Update 10 April 2020
I have written out a formal description of the model, and in doing so I have tidied up and altered the notation a little. This is in Word document which can be downloaded below. I have also put up aa new Excel version matching the formal description, in an Excel file also downloadable. This shows specimen projections on certain assumptions; as always they are not forecasts, only “what ifs”.
Update 1 April 2020
I have uploaded another file below. This takes the V3-0 model, applied to the UK as before, in sheet UK02A, with r(t) = 0.2907 throughout. I have made one important change, in altering the dates in col B to be two days earlier than the ECDC dates. The ECDC dates are one day later than published UK dates, which are roughly one day later than the actual events. I am now trying to represent the date of actual events.
Next, in sheets UK02B to UK02I I try out changing r(t) from 21 March onwards to a new value. I am trying to represent the changes in behaviour, which probably started a little before the “lockdown”, which I take as starting on 24 March. No one can know the exact effect of these measures, nor quite when thy might show an effect. But in the model, as inn reality, the effects are delayed. With trials using different values, one can compare the outcome with these trials, and see which best fits the recorded observations.
In UK02A I keep r(t) unchanged at 0.2907, to represent the base case. All the others show just the same numbers as UK02A until 21 March, and differ only after that date. In UK02B I set r(t) for 21 March to 0.25, a smallish reduction. In successive sheets I try, in C 0.2, in D 0.15, in E 0.125, in F 0.11, in G 0.10, in H 0.05, and in UK02I zero. This last shows the direct effect of the delay in the system, even if there were no more infections on or after 21 March. it has already been invalidated by events.
The values of 0.11 and 0.10 are include, because the implied values of R0 in the model are 1.096 and 0.996, just above and just below the critical value of 1.0. Below this level, the disease starts dies out at once; above it, and its goes on up to some peak a bit later.
There are many numbers that one could use to compare the projections with the outcome, and there are delays in likely changes. I have put comparisons of New Cases, Total Cases, R7 for Cases, New Deaths, Total Deaths, R7 for Deaths and, and the Deaths Percentage in the sheet “Summary”. It is really too soon to be making these comparisons, but it does look as if the numbers of New Cases are recuing, There were two very low days on 21 and 22 March, so the Total Cases seem to be at a lower level than New Cases now. The former lies close to U02F with 0.11, the latter between UK02C and UK02D with 0.2 and 0.15. But so also does the R7 for Cases.
For Deaths there is a different picture. the numbers announced in the UK on 31 March. 381, make the number of New Deaths and of Total Deaths higher even in UK02A. So the jury is till out, until more data comes through.. Additional points to consider: according to the news media there seems to be a shortage of test kits; this might result in fewer cases being recorded, There are also reports that the NHS has been recording fewer deaths than ONS has. If this is being corrected, the NHS published deaths may rise.
Finally, note that number of Total Deaths in the projections is still very high. Even with r(t) = 0.1, as in UK02G, it comes to about 50,000. One has to get the rate well below that for the number of deaths projected to come down.
Further update 30 March p.m.
I have been able to construct a new Version 3-0, and write notes on it quit quickly. The new Version and the Notes are in the files below. The model is more realistic in that it allows for many cases of Covid-19 not being reported, and also allows for delay before those infected notice that they are infected. The disturbing result is that, using the values of the parameters that best represent the course of the infection before the recent measures of social distancing and isolation were implemented, the projection with no change in behaviour gives almost one million deaths by the end of May. It is this sort of result that makes an author hope that someone can find an error is his logic or calculations. Please try to find it.
Update 30 March
I have done some investigations into what happens when one changes the values of some of the parameters. These are described in the Word file “CV19Model Notes on UK_04_05_06.docx” and in Excel files “Cv19DailyModel_UK04”, “Cv19DailyModel_UK05” and “Cv19DailyModel_UK04 ” which can be downloaded below. UK04 shows what happens when the number of initially healthy, H(0), (or the susceptible) is changed. UK05 shows what happens when the infection rate, r(t), is changed after 21 March. UK06 investigates the possibility of “hidden cases”. This last suggests that the model could be exposed a little,and work is under way on this. But these results are interesting in themselves.
First posts: 29 March 2020
The first basic model
I have produced a system for modelling Covid-19 in daily steps in one Excel worksheet. I describe the model in the Word document “Cv19DailyModel_V2.2.docx”, and the model is contained in the Excel file “Cv19DailyModel_V2.xlsx”.
My first trials on UK data
I have then done first trials on the UK data, which I describe in “Cv19DailyModel-UK02.docx” and show in the Excel file “Cv19DailyModel-UK02.xlsx”. I use a file of data extracted from the European Centre for Disease Control (ECDC) website and processed into a tidy form shown in the file: “COVID19-ECDC-2020-03-24.xlsx”. Note that ECDC is a day different from the UK. The latest data in their file is dated 24 March, but published in the UK on 23 March, so is really perhaps cases and deaths noted on 22 March.
Make use of this if you wish, and pass it on to anyone that might be interested. My actuarial approach is very similar to that of the epidemiologists, but adds duration since the start of infection as a feature. It is easy to recalibrate for any other country.
I must emphasise that I am not making specific forecasts. I am providing a tool that allows others to try out the results of hypothetical changes in what seem to me to be a plausible set of initial assumptions. If I made it much more realistic by stratifying the population by age, the model would need a full programme, which would be much harder for people to play with. Those who are happy changing the Excel may modify it as they wish. Others may just try altering certain values. My only request is that you recognise the authorship, and do not try copyrighting it yourself to prevent others using it.
File links below
29 March 2020