Dear Martin,

Please give me a couple of (or a few - I am really busy at the moment, fortunately) days to produce a clean version of it. If you have done a little mathematical and physics modeling you will probably understand why I chose to do it the way I do ... I have another version of it that produces a steady state ... etc. I will try to pack it all there, depending on how much time I have. Once you understand how to operate on the basic equations, you can get to model pretty much any effect which depends on susceptibility (lockdowns, lifting lockdowns, more or less strongly, etc.). I will attach the notebook in a reply to your reply directly (there is no space up in the main sections). Alternatively, I might start a new section for this purpose ... when ready, I will let you know.

Best, Enrique Enrique

Dear Martin,

Attached to this reply is a quick and dirty notebook which shows how to make S grow ... the change is in the equation for S'. If you uncomment the commented factor, you can also bring it back down to zero quickly ... sorry I don't have time to make this better looking. Hope this is helpful.

Regards, Enrique

Attachments:

Martin 12 may 2020.nb

In this section I discuss an SIR-like model which works almost as well as the SEIR model, at least with the Chinese data. I will have to investigate whether I can make it work as well with other data sets.

Our SIR like model is described as follows

s'(t) = - Beta * s(t) * i(t) / p,

i'(t) = Beta * s(t) * i(t) / p - Gamma * i(t - n),

r'(t) = Gamma * i(t - n)

The function s(t) is the number of susceptible people (the people that can get exposed to the pathogen) at time t; i(t) is the number of people who are infected and infective; r(t) is the number of people who have become resistant to the pathogen: they have recovered and developed immunity or died. Now the parameters.

beta is the rate of infection or "force of infection", gamma is the removal rate, and n is a shift parameter used in part to line up the curves (see a bit of an explanation of this in the SEIR section). The parameter values are in the titles of the pictures for each country. In general we assume i(0)=1 unless stated otherwise in the model label. Also, s(0)=p, and r(0)=0, unless otherwise stated.

We present in the picture, an SIR model and its parameters that fits the Chinese infection data. With time, I will try to fit the SIR-like model to other data sets. I also plan to investigate the known analytic solutions (although I need to understand how the shift parameter changes the classical solutions) in order to attempt automatic fitting via computational optimization in Mathematica. This is work in progress.

August 30: Positivity rates picture updated. For the time being, we will no longer update this picture

August 10, 17: Positivity rates picture updated. Mexico, which has a huge positivity rate, aside, Sweden still has a very high one.

June 15,22,29: Updated June 29, July 13, 27. The positivity ratios picture will not be updated again until August 17, or then again, only occasionally. All countries in the picture have brought this ratio down over time, except Mexico.

June 1,8: Positivity rates updated

May 21: The positivity rates file will be updated on Mondays from now on. It is updated today.

April 21-May 12: The SIR models will now only be updated occasionally. The positivity rates picture will be updated daily.

April 19-20: Finland updated. A new SIR model for Finland with a different recovery schedule is included. We include now the current positivity rates for various countries (number of cases/number of tests)

April 18. There is now a SIR model for Finland which uses JHU data

April 14: I have added the SIR model for Italy. I will not update this daily. Later, I will make a notebook available for this model. If I am able to complete it, I will also have in the notebook a program to find the parameters to fit the model to the data - but I cannot promise I will have that, at least not soon. Note the forecast is somewhat more optimistic, both in regard to total susceptibility and duration of o

Attachments:

FiSIRmodel6bfr.png

FiSIRmodel6c.png

ISIRmodel6.png

SIRmodel6.png

SPositivityRatesToday.png

Hello,

Is it possible to post an animation, and how?

Thanks.

Hi, can you post the code you use about how the fitting is done to obtain beta,sigma,gamma,Ishift,Eshift, for example for some random data of {day , infected, resistant} ?

Thanks.

Hello,

I will post this soon for the SIR models.

