Alan,

The zip file contains the package. If you use Install... in the file menu to load it, some limited documentation will be available. The file ReedPaper.nb uses the package to check all the math in the Reed paper noted above. DPLNTest.nb shows some of the basic functions, and calculates LMoments.

I couldn't get the zip file to load as an attachment, so I put it here:

Let me know when you have it and I will remove it from the link.

It was written nine years ago for version 8. I did test it with the two notebooks above and it still seems to work. Let me know if you have any questions

Attachments:

DPLNTest.nb

ReedPaper.nb

Clearly the data are showing a different structure late in the epidemic than was seen in the early data. I was looking yesterday at Massachusetts' website where they have the case rates per 100000 for each town. The case rates vary wildly with very high rates for low income areas where there are large immigrant populations and English isn't spoken in many homes. Today MA just made its raw data available. MA Data I have just started to look at it, but it is detailed enough that it might provide some insight into how the data unfolded, I am thinking right now that since most of the control measures required dissemination of information, different communities processed the information at different rates, with the low income areas lagging very badly in some states. If that is true we may have a mixture of models, but we may yet see an exponential tail if quarantine measures eventually succeed.

I lived in MA for forty years so I am familiar with the communities. I now live in AZ where the counties are very large, but the two counties which are mostly Indian Reservation have per capita case rates seven times higher than the rest of the state. NY and NJ maybe just starting to approach the tail behavior, I think it will be exponential, but rates are still varying daily when I fit the last ten days of new case results.

For the fun of it, here is a notebook looking at a fit using an exponential startup with a linear tail:

Brian,

If you use Mathematica I wrote a package for the lognormal double Pareto distribution. It was written for version 8, but I checked it this afternoon and it still works. I used it to go through every equation in that paper in a notebook. Let me know if you would like those files.

I was originally interested in it for stock market volatility measures, but they are modeled as well by the generalized extreme value distribution with one less parameter.

Bob

There is another interesting distribution with two power tails, the double Pareto lognormal distribution. I have it implemented in Mathematica if you would like to fool around with it. The paper describing it is attached.

Attachments:

dPlN.3.pdf

Dietmar, forgive me for pigeon-holing this response as an "Argumentum Ad Vericundiam"; specifically, the physics teacher approach. This joins "aerodynamic lift as impulse/momentum of air molecules with due attention to inhomogeneities from micro gusts" and "aerodynamic drag as a function of velocity" [in this case because the calculus offers reasonable solutions, where "drag as a function of velocity squared" needs one to fall back on iterative methods for decent sizes and shapes.] Which is a slightly derogatory way of suggesting that there are emergent properties in building blocks where chaotic conditions are subsumed. No insult intended....

Brian W

Day #48 of my data was rather like that: a step increase when NY State and several others decided to include non-hospital cases and fatalities in their totals. I excluded two prior days data - which is itself a kind of massaging of the input - an original sin of sorts.

Robert, you asked about my data source. I use Worldometer.info. It provides data for a number of countries, and includes US by state - but no smaller divisions. I collect it daily. With a little effort, you can recapture prior data by clicking data points, as I recall https://www.worldometers.info/coronavirus/country/us/?fbclid=IwAR0fWg98jsNTSUX0_f2s0s7xnPE1pdClvUGyphK7YC_l24od34kMmdnLyGE

Brian W

Agreed. Most of the epidemiologic models are focused on the dynamics of the disease and recovery rates. Effective quarantine, however, can shut down an epidemic very rapidly, and the epidemiologic models also require transmission assumptions that are not known for a new disease.

I do suspect the S. Korea finale was unusual, and that we should be able to predict the rate of decline from the data pretty soon, then modelling with the tail behavior of statistical distributions might prove useful. If the decline is faster than exponential, then the gamma distribution offers the range of tail behaviors between exponential and normal distributions, if it is slower, then a Pareto distribution could model it. If there are recurrent new small outbreaks, which might be what happened in S. Korea, then no mathematical model will succeed.

Dietmar,

I am not sure what to make of the dynamic the moving average introduces, but mid March most of the datasets show a jump in cases for one day, but that wouldn't affect deaths. I think probably the cyclic phenomenon is introduced by using a linear moving average on data that is more or less growing exponentially. I think if you want to do smoothing that is consistent with the data, you should try an ExponentialMovingAverage, I have also looked at the parameters as the data accumulates. The L parameter in general keeps growing, while k decreases. My best answer is that the early data doesn't have any particular shape to it, rather it is chaotic -- you could fit most anything to it. There is probably good reason for this. Public health officials were playing catch up; the spread of the virus was much wider than was realized. So initially there was a mix of unrestrained viral spread and minimal quarantine. So we are seeing a situation that doesn't fit any model. Eventually the quarantine effort becomes more effective, but it may not become perfect and behave as a leaky quarantine. In the end the South Korea data, showed a persistent growth not at all consistent with the logistic model. Notebook attached.

Another interesting way to look at models is that the logistic model is the same as multiplying L times any unimodal statistical distribution. I am also attaching a notebook showing what fitting with that idea shows. Basically if we want to predict when it will end, The tail behavior of the distribution is what is important, and it is the right tail that will be important, because it doesn't really matter if timing screwed up the initial data accumulation. The ultimate conclusion I think is that you cannot make predictions unless you know in advance the shape of the entire curve. If the tail behavior is exponential, then the logistic distribution should eventually succeed, but it wasn't in S. Korea.

One thing that probably is predictable is that once there is a clear decline in case rates, it should eventually end, and we may be able to detect the shape of the decline. Right now we are probably still too close to the peak in the U.S.

The attached notebooks are not annotated, but you clearly will be able to understand them.

Attachments:

SKorea.nb

StatidticalModels.nb

No, the fits are done with the cumulative case data, then the fit is used to predict what the daily rate changes would be according to the fit of the cumulative case data up to that point. Growth rate changes are what will be noticed in the news. The fits raise the interesting question that the early data may not be very good as the rates derived from the daily log differences of the reported cases do not fit any pattern. Here are the plots for New Jersey including today's data.

Using the default fitting, which uses Norm to minimize the fit residuals, the early data are not adding much to the fit as shown by the last log plot. Either the dynamics are changing after the peak or the early data are not very good. The plots of the early daily rates show a chaotic pattern suggesting the problem may be with the early data. Using the default fitting method to the cumulative data limits the influence of the early data points, so hopefully the predictive value gets better with more data, but the data from South Korea looked a little peculiar at the end, but the late case increases were not huge even though they failed to fit the model.

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