is not something we have seen at this scale before, either, so I'm not too surprised. However, I am still hopeful that we can better understand the cause and model the core behavior so that we plan a better exit out of this situation we have found ourselves in, without having to wait for vaccines or cures for something as hard to defeat as the common cold.

Local Saturation Theory

France continues to adjust their numbers to align with international standards, leading to a significant jump in reported cases today. That is not an actual growth in cases, just continuing to move to a new baseline. The change from yesterday is left blank today while they sort things out. Iran is coming in a little low today, but that could be noise. The other countries are all coming in as expected.

Switzerland continues to see a decline in active cases. This is from a roughly constant number of new infections being overwhelmed by a higher rate of people recovering. If the linear model continues (i.e. the number of new infections stays more or less constant) their active cases should continue to decrease more quickly over the next week, before slowing down to asymptotically approach a steady level where people recovery just as fast as they are catching the disease. Which is about as weird as it sounds. To take this out to the extreme, if nothing changes, that means that about 1000 new Swiss will be infected every day. The population of Switzerland is about 8.6 million people. Even if we are undercounting the number of cases by a factor of 10, that means it will take 8,600,000/10,000 = 860 days = 2.35 years for the entire country to be infected. That assumes that it hasn't mutated enough by then, so that a previous infection from COVID-19 means that your body is still immune to COVID-21.

The US isn't quite at a linear growth model yet but with a majority of the states with the most infections all imposing comprehensive stay-at-home orders, things are getting closer.

So what is causing the linear growth? According to research on Ebola from Maria Kiskowski and Gerardo Chowell, the transition from exponential to linear growth is expected as the disease is "saturating" the local communities. The disease then passes through in a "wave" or a "bubble" through the community, infecting new people only along the leading edge of the "bubble" while the interior is fully saturated. This would imply one (or more) local areas of high infection that are slowly working their way through a region. Events could happen where a bubble hits a kind of connectivity dead-end, or a particular connectivity crossroads could split off a second bubble, but the basic linear behavior is maintained.

If we follow this theory for what we are seeing with COVID-19, you can't consider a city or a neighborhood as a bubble. One idea then is that stay-at-home orders are reducing the number of interactions enough that these bubbles can form a saturated area in a fairly small region. Perhaps a single household is only interacting with a very few number of other households now, and a bubble can form out of just a few such households. A full cut-off from the rest of the world would cause the bubble to collapse, with nowhere left to go. However, occasional forays into 'necessary services' such as a grocery store could provide just enough exposure to inadvertently cause one more temporary link.

My problem with this theory applied to COVID-19 is that there is not just one bubble. Most states do not have any significantly high saturation counties, and nearly all infected states have at least a few cases in more than half of the counties. So that would mean bubbles would have to be tiny, and saturation areas would be being created and dry up all the time. That they would continue to cancel each other out equally is awfully convenient. It also implies that the type of stay-at-home order doesn't truly matter to connectivity, since not all states and counties are implementing the same thing. While this seems plausible for some epidemics, it doesn't seem like the full story for this situation.

Let me know your thoughts and ideas in the comments below. I'm interested to see if we can crowdsource an answer to this. Because I don't know about you, but I don't have 2.35 years of pasta stockpiled.

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.

Good News from Switzerland

Big news today- Switzerland has managed to REDUCE their number of active cases of COVID-19! It's too early to tell if this will be a regular trend yet, but it is an amazing accomplishment for their management that they managed to turn from an average 25% daily increase just a week ago to the point that there are more people who recovered than tested positive.

Other Western European countries are showing steadily low rates of growth, with Spain getting extra recognition for finally getting down to 10% growth after very recently looking like they would be the next place to break down like Italy. Speaking of Italy, their number of cases is still growing, but they have managed to maintain their growth rate under 10% for six straight days.

The US is not looking nearly as good. Today, they crossed 100,000 total confirmed cases. While the national growth percentage continues in the low 20s, that is mostly due to NY and WA keeping their numbers lower. Discounting those two states, the rest of the nation is still averaging 28% growth rate for the past few days.

Below, you can see the growth rates for individual states for today and the average over the past few days. For reference, I have also colored the state names based on whether they have a Democratic governor (blue) or a Republican governor (red). This is relevant, since Trump has increasingly made it clear that he will direct aid and supplies based on how much he likes the governor and how much they toady up to him.

Also, to the right you will see if the state has a shutdown/shelter-in-place order. In many states where the governor has not called for a shelter-in-place, local cities and counties have issued their own. Sometimes, if your president and your governor are unwilling to lead, your county commissioner has to be the hero. You may notice some correlation with the governor's party affiliation here (or mayor, in the case of DC). There is also some good correlation with the average infection rate (though some states have issued the order too recently to have had an effect on confirmed cases).

Here are the numbers for today. US and Canada numbers are still preliminary, and may change.
The good news is that the European numbers are staying low, like we saw earlier in the week. Social distancing and shutdowns can work, if we all agree to follow them.
The United States today hit the grim milestone today of having the most number of cases - total cases and active cases - in the world. Almost half of that is concentrated in NY. So, as we see shutdown procedures starting to have an effect in the Big Apple, it is driving the nation's infection rate to 23%. This is a huge improvement over the 35% daily growth we have seen. Unfortunately, the other states have not gotten the memo yet.

I noticed there was something odd in Europe's numbers a couple of days ago, but now I have three consistent days to show that they are not a fluke. Across the Western European countries, their daily infection rates just plummeted. Here's a list of the countries that I'm tracking, showing their average daily growth in infections from two weeks ago, compared to last week, compared to the past three days:

Germany: 30% -> 25% -> 10%

France: 26% -> 17% -> 13%

Spain: 37% -> 22% -> 17%

Italy: 18% -> 14% -> 8%

Switzerland: 30% -> 25% -> 13%

Netherlands: 20% -> 20% -> 10%

These numbers could well change in the future, but this is what is happening right now. Note that we don't need this daily infection rate to get to zero to make this pandemic manageable. Given what we are seeing for average times to recovery, something in the 7-10% daily infection rate could actually lead to a reduced number of active cases, as people can get better faster than new individuals are being infected.

The UK is noticeably missing from that list. Their policies are keeping them in the low 20% range (actually 23% last week, 19% the past three days) for now. In the US, we are seeing lower rates for NY, CA, and even FL, even as the rest of the nation is accelerating. 

The lesson here: stay safe and stay isolated, regardless of whether your President/PM/Governor is ordering you to or not, and we can see things turn around.