dealtn wrote:spasmodicus wrote:
NOTES ON THE MODEL
The first parameter to decide is the % of the population which have immunity. There are various ways of going about this. Assuming an IFR of 1%, with the cumulative death figures since the beginning of the pandemic, for everone that died, 99 would survive , so in a populationof 56.3 million, 9.4% or over 5 million would have survived and might have become immune as a result. The ONS publishes figures for population antibody testing which suggest that in Nov 2020 a median figure of 8.7% of the population had antibodies.Their table also shows some decay of those showing antibodies from 7.4% to 5.5% between the first wave in May and August 2020.
The ONS figures exclude those under 16 and there is also uncertainty about the role antibodies play and how general immunity lasts in survivors of covid19 infection. I concluded that 10% would be a reasonable (minimum) figure for the immune population .
it looks like you are using a susceptibility rate of 100%.
As you say the first parameter is to decide the % of the population that have immunity. Your assumption appears to be that immunity only comes about if you have antibodies, which might arise presumably either from prior infection, or vaccination. As such your model is looking at a 100% susceptibility worse case scenario. Maybe that is no bad thing if that's what is intended, but if it is for practical, or predictive purposes, might it not make sense to relax the 100% susceptibility start point, and consider other immunity groups?
Hi there,
In a word, no. It's difficult to go into all the nuances of such in the relatively short notes that I wrote. For simplicity I am using a model in which parameters like RT and IFR do not vary with time, which restricts modelling to periods when these things don't change much. It would be quite simple to make these parameters time variant, as inputs, but one would have to decide how to vary them, increasing complication and the danger of over-fitting. In the model, the susceptible population is calculated from a fixed starting value, e.g. 90% of the population, which had to be decided on some rational basis. Whilst I accpet that some areas of the country may have achieved herd immunity, we are talking here about an average over the whole of England. One could,of course, run the such a model region by region, but this would require a lot of extra work in finding and setting the appropriate parameters.
So the initial estimate that I used for the whole of England is that 90% of the population P was susceptible on Dec 2nd. The susceptibility s, i.e. those not infected, immune or dead is recalculated at each time step j as
s(j) = P – c(j-1) – B(j-1) – X(j-1)
where
P is the total population
c(j) is new daily infections
B(j) is the cumulative number of covid recoveries since j=0
X(j) is the cumulative number of deaths since j=0
now, daily infections are calculated from
c(j) = Rd * s(j) / P
where Rd=RT/Di is a simple (constant) estimate for the number of infections that an infected individual will cause in a day.
RT is an estimate for the R value under current lock down conditions and social behaviour, averaged of course over the whole of England.
Di is the no. of days that a person remains infected (see previous notes).
Every day, some people b(j) will also stop being infected, or an unlucky x(j) will die.
b(j) is calculated as
b(j) = c(j-Di) i.e. everyone infected Di days earlier got better (less those that died)
this is added to the cumulative recoveries Bj), i.e.
B(j) =B(j-1) + b(j)
similarly x(j) is calculated from the IFR and the number of infections XD days before, i.e.
x(j) = c(j-XD) * IFR
XD may be interpreted as the number of days before the average victim dies, after becoming infected. This is a pretty rough estimate, but doesn't affect the overall progression of the pandemic hugely, because x(j) << s(j) so the people taken out through dying don't affect the new infections much.
X(j) = X(j-1) + x(j)
we now have everything needed to calculate the number susceptible s(j)
The progression of the pandemic is an interplay between RT and the ratio of s(j)/P, i.e. those susceptible / size of population. Once s(j)/P becomes less than 1/RT, the pandemic will die out. Putting this another way, if there about 10% herd immunity, RT has to be above about 1.1 for the pandemic to grow. At the beginning of the pandemic, I read somewhere that R0, as conventionally defined in pandemic modelling, was 2.5 to 3 for this coronavirus. The new variant probably has an R0 of 3 to 4 it would seem (has anyone found any stats on that?). RT=1.35 seems to be appropriate to lockdown version 3 and the general (lack of?) adherence to it.
Yes, this is a pretty rough model. To test the effect of initial immunity I tried the following
test a) 0 % intial immunity, RT=1.36. Infection rate on Jan 8th is more than double the ONS value of 1.2 million. To get a reasonable fit, e.g. rms error in fit to infections less than 3%, RT has to be reduced to 1.22
test b) 20% intial immunity, RT=1.36. Infection rate on Jan 8th is about half the ONS value of 1.2million. To get a reasonable fit, RT needs to be increased to 1.54, to give an rms error of about 3%.
The model behaved more or less as I would expect, which is reassuring. In all cases it is tending to overestimate the infections earlier on in December and approaches the ONS value both at the beginning and the end of the period for which the stats are available. This suggests that RT is actually varying slightly with lockdowns and as Christmas came and went.
There should be at least some degree of herd immunity, but I think that 20% initial immunity (averaged over all of England) is probably too high as it requires an unrealistically large value for RT.
Next step I will maybe add some graphics to the spreadsheet. I might also turn the spreadsheet's algorithm into a proper program, e.g. Pascal or Python, where it would be easier to run sensitivity tests, if I can be bothered.
regards,
S
