Notice: because Dr. Frank’s posts are now available up to the minute by clicking here, we will not be providing daily updates of his exemplary work. Readers are encouraged to join the Dr. Frank’s Models Facebook group. We look forward to working with Dr. Frank on future projects.

Dr. Douglas G. Frank is the scientist/mathematician behind this blog.

Dr. Douglas G. Frank

Dr. Frank received a B.A. in Chemistry from Westmont College in Santa Barbara, California. He qualified for his doctorate at the University of California, Santa Barbara before transferring to the University of Cincinnati in 1986 as part of the Ohio Eminent Scholar program. In 1990, he received a Ph.D. in Surface Analytical Chemistry. After graduating, he formed “ADAM Instrument Company, Inc.,” named for the new surface analysis technique he discovered during his graduate studies. The “ADAM” technique brought him international acclaim, and his work was featured in several scientific books and international journals, including cover articles in Science and Naturwissenschaften. He has over 50 scientific publications, and is internationally regarded as an expert in Auger spectroscopy.

Dr. Frank currently serves as the Math and Science Chair at the Schilling School for Gifted Children in Cincinnati, OH. He is also the President of Precision Analytical Instruments.

Good news about chart and commentary availability:

A facebook group called “Dr. Frank models” has now been established. This will enable you to follow the thought stream of Dr. Frank, and to receive the models as they become available. This blog will remain as the “reader’s digest” for those who do not have Facebook or prefer a “once-daily dose.” The Facebook group can be accessed by clicking here and then requesting membership in the group.


Based upon the data and the 3WM, we should see total deaths tomorrow [reference to April 2] in the US approaching 800. Keep a level head. It’s going to be like that a few days, we are on the peak.

“Someone already knows that.”

When I published my discovery in 1990, our research group got a lot of media attention. During the summer, the local school district sent teachers to our lab to observe and learn the “scientific method.” It was nice having the company.

One day, a Jr High school teacher asked me, “I’ve been watching you do these experiments all week, but I can’t figure out why. Why don’t you just look it up?”

I told her that I was exploring something that had never been explored before.

She answered, “Nah. Someone must have done this before. Someone already knows the answer. You should just look it up.”

Never going to make discoveries that way.

=== So when all of this is over, there will be a bunch of so-called experts claiming that they knew all along what was going to happen. Where is their timestamped prediction?

This has never happened before in the history of the United States. I am exploring it. No one knows the answer yet.

For State graphs, click the state below:

Each state is updated daily, or as often as Dr. Frank supplies an updated chart.

Alabama     Alaska     Arizona     Arkansas     California.    Colorado     Connecticut     Delaware     Florida     Georgia     Hawaii     Idaho    Illinois
Indiana     Iowa     Kansas     Kentucky     Louisiana     Maine     Maryland     Massachusetts     Michigan     Minnesota     Mississippi     Missouri
Montana     Nebraska     Nevada     New Hampshire     New Jersey     New Mexico     New York     North Carolina     North Dakota     Ohio    Oklahoma     Oregon     Pennsylvania     Rhode Island    South Carolina     South Dakota.    Tennessee    Texas     Utah     Vermont.    Virginia.    Washington     West Virginia     Wisconsin     Wyoming


Publisher’s Note: All words below this line are compiled from Dr. Frank. Other than ordering and organizing the comments and the charts, we (DPH) have not edited any of his comments because that would be out of our area of expertise. The comments were given in a social media context, so they have a friendly, conversational tone.

“Note to Self”

The rise of an epidemic is easier to predict than the tail, especially because it spreads clandestinely, so it behaves statistically more ideally at the start. The tail is harder because how many rest homes get infected is tricky to guess ahead of time, and some states pay attention to stay at home orders, and others do not.

What’s cool about the 3WM [three wave model – see below], is now I have the tiger by the “tail” (haha). Since I can adjust the parameters of the tailing edge of the peak to match what we observe, I should be able to make an accurate forecast, and we will have a good model with which to judge our progress. (Use a drunkards walk)

“Bayes’ Theorem”

Say you have a jar full of a mixture of 10,000 black and white marbles. What is the proportion? How could you find out?

Well, you could count them all. Tedious. Or you could *assume* you know the ratio. For example, you could assume it is 50/50. Reasonable. Then start taking marbles out, one at a time, and revise your proportion as each marble comes out. Assuming they are well mixed it doesn’t take long to converge on the right proportion.

Using this approach, one can calculate anything. And I mean *anything.*

Make a starting assumption, and then start revising it with data. The starting assumption doesn’t even have to be close.

Latest World and USA Model:

(Note: Click on the heading to get a page of past charts for that heading)

Three Wave Model (new)

“Three Wave Model I”

First, let’s describe the model. The Three Wave Model combines three USA epidemics into one. All three waves are derived from the reported data, with assumptions that have evolved as I have been tracking the data using the scientific method and applied mathematics.

It is not mere “curve-fitting.” It is a constrained model which uses parameters and intuition derived by modeling this and other epidemics. Constrained means not only the values, but also the shapes, timing, and parameters must all line up.