Dear Roberto,

I would be glad to try to come up with a model for the Northern regions of Italy, if I can find the data. Alternatively you can tell me where to find it, that is probably a better solution, or you can send them to me (I can tell you how, or you can post them as Mathematica lists of numbers in your reply). I can curate it by hand if necessary. Please bear in mind that it is difficult to really know how the parameters are going to end up in the end until the the I curve has peaked. Although I have learned to make educated guesses for this disease now. So I think I might be able to get something that is close to the reality. Glad to help if I can. Personally, I don't think the outbreak in Italy is going to get much worse than is implied already in the posted model, in terms of total number of cases (susceptibility). Best, Enrique

Here is a picture with the fatalities per million of Italy, UK, Sweden, and the US; and now also for Brazil and Mexico. For the US, for example, this translates into a forecast of about a total of 146 000 fatalities during the outbreak. We will update this on a daily basis but keep the June 1 forecast. For Brazil, it translates into over a million fatalities in the long run, and the fatality per million forecast will surpass that of every other country. This model will be hard to verify as there is not yet enough data (the data was restored last night). Only time might tell what happens in the end.

September 14,20,27: pictures updated. We need to update the models for Brazil and Mexico. Next update in about two weeks.

September 5: Brazil's fatality rate is now higher than that of the US, Sweden, or Italy. IHME updated its fatality forecast for the US for the end of the year to something above 400 thousand as the most likely estimate.

August 28: Brazil's fatality rate has now caught up with that of the U.S. and is headed higher. Shortly it will surpass Italy and Sweden. IHME now forecasts around 317000 fatalities in the US by December. Unless there is another spike in the number of cases, we doubt that the number of fatalities will be so high. They should start declining soon. We hope to make a new model for the US soon.

August 24: The IHME forecast of fatalities for Sweden and Brazil (see below June 14) by 4th August was way to high ... our forecasts are almost exact. In particular, our forecast of the Brazilian fatality rate dates back to early June and it is our most exact model. The number of fatalities is nowhere near what IHME predicted. However, the estimate for the US in October will turn out to be, most likely, low, as there was an upsurge. We have not calculated a new model for the US, but there should be enough data to do that in the near future.

August 9: We see clearly a second wave in the making in Spain, and probably elsewhere in Europe. We will compute a new model for these waves when there is enough data. The weekly total in Spain this week is back to levels of 25 April.

August 1: There is now growth in all five big EU countries, and quite a bit of it in Spain

July 27, 28: We continue to observe growth in the big 5 euro countries, especially in Spain, France, and Germany. Italy and the UK seem under control. We have adjusted the model for the fatalities rate for all countries except the US. Our forecast for these countries have changed relatively little. There will be a substantial change for the US when more data is available. The date when Sweden's rate becomes larger than Italy's has been pushed back by three days, from the 12th of August, to the 15th of August.

July 18: We observe new growth in the big 5 euro countries, and an uptick in the fatality rate in the US.

July 7: IHME forecasts 88000 deaths in Mexico by October. Our forecast is slightly under 80000.

July 3: Our forecast for the number of fatalities in the U.S. for the 4th of July is within the margin of error of our calculation (about 3%). It is slightly lower than the actual. We will keep the same forecast for the HGHI forecast for September. On Monday we will provide a more specific number, but we guess it should not be more than 160000 (vs the 200000 of HGHI) - more on this on Monday. We note, however, that there is an uptick in the fatality rate, so it is hard to know exactly what will happen. Hopefully the situation in the U.S. stabilizes and improves soon.

July 1: We are adding the weekly totals for the five big Euro countries since May 2. We exclude the current week. Weeks run from Sunday to Saturday.

June 26: There is a significant uptick in fatalities in the U.S. Our model might have to be recalculated if it persists

June 23: This section will be updated in the mornings with the results of the previous day.

