Hopefully it is easier to read now and clearer. I specify where to find material in various sections. It is possible to model the data by assuming a low enough susceptibility. I explain in the discussion with Robert Nachbar below why the main effect of containment measures is to lower the effective number of susceptible individuals when it is imposed relatively speaking, at the beginning. Using this idea, it is also possible to model what could happen if you lift restrictions too early by letting the susceptibility increase see picture and you then reintroduced them this is not a forecast, just a possible scenario. Regardless of these considerations, our model allows us to understand how the disease evolves in time.
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Hopefully it is easier to read now and clearer. I specify where to find material in various sections. It is possible to model the data by assuming a low enough susceptibility. I explain in the discussion with Robert Nachbar below why the main effect of containment measures is to lower the effective number of susceptible individuals when it is imposed relatively speaking, at the beginning. Using this idea, it is also possible to model what could happen if you lift restrictions too early by letting the susceptibility increase see picture and you then reintroduced them this is not a forecast, just a possible scenario.
Regardless of these considerations, our model allows us to understand how the disease evolves in time. We also include 1 a picture that illustrates what could happen if you lift restrictions too early only to reintroduce them we explain ahead , in this section, 2 a picture of a steady state solution in the Finland section; 3 a notebook in a response towards the end of the post.
It has two advantages over the SEIR model: a the classic SIR model has analytic solutions, so straightforward somewhat computational optimization can be carried out to estimate the parameters - although our equations are not the classical ones as they have a delay; b it yields for the data we are trying to model values of R0 that are congruent to the observed ones, 5.
If they are asymptotic to a positive value, that means there is a herd immunity effect - the value of the asymptote being the number of people who remain susceptible under containment that will not get infected; moreover, we can see that the susceptibility curve is very close to its asymptote near the peak of the infections curve I in the diagrams , so that targeted testing is warranted as an effective measure of containment at that stage.
That section contains a model of China final and a model for Italy that is not updated , and a two models for Finland, one using the JHU data instead of the Finnish authorities data, and another one using another recovery schedule using an estimate based on the scant recovery data for Finland. Also in the Finland section there is a SIR model that shows what it looks like to reach a plateau or steady state, rather than a peak; the equations for this are necessarily different than the simple SIR equations given in the SIR section, where there is yet another model for Finland using JHU data and another model for Finland using a different recovery schedule.
The newer models are adjusted quite frequently, especially with respect to the number of susceptible individuals, as they continue to grow. They tend to stabilize about three weeks after control measures have been in place. After the I curve peaks, it is possible to begin to get an idea of how long the outbreak will last. Now the parameters. Here, we are operating with a delay, we call an individual infective when it gets detected a case and continue to consider it infective until it gets "removed" when it has recovered or passed not when it gets caught.
The parameter values are in the titles of the pictures for each country. There is an intuitive explanation for that. That would give us the R0 that is being measured my understanding is that R0 was estimated on DETECTED number of cases - but if this is wrong, then my explanation for the disparity is not correct.
Recall, the basic reproduction number R0 is constant. The R0 numbers obtained in the SIR models discussed in a separate section are congruent with the values that are proposed in the research litereature more about that in the SIR section.
In a response, towards the bottom of the post, you can find two alternative models for Finland. In yet another response, the notebook with the model for China, a final version of which will be made available soon, including the SEIR and SIR models. In addition, we have in this section, the picture which shows what might happen if restrictions are removed too early.
In the Finland section, a steady state solution and document with daily tallies per million inhabitants and a document with daily tallies smoothed per million inhabitants for several countries. We have estimated this data, sometimes extrapolating from available data, sometimes using an estimating function based on average rates from countries that do provide the data, etc.
It would take too long to discuss what we have done in each case where recovery data seems to be missing or partial. We explain the Finnish case. There is a delay in the release of the data of days. According to the medical chief of staff of the infections diseases clinic at the Helsinki and Uusimaa hospital district, it was "important to define what people mean when they talk about recovery", and that "eventually it would be important to compile statistics to better understand the disease" and "was taking the numbers with a grain of salt" noting that "the criteria undrelying the data are not always clear and they are not always the same in each country".
He also said that "tracking recovered patients was not a top priority". The number of recoveries given at the JHU site is a lump sum. We have serialized this datum into data according to how cases might have arisen in time to obtain a model. The recovery period appears to be 17 days or so. After a new estimate published on April 16, we have corrected this time series and it now has a delay of 15 days.
We are modifying the Finland section to reflect this new reality and will explore other models. If no new data is provided, we will continue to extrapolate from our series the next elements according to a best fitting model to the data that we have. For Finland we used to provide additional information which can be used to verify the model, such as the peak date for the cumulative mean of the number of daily new cases but this has been removed.
