Hi Enrique,

Could you possibly let me know how to apply and lift restrictions? Many thanks, Martin

Hi Enrique,

Could you please post the notebook for your Spain model? I am curious as to how susceptibility changes as regards lifting restrictions.

Many thanks, Martin.

Thank you!

Enrique,

Attached are two notebooks which I am currently using. The fitting methods are embedded in the report procedures. When fitting to cumulative cases, the large numbers in the latest data dominate the fit, but they are probably also more accurate. At the end of the CDCData.nb there is a method for fitting the first differences of the cumulative data to a first difference formula derived from the logistic model, if you also want to see more directly the influence of the early data. We are about to enter the tail phase of the epidemic in the U.S. The model requires an exponential decay in the tail. That didn't happen with South Korea, but the China data did fit. If the U.S. data doesn't follow the tail behavior, that would indicate the model isn't valid.

Bob

Attachments:

CDCData.nb

KeyStates.nb

This post has been listed in the main resource-hub COVID-19 thread: https://wolfr.am/coronavirus in the section Computational Publications. Please feel free to add your own comment on that discussion pointing to this post ( https://community.wolfram.com/groups/-/m/t/1888335 ) so many more interested readers will become aware of your excellent work. Thank you for your effort!

In this section I attach the notebook. I will update it soon and add new features to it, such as the effect of lifting restrictions early.

August 30, September 6: the pdf document has been updated. Next update at the end of September

June 29, July 5, 13,19, 27, August 2, 10, 17: the pdf is updated. Next update in a week

This notebook is also posted in a reply to Kaurov, above the Scandinavian countries section. I will post (hopefully soon, I did not get it ready June 22) the new notebook for adjusting parameters automatically in both sections. In the reply to Kaurov, there is also a pdf document with daily new cases curves for several countries, now also attached here. This will not be updated until August 17, or then again, maybe only occasionally. It is interesting to see that some countries which brought their numbers down initially have settled in a plateau of a small number of daily cases, without being able to completely eliminate the v

Attachments:

SEIR model C 27 ...nb

SmoothDaily.pdf

Dear Zach,

Thank you very much ... indeed, my intent was to be "utilitarian" in the sense you point out, and I think I have more or less succeeded ...

But let me tell you, you have outdone me now, and I would almost like to ask you for help. If you have kept the daily slides, then please keep saving them ... and later, perhaps you might make them available to me as a time sequence ... I ask because I have not kept track of the parameters as time goes by, (lack of foresight on my part) and indeed, that would be very useful information for the sequel of this work which has to do with the methodology for fitting the parameters. As you mention it, having the history of those fits would go a long way towards completing that work; but I have not been keeping track of that history (because basically, my methods work well for the time being, and lack of foresight combined with lack of time to make a tool or keep track of all of them on a daily basis). With all that said, I hope I we can stay in touch and correspond about this matter later on. Once again, thank you for your good words, and I will try to keep making this as useful as I can to the community at large. If I only had the resources, I would do this for all countries. But for now, this will have to do.

Best regards, Enrique Garcia Moreno E.

You are right of course in what you say, but there is a caveat ... in using computational methods, I have had to "intervene" and fit to just part of the data, rather than the whole set, when the data was "messy" and behaved unpredictably ... I have since developed a more or less systematic way of doing these interventions with the aid of the additional parameter in the model (note it is not exactly an SEIR model). My line of thinking is that computational tools aided with further AI computational tools that emulate the actions of the interventions would be the ideal way to go about this. If I ever get time to do it, I will try to build such a tool. But for now, my approximate methods, the use of a more complicated model to get first approximations, etc. will have to do. Perhaps later we can discuss how one might do the fits with established methods in Mathematica ... if you are up to doing something like that. There are a couple of us now working together on this ...

Hi,

This is not possible at the moment, it is incomplete work. At the end of the day, I do quite a bit of fine tuning based on my by now fairly good understanding of the behavior of the parameters. You also need to make judgement calls regarding the quality of the data, and on what part of the data stream you want a better fit. For example, in the China model I fit to the tail first (once it was there), and then adjust to fit the front part of the data stream. This is, at the moment, an interactive process. I hope to have useful tools to do this later on more or less automatically. But that is quite a bit of development work.

Dear Enrique,

the data are here, for the regions update every day at 18:00

https://github.com/pcm-dpc/COVID-19/blob/master/dati-regioni/dpc-covid19-ita-regioni.csv

The regions are all there : the most hit, as you know are Lombardia, Emilia, Veneto, Piemonte, but the disease is spreading allover the country.