June 14: IHME (https://covid19.healthdata.org/brazil, and https://covid19.healthdata.org/sweden) forecasts 165590 fatalities for Brazil by August 4th, and 8534 for Sweden. Our model forecasts 6012 for Sweden and 100484 for Brazil. It will be a good check for our modeling. A similar projection for USA extending to October 1 forecasts 169890 fatalities. If the country avoids a second wave, our model indicates that there will not be more than 150000 fatalities. The IHME model for USA can be found in covid19.healthdata.org/united-states-of-america. We will follow this closely, always hoping for the best.

June 9: University of Massachusetts, reports the Guardian on June 9, forecasts 130000 fatalities by 4th of July in the US. Our model's forecast is 128550, slightly lower. Our MARGIN of ERROR is +/- 3000. If w

Attachments:

FatalitiesPM.png

weeklyTotals.png

Hello,

The published notebook is not my working notebook. I am on travel for a week. I hope to get to updating the published notebook then, apologies, and thank you.

I have updated the notebook again, there was a mistake in the notebook I posted before.

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

June 15,22, 28-29, July 5-6,13, 19, 27, August 2,10,17: PDF documented updated with weekend data - no country seems to be succeeding in eliminating the virus completely. NEXT UPDATE in a week. The June 22-promised notebook for fitting parameters automatically is not yet ready, I will try to get it ready this week, but I can't promise, except it will be ready some time. --- The pdf document will now be updated only on weekends. It is interesting to note how many countries which have brought the number of new cases down have settled on a small constant number of daily cases below which the numbers are not dropping, for example, Austria and the Czech Republic. Some countries have been unable to bring down their numbers at all, such as Poland. Some countries have just peaked (US, Russia), and some countries are growing (Mx, Brazil).

June 1,8: Updated. In this section soon there will be two new notebooks, pertaining to the automatic determination of parameters, and to making susceptibility grow.

May 20-21: The pdf with smooth daily tallies now includes tallies for Russia and Brazil, in addition to all the previous 20 countries. This file will be updated on Mondays only

May 12: The pdf with smooth daily tallies has moved here. Soon I will post a notebook with the

Attachments:

SEIR model C 27 ...nb

SmoothDaily.pdf

I made a notebook deriving the logistic model from your SEIR equations with a simple assumption.

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Dear @Enrique Garcia Moreno E. thank you for sharing! Could you please share the notebook with Wolfram Language code? You can attach it or embedded into the post.

Attached, I tried embedding it and even though I am signed into the cloud, I can't do it.

Excellent to get this kind of confirmation ... later tonight, EET, I will post an update with the equations and parameters of the model, which changed a bit last night, for the last time I hope. I can now really calculate R0 classically, and it is different, unfortunately higher ... Thanks for sharing this.

Dear Robert,

Would you happen to have a brief notebook which implements your logistic model based on the number of cumulative cases? I would be very grateful to you ... I need this because I have a case where I only have the cumulative cases data. I thank you in advance for your help.

Best regards, Enrique

Robert,

Thank you very much!

Best, Enrique

Thank you, it's much more clear now.

As a teacher of graphing and visualization, might I make a proposal that you properly label chart axes with units and also label charted values (which colour is which variable).

This makes it much easier to instantly understand the graph.

Thanks for your efforts and sharing them.

Enrique, can you confirm the following:

1) You based your model on Mainland China Infections numbers from Johns Hopkins CSSE (JHU)

2) Your graph is then a model run on the Helsinki population with 90 000 susceptible being the starting number for the population

3) Blue is the number of confirmed infections (?) still ongoing (?) (i.e. not total, not cumulative all, not daily)?

4) Magenta is the (percentage?) of recovered (axis scale missing?)

Did I get this right?

Thanks for the model!

Your points about the susceptible population size are well taken. Unlike modeling chemical reactions in a beaker (the math is the same), one cannot assume instantaneous and uniform mixing in epidemiological models. The population size does have an effect depending on the type in incidence used for the force of infection, so one cannot always simply scale the model for larger or smaller populations.