We have replaced it with a pdf document which has daily tally per capita per million inhabitants information. In the SIR section there are two more models for Finland, one using JHU data and another one using another estimate for the recovery schedule. The US model now uses an alternative recovery schedule based on an average of the recovery schedules of countries which are providing these data, as the US recovery data seems lower than it ought to be.
We also use an estimate for UK data which is not available. Some countries have changed the way they count in the middle of the process, and we have adjusted for this or not as we see fit - again, it would take too long to discuss this. For the most part, we use the data that is available and take it from there. One can appreciate that contrary to what some European authorities are suggesting, hardly any European country is ready to move forward, according to this parameter, especially if we compare it to the role model country for exit strategy, South Korea.
It is a useful diagnostic of where a country stands in the process. April updating. Spain went back to work ten days ago. We see new growth and forecast it will continue so I changed the text above to improve it and hopefully make it more readable, and highlight important issues. The problem with Finnish data is that the entire time series gets corrected every day, not just the last day. While this makes for accuracy, it makes modeling difficult.
I will alternate with the usual SEIR model. There is yet another model for Finland using another estimate for the recovery schedule. This should be a useful diagnostic. I will start keeping a history of these data from now on I only have a history for Finland.
This section now has the SIR model for Finland, we believe it is a more accurate model for the time being, and based on a just published estimate of recoveries. Our extrapolating function seems to be working quite well and we have adjusted it to reflect this last change. The German model in its section is now an SIR model. In the Finland section there is a SIR model in which a plateau, rather than a peak, is reached April updating.
Today I will show an alternative model for the USA that uses an average recovery rate obtained from other countries rather than the reported data, which seem low understandably so, it is not a priority to test people who have tested positive and are recovering at home.
The daily tally in the US has slowed down somewhat, which would lend credibility to the model, which shows the infection curve getting close to a peak.
Also, in the previous model, the number of susceptible individuals was probably too high. I will compute an estimated peak date tomorrow based on this model. April Today is the last day the China model will be updated April I will add SIR models for other countries as well. I am working on an optimization program for the SIR model to further automate the determination of parameters - if I ever complete this it will also be in the SIR notebook eventually. On another note, I plan to add a section with models for other Scandinavian countries as soon as I have time.
Today I am adding a response with a section which discusses the simpler SIR-like model which I managed to make work almost just as well as the SEIR model, although it is somewhat more difficult to get it to work. The SIR-like model has the advantage that analytical solutions are known for SIR models which might be modified for our specific instance of the model, and in the case of our investigations, it yields an adequate value for R0 without the need for any further explanations.
The daily tally pdf document in the Finland section is now per million inhabitants. Again note the disparity between European countries wishing to pursue an exit strategy at the moment, and South Korea, the role model country. I have added a picture which explores a scenario in which Spain lifts restrictions as it has announced today.
We are able to model with some mathematical ingenuity the effect of this on the S curve, and subsequent effect on the number of infections. We hope this does not happen, but it might. April 10; updating. I added some explanations in the text and a picture that illustrates what could happen when restrictions are lifted too early and then reintroduced - this is not a forecast, just a plausible scenario. I moved the notebook to a new response at the end of the post.
April 9: updating. In the Finland section there is a pdf document with the smooth version of the daily tallies for several Euro countries, USA, and South Korea. The Italy model seems very stable now. April 8. Today I added, in the main part of the text above, an "intuitive" note about the basic reproduction number R0 in these models and why they are about an order of magnitude larger than the measured rates.
April Updating throughout the day. France and UK models temporarily suspended due to missing or inconsistent data, until more data is available April 5: Updating throughout the day. There is a new model for Austria in the Europe section.
April 4: Updating. The model was adjusted a tiny bit. In the model for Italy there is now a possible end of outbreak date, should the data stay on the curve. It is calculated using a threshold, which I do not explain here.
April 3: Updating. As expected, there is no recovery data for Finland so we are extrapolating from previous data. April 2: Updating.
The model is almost the same as before. There was a lack of recovery data, but yesterdays the recovery data was released as a lump sum the number of recoveries. We have stretched this datum into data according to how cases might have arisen in time to obtain a model. The recovery period appears to be 17 days or so see more detailed explanation above. Italy seems to be reaching the peak of its I curve at around the time I estimated it would, maybe a week later.
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Not as much as you might think. The simulated data here contrast policies that isolate people who test positive using four different assumptions about the quality of the test. Even a very bad test cuts the fraction of the population who are ultimately infected almost in half. In this post, I use it to compare the economic and social cost of two policies that are equally effective at containing the virus. What the simulations show is that if we use a test to determine who gets put into isolation the fraction of the population that needs to be confined and isolated is dramatically smaller.