I would have many questions to ask you.

1: what the parameters (beta, sigma, gamma, p, Es, Is) ? 2: apparently is no lockdown effect/date in your model ? 3: I am puzzled by the Wuhan case: how did you optimized the parameters ? the early data are clearly filled with artefacts. Did you fit/optimized using the later date ? The model is robust enough to find its way through the overall behaviour, unsensitive to "strange" data ? 4: would it be possible to split fatal from recovered ? 5: could we have some discussion through private email or phone ? if so my email is roberto.battiston@unitn.it please reply there.

thanks again for your very interesting work , it is exactly what we need.

Roberto

Enrique, I am writing from Italy. I am a physicis. Today I was trying to see how to adapt SEIR to the OBSERVED value and I found your work ! Impressive. Congrats. You know how bat the situation is in Italy. Your work would be extremely useful. I would like to suggest to additionally test your work on a couple or regions in Italy: Lombardia, Emilia Romagna, Veneto and Piemonte. In this way you could spot if there are or not differences in the fitting parameters. Would it be possible at all ? Please reply me asap: time is of essence. With my best regards Roberto Battiston

Dear @Enrique, is the notebook you attached to your head post the latest one? I see many images in your post that are not in that notebook. It would be great to see your most complete recent notebook for all things that you demonstrated. Thank you for sharing, this is very nice!

Hi,

I have updated my notebook (March 20) with the China model and data. I will add the other models when they stabilize.

Dear Vitaliy,

I realize it is about two months now and I have not updated my notebook. As soon as I get the time (that is, I go on holiday, in about a couple of months), I will at least publish another notebook which has two things. A different kind of model that I have been discussing here (I call them "TRUE" models, and it is what epidemiologists usually use) which is very good for forecasting once there is enough data, and also, a set of programs to find optimal fits of the models to data automatically. This notebook will contain also the utility program from which you extract the forecasts, to get the pictures I have at the bottom of the post and which I added just a few days ago. I will not publish a notebook with fits to all data sets and gives you all the pictures ... but at least one, so that people can then do their own with the data they might be interested. I will also update the old notebook to contain both SEIR and SIR models of the kind I have been discussing since the beginning, with ONE data set. These, unfortunately, are harder to fit, and I don't have yet a fully, or almost fully, automated way for fitting the parameters; I have a set of "rules" which I apply to get the fits, and they work most of the time, but not always, and they are not fully automatic in the least bit. Apologies for the delays and the limitations. I am up to my ears with work. During lockdown, I have been busier than ever!

Regards, Enrique

Thanks!

Right, I apologize I haven't done it, it has taken me a while to get this right. I will try to get it done by tomorrow, or else, I will be on travel and it would have to wait a week. The trouble is I am using approximate data or data straight out of websites, rather than the github data, which I don't find I can computer read. But an approximation to the nearest hundred works fine (as in the JSU CSSE website itself). I will explain in the notebook with more detail other aspects pertaining to the data, such as a correction for the counting methods change. I am now modeling the Italian outbreak but it is to early to settle on any parameters ... but I think, and I want to believe, that it will be controlled ... apologies again and thank you for your patience.

If I am understanding your graph correctly it looks as if it is predicting the end of the epidemic at about the same time as a simpler logistic model based only on cumulative cases. In the graphs below day number 1 is the first data day on the JHU database, 1/22/2020. Also the curve looks like it is becoming more symmetric like the logistic growth model.

Well, it seems our forecasts were quite on the dot ... there were no new infections yesterday, March 19, aside from imported cases. I am now trying to extend the models to other countries and get a feeling for how long this will last. It seems to me that the dynamics of this disease are pretty much the same everywhere, except for the volume (effective susceptibility). It seems that, from the moment that the number of cases starts climbing to there being almost no cases is at most about four months. Let's hope this is the case. The important thing is to then guard against other waves. I hope the whole world keeps its guard up.

Enrique,

Attached are two notebooks which I am currently using. The fitting methods are embedded in the report procedures. When fitting to cumulative cases, the large numbers in the latest data dominate the fit, but they are probably also more accurate. At the end of the CDCData.nb there is a method for fitting the first differences of the cumulative data to a first difference formula derived from the logistic model, if you also want to see more directly the influence of the early data. We are about to enter the tail phase of the epidemic in the U.S. The model requires an exponential decay in the tail. That didn't happen with South Korea, but the China data did fit. If the U.S. data doesn't follow the tail behavior, that would indicate the model isn't valid.

