Building models II: Parameters

In part I of this blog series I tried to explain the SIR model, but I now need to try and use the science and medical literature to work out some sensible starting parameters. (Basically, I need some numbers to tell me where to start.)

The SIR values?

We need to try and find values for S, I, R, a and b so we can get going on just the equations part.

To start (on day 1 of my model) I’ll assume that S = 60 million (the population of the UK), I = 1 (a single infected person) and R = 0. I’ll design the spreadsheet so different options of S and I are available (should I want to model Scotland – S=6 million – or India – S = 1400 million).

So, that just leaves a and b. I can get one of these from Ro, as long as I know the other. In this case I’m going to calculate b first. But…and this makes me sooooo much of an Armchair Epidemiologist…I’m going to use some physics to help me!

In part I, I mentioned this equation:

How fast the number of Recovered people changes per day = b x I

Is what we call a ‘first order rate equation’ and we see it all over different science disciplines (in chemistry it controls how fast reactions occur, in biology it specifies how quickly a drug is metabolised by the body, and in physics it tells us how long radioactivity lasts). Nuclear physics tells us that the average lifetime of a radioactive atom is given the equation:

Average lifetime = 1/rate constant

See this wikipedia link for details, but there are many other sources.

So for my SIR model this means that

Average length of time a person is infected for = 1/b

and so:

b = 1/Average length of time a person is infected for

(This article confirms this equation in relation to the SIR model).

While, I’m building a model for Divoc-91, I’m using estimates from the current Covid-19 pandemic, and fortunately, the average length of time a person is infected with Covid-19 has been studied. It appears to be around 20 days (or at least this is the median duration of viral shedding). Another paper reports the time between illness and recovery (or death) has been reported as being between 19 and 25 days.

However, that’s not all. The values in these papers are derived from those that are ill enough to go to hospital: and those that go to hospital are in the minority. (More of that below.). It appears that most people recover within about 7-14 days (worldometer report here). The UK authorities are currently asking people to stay isolated for 2 days after they have their last fever, and so we can assume that they may still be infectious for that time.

I also need to consider the length of time someone incubates the virus for (that is they are how longs someone is infected). This value varies quite a lot: a report from the US in early March 2020 (accessed in early April) proposed a median of 5 days; the covid worldometer webpages reported in mid-March 2020 (accessed in early April) studies of 3, 5 and 6 days, while reporting that the incubation period could vary from zero days (which seems surprising) to a month.

So if we consider that 80% of the population have mild disease and so will have 5 days for incubation, 10 days of mild symptoms and another 2 days to become ‘non-infectious’ (and so move from ‘I’ to ‘R’ in my model) we have 17 days.

If the other 20% of the population has significant symptoms and so will have 5 days for incubation and another 20 days to become ‘non-infectious’ we have 25 days.

So, if (0.8 x 17) + (0.2 x 25) = 18.6 days, so I’ll use 20 to keep it simple.

So if b = 20 days, then this means that b for my model is 0.05 (day-1) and that a for my model is equal to Ro x 0.05.

What the number of infected people mean for hospitals and mortality?

Knowing the number of infected people is only part of what I want to do with the spreadsheet. I want to know what the impact of all those infected people might have on the health service. We know that there are limits to the number of bed on wards and in intensive care units. There are also a certain number of infected people that will die, and so we can work out the mortality rates.

So where will I get these numbers from?

One of the first paper I looked at was this one from the Imperial College: this was supposedly the paper that changed the governments mind from a ‘herd immunity’ approach (allow the virus to spread and rely on immunity to build in the community to slow the spread down) to their ‘lockdown’ approach (restrict the movement of people to slow the spread and reduce the number of people catching the virus). This paper suggests that 40-50% of people that catch the virus show no symptoms at all (so called ‘asymptomatic), that 4.4% of cases need hospitalisation, that 30% of those cases need critical care, and that 30% of those that need critical case die. That gives and overall mortality rate of just under 1%. The paper also suggests that the average stay in ICU is 10 days.

[A few days ago, Covid-19 vlogger, John Campbell summarised on a new German study that provided antibody data indicating the number of asymptomatic case may be seven times higher than the number of symptomatic cases.]

However the mortality rate seems to vary from country to country (probably as a result of different testing regimes, different groups of people within an infected country, and different methods of counting deaths).  This article (in the Spectator) points to variabilities between 10% (in Spain and Italy) to less than 1% in Germany. Currently in Scotland the number of deaths/number of confirmed cases is 16%. However, we know that it’s mostly only patients in hospitals that are tested for Covid-19, so if only 5% of people infected with Covid-19 then the real number of infected people is 20 time higher than the published values (which is in keeping with recent statements from the Scottish government).

I’ll run the model with between 5 and 20% of patients needing hospital treatment, and 30% of those patients needing critical care, and between 30% and 50% of those in ICU dying. (The 50% value comes from a comment I had from a frontline doctor friend of mine.)

In terms of NHS capacity, some other numbers are needed. The number of UK hospital beds at 140,000 can be found here. I’ll assume that 80% of those could be freed up for Covid cases and I’ll assume that the average stay in hospital is 2 weeks long (14 days).

The target number of ICU beds has been widely touted in the UK as 30000. The level is normally 8000, so I’ll assume that 25,000 is the number of ICU beds available to Covid patients.

Spreadsheet design

Now that I have the numbers I need, I need to design the spreadsheet. There are quite a few things I want to design into the spreadsheet.

  1. To make sure I can alter the Ro value (so I can see if I can understand what happens to the spread of Dovic-91 when the Ro value changes at different points in time).
  2. To be able to alter the population sizes.
  3. To be able to alter the proportion of people needing hospitalisation, and the mortality rate, and know when the number of infected people exceed these values.
  4. I want a dashboard-like effect, so I can see the effects of any changes I make without having to replot data, or sort through lots of tables!

Spreadsheet size: as I design my spreadsheet I only need 100 data points. However, the pandemic might last longer than 100 days, so I need to be able to alter the ‘time unit’. So for example if I wanted to monitor the course of the pandemic for 3 years, I would set the time unit to 10 days. However I need to make sure the calculation for b (and so a) changes accordingly.

During the design I had to make sure that the numbers of infected people I was counting were those that had become infected during a single time unit and not the number of people that were infected during during a single time unit: the two numbers are different.

And so to spreadsheet!

In the next post I’ll go through the spreadsheet equations and the design.

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