That is a lot of constraint. We are talking thousands of values. And when the model is working right, it converges to what we observe, I don’t have to force it. That’s how I know when I am on the right track.

The graph summarizes the results of the Three Wave Model. The vertical axis is deaths per day, and the horizontal axis is calendar date.

The graph also combines real data with the models so you can compare. The dark black circles are the actual number of reported deaths in the country each day. The red dots are the actual number of cases reported in the country each day (scaled down to fit by a factor of about fifty, so you can compare shapes).

The first wave is the one that worked really well for about two weeks. It describes the general spread of Covid-19 throughout the US. (light grey peak)

The second wave is New York (light blue peak). While tracking the first wave, the data suddenly deviated from the model, so we knew something was up. Yes, it was New York. It now gets its own wave, which is now fairly stable and tracking well (see earlier posts today).

The third wave is the new wave that has just begun in the last couple of days (light pink peak). Everything was going swimmingly, combining the first two waves, when suddenly we saw a new inflection point. Again, deviation from our model showed us that something was up.

And something certainly is up. The East is experiencing a new round of infections. Some people here have suggested a Spring Break effect, others evacuation from New York. Whatever the cause(s), there it is, plain as day, in the data. So we need a new model so we can track it too.

Most of the rest of the country is looking good, and as I’ve been doing the daily updates on the states I find myself in an increasingly giddy mood (not good for science), but then I do updates in the East which balance me out again.

If we didn’t have a model to compare the data to, we would not be able to recognize what is going on. Our model has been hard won, and now it is delivering the goods. You start with some reasonable basic assumptions, use the scientific method to test and correct your assumptions, and steadily develop a model that accurately describes what you observe.

The sum of the three waves is the biggest peak in the graph (dark red peak). The units are deaths per day. If the model is working, this peak should match the reported values each day. If it doesn’t, something is up, and it is “back to the drawing board.”

The big red peak is the “Three Wave Model”

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The Three Wave Model (3WM) will always work well for the left half of the peak, because that is history that doesn’t change.

The right half of the peak is what everyone wants to know. How long will the tail be? Usually they stretch out a bit, due to additional “break-out” infections (such as in nursing homes).

How do we know the third wave is correct? We don’t. So far, we’ve only had a few days of data to characterize it. Covid-19 spreads clandestinely at first, so can emerge suddenly. (Really it is just hiding in the noise until it is big enough to stick out.)

And just like the membership on this page (see the graphs I’ve posted), there is an exponential phase at the beginning of every wave that eventually changes over to “bell curve” behavior. Three data points are not enough to know very well.

Fortunately, we don’t just have to make the shapes line up in this curve. We ALSO have to make the shapes line up in other graphs (like the deaths projection)… and they have to remain internally consistent (reconcile). So there are lots of ways to know quickly if the third wave needs adjusting.

I will likely need to tune it for the next day or two (like in the membership graph). Then, we will leave it alone for a while. Of course if the data coming in are noisy, then we will have difficulty learning the parameters of the third peak. Garbage in, garbage out.

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We have removed these models from this page because they have been replaced with the “Three Wave Model” above. Click on the subtitle above this line to see the previous “Big US Model” historic charts.

Covid-19 “US Deaths Tracking”

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Worldwide Models

Note: for worldwide models we are only adding the latest graphs. We are not keeping updated records and comments as we are doing for the states.

Covid-19 “Quick Look at Australia”

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Covid-19 “Quick Look at France”

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Covid-19 “Model Update for S Korea”


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Covid-19 “Italy Model Update”

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Covid-19 “Quick Look at India”

“Playing with matches in a room full of dynamite” is how I feel about this one. I hope I’m wrong, but it looks like the initial infection is complete (red hump), and the real epidemic is about to begin. Notice how the model was tracking the initial infection on the dot… If this were not merely a “quick look” we would have included probabilities based upon population densities, etc. and tried to predict the next, large peak. But there are only so many hours in the day.

There is a really useful map of parameters for just about every city in the world. ( It contains the types of parameters we need to predict how high the next peak will be. My earlier “models” drew on these.

And the numbers are likely to be very big. 

I find the Indian people to be lovely. Some regions have a marvellous tradition of brilliance in math as well.

Dear God, please grant them peace through this ordeal.

More on this as it develops. For now, praying.

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Covid-19 “Model Update for Iran”

Just added data, but also marked the inflection point in the graph. When things settle down, I can teach y’all the calculus that makes inflection points quantitatively obvious (not just visually).
The inflection point is where the progress of Iran’s epidemic suddenly deviated from a model it had followed dead on for weeks. So we suspected a significant second infection.

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Covid-19 “Model Update for the UK”

Sounding the alarm. Just added data. Cases up, in tandem with deaths. Time to pray for a small infection.

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Covid-19 Model Spain


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Covid-19 Model for Cambodia

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Covid-19 Model for the Philippines

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Covid-19 Model for the Netherlands

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