Bob

Attachments:

CDCData.nb

KeyStates.nb

I would like to add that in China the situation was strictly controlled until the end, so what had to stay constant (susceptibility) did ... I am afraid that elsewhere this will not happen. I already have a model which allows for the growth of susceptibility. It is beyond the scope to discuss it here, but I plan to make a notebook available in the aftermath, when we have the whole picture. So when restrictions are lifted, susceptibility, the parameter that controls the size of the epidemic, grows again, maybe ever so slowly, all depending ... all the measures put together work to determine that one quantity, which is the overall most important quantity.

Enrique

Thank you, your points are well taken and I will do so in the next round ... there are already some updates and some changes and some things that I still need to clarify (or else I am creating confusion) ... but I think I will wait a few days and see how close future data is to the model without having modified it further ... I am hoping in the end to have an accurate description of what is observed by the data and I hope to publish the equations that give the model. Thank you once again.

Hello,

No, there is a bit of misuderstanding I see. I answer your questions the best I can one by one:

1) That is correct ... here is the webpage with their data and graphics: https://gisanddata.maps.arcgis.com/apps/opsdashboard/index.html#/\bda7594740fd40299423467b48e9ecf6

2) No, my model is a model for a susceptible population of 90000 (it can be from anywhere). It fits the JHU data. If you read the rest of my post, you will see that I explain the main effect of the containment measures is to reduce the effective susceptible population. So although a city population might be 11M, with the containment measures, that number dropped to an effective susceptible population of about 90000. It is a model of the JHU data. There is no data for Helsinki yet, as we don't have any spread here at the moment. I hope this is clear enough for now.

3) Blue is the number of infections at a moment of time. It is time dependent. It is what is called the I curve. It is NOT the total number of confirmed infections; you need to think of it (I won't go into the math details) as that number minus the recoveries and fatalities in your data, to get something similar to the red dots.

4) Magenta is the NUMBER (NOT percentage) of cases that are resistant, essentially those that are not infected any more plus the fatalities, among the 90000 susceptible ones. Horizontal axis scale is days, that is standard.

You have to keep in mind that this model works provided that things stay in check and there are no new paths of infection ... otherwise, you still have a huge uninfected population out there which could become susceptible. The difficulty now is to keep this under control.

So no, I apologize but you did not get it right, but hopefully this helps. I hope this is useful. Especially because it has been hard to get a good fit for the data. Thanks for your questions.

You are of course right about all of what you say. My goal was to try to find a phenomenological reason with implications about the values of the parameters that would make it possible for me to model the data, and I succeeded in that sense. I agree with you in that one has to be cautious about the scaling. But for the data that we have, we have some kind of model. I will update this data daily and see if it continues to fit the model. I expect it will and what the model predicts is in line with the expectations of some epidemiologists.

Right, real data is messy ... you have to do the best you can with it. I think that what I did somewhat works.

About recovery, it should mean "not infectious" given the tests that the patients need to go through to be discharged. However, there are cases documented when discharged individuals test positive again later on, leading to the suspicion that the disease might be biphasic in some instances.

And correct, each focus will probably need its own, if similar, parameters. I am tracking several of the new foci, and we will see what happens with time. Unfortunately, it is harder to get data broken up into clusters of interest. Surely it can be obtained, but it is not easy to get it.

Look at the equations AND the initial conditions (for S). What you see (relative to the y axis) is what you get ... (I should include a plot with S) ... hope this helps; if not, we can come back to it at some point.

Thanks for adding the additional detail about the model, it helps one understand the results better.

It looks like you are using the data from Hubei, based on the large number of confirmed cases. However, when I downloaded the JHU data from github (https://github.com/CSSEGISandData/COVID-19/tree/master/csse_covid_19_data/csse_covid_19_time_series) last evening, I got similar but noticeably different values. What is your data source, and what preprocessing did you do? What was the adjustment that you made?

What assumptions did you make when choosing a susceptible population size of 180,000? The population of Hubei is very much larger:

In[22]:= WolframAlpha["population Hubei China", "Result"]

Out[22]= Quantity[58160000, "People"]

It will have a dramatic effect on the dynamics, especially if one uses mass action incidence for the force of infection as opposed to standard incidence. Which did you use? Models for the SARS epidemic used mass action incidence.

I have found these data quite challenging to model. My current model has 7 compartments and includes quarantine. It's still very preliminary, and I hope to have it ready next week, but here is a peek at its structure (as a Petri net);