I found an article in The Guardian about the change in counting methods in Hubei. Real data are invariable messy. Understanding exactly what the data are is crucial to modeling them accurately. For example, does "recovered" mean no longer infectious in the epidemiological sense, or does it mean asymptomatic in the medical sense? And, of course, there is always the question of unreported cases to deal with in these retrospective analyses.

I'm quite certain that one will have to model each focus or cluster of infections individually. The derived parameters should be comparable, but are most likely not transferable.

I did not understand your comments on population scaling. If one scales S,I,E,R by p, keeping \beta,\sigma,\gamma fixed, p-dependence drops out of the equations(as it should). So, solving for population scaled equations is mathematically equivalent to solving keeping p as a parameter. thanks, hari dass

Hello,

Let me try to explain how I tried to model these data and try to address some of your questions, not necessarily in the order they appear. First, I decided to work with the simplest model possible, to keep the number of parameters as low as possible. Otherwise, estimating the parameters gets quite difficult. It seemed to me that the basic SEIR model should work, if only we modified it to reflect the situation at hand. Then I made some thought experiments to try to make these modifications and parameter estimations. Let me explain a couple of these with an example.

The first thing I tried to understand is the effect of the containment measures enacted on the parameters of the basic SEIR model. I illustrate what happens with an simple example. Suppose you have ten families with four members each and that you all of a sudden detect three infected individuals, each in a different family. Now suppose you then immediately put each family in a different house, and you prevent them from having any contact with each other. The effect of this is essentially to remove 28 individuals from the susceptible group, which would otherwise be 40. Only 12 individuals, those families with an infected member, are now susceptible. Understanding this told me that I might have a chance of modeling the data if I lowered the susceptible population to an adequate number. I then used further information about the other possible value of the parameters, including information about maximum observed incubation periods. And then it was a matter of some fine tuning to get the right fit for the I curve. I observed that even if the actual data was only a sample of the real number of infections, the shape of the curve should be the same with the real numbers. You just multiply by a constant factor. So say that, as suggested, there might have been ten times (or twenty) times the number of cases as reported, then the susceptible population would have been 1800000 or 3600000 and likewise, you would have ten times more infections, recoveries, etc. The key is getting the shape of the curve right.

In the model, I tried to get the I curve right, and then, modify the equation for R with a root factor to account for a delay of the onset of recovery that fits the recovery data (it takes a while for people to recover).

Now about the data. I started early on, and I only had data that I could copy by hand ... and then I just kept doing it that way, among other reasons, because it forces me to see what is going on, rather than just letting the computer read it. From the 13th of February on, I subtract a constant of about 14000 to the JHU data, because of the change of counting method that lasted only a few days and the leap in the numbers. This is the simplest quick and dirty fix that keeps the data in line with the trend and nullifies the leap in the count which occurred on that particular day.

I am quite confident that the model is basically working well. As the days go by, I will continue to check it and adjust it.

If you think about the most difficult issue regarding action to contain the spread, given that you can't test extensively, is to decide when you have a large enough number of detected cases to warrant some containment action. It is a difficult decision to make, and the better data and more extensive testing you have, the better prepared you are to make a wise decision. It would be very useful to have a very good estimate of what the real number of cases is given a number of detected cases that you might have as a sample. We don't really know. Some literature has suggested that it might be 20 times the number of detected cases.

It might be that in the new foci of infection, the model might have to be modified to fit the circumstances of the situation. I am following several foci. I did model the cruise ship situation, and the with almost the same parameters I managed to obtain numbers that were quite very close to the actual numbers while that outbreak lasted. When I have time I will revisit that model and see if the improvements in the model here can give even better results. I need to consider all days, I don't have all the days.

I hope this answers some of your questions. I hope I get around to cleaning the notebook and providing all the details ... but in a nutshell, it is the simplest possible SEIR model you can think of, almost.

Very nice!

Please publish your notebook. I'm sure others would like to see the ODEs.

Bob