Building models IV: Findings

In part I of this blog series, I tried to explain the ideas behind doing the whole modelling exercise, the maths of the SIR model and Divoc-91; in part II I started to use the published data on covid-19 to get some sensible numbers (or starting parameters) for my model; in part III I put the spreadsheet together and now, in part IV, I work out what the model for the fictional virus, Divoc-91, can tell me.

Test 1: Different Ro values

I wanted to run the model at the suggested Ro values without an social interventions (see this paper which suggests the Ro value in the UK before control measures was 3.5 to 4), and go down to and Ro of 1.1 looking at death rates, % of the population infected and the amount of time it takes for the pandemic to peak. I’ll set the fraction of infection that end up in ICU as 2%, and the mortality in ICU as 50%.  For plotting (and frankly sanity purposes!), I’ve put the death rates as a percentage of the maximum death rate (which at an Ro value of 4 the number of deaths comes out at about a staggering 1.3 million).

The plot is here:Fig 1

This suggests to me that if we want to reduce the number of deaths, we need reduce the rate at which the disease spreads (the Ro value), and this will have two effects: firstly, reduceing the overall population that gets infected, and secondly allowing the healthcare system to cope. Unfortunately the Ro values need to be less than 1.5 if we’re to see the big reductions in death rates, but there does seem to be a sweet spot at Ro of 1.5 where although 60% of the population catch the disease, the death rates drops to 30% of the maximum and the peak occurs around the (relatively low) 2 year mark.

There is another factor that’s interesting: the death rate (the number of deaths per day) as compared to the natural death rate. Here’s a plot of the number of deaths in each model divided by the number of days until the epidemic peaks, and then divided by the natural death rate. A value of zero on this plot gives the same number of deaths from Divoc-19 as from natural causes. The UK’s average death rate is about 700 people per day.

Fig 2

So at an Ro value of 1.5 we get a slightly smaller number of deaths (on average) from this virus as from natural causes.

Overall, I think Test 1 shows that reducing Ro values to 2 aren’t going to help much. We need to get the Ro value (which remember is the extent to which we spread the virus low) down to around 1.5. That extends the epidemic for 2 years, but reduces the number of people that catch it and reduces the death rate to something closer to the UK normal average death rate.

Test 2: Improving treatments

Test 2A: Improving ICU outcomes

I’ll improve ICU outcomes by changing the mortality rate to patients going into ICU from 50% to 25%, 10%, 5% and 1%; then increase the number of ICU beds from 25000, to 35000 and 45000; with an Ro value of 2. The idea is that the UK could either set up more ICU beds (although that’s not easy because they need staff and equipment) and develop better ICU treatment regimes that improve patient outcomes. The plot comes out as shown below (sorry the axis values are a bit small – I’d mark a student down for that!!)

Fig 3

It seems to indicate that increasing the number of beds doesn’t help much until the ICU mortality rate starts to drop. Although it is worth mentioning that in this model a faster turn round of ICU patients is the same as having more ICU beds. (For example, reducing the ICU stay of a patient from 10 days to 5 days will double the number of available daily, ICU spaces.)

Test 2B: Improving pre-ICU treatments

What happens to mortality if we can reduce the proportion of patients needing ICU treatment: perhaps some of the clinical trials which are on going produce drug therapies that achieve this. Again I’ll use a Ro value of 2, set the fraction of ICU patients dying at 25%, and change the number of infected individuals needing ICU treatment from 5%, to 2%, to 1% to 0.5 to 0.1%. I haven’t shown the data here because it turns out there’s a straight line between mortality and the fraction of people that need to go into ICU. It appears that any kind of therapy that keeps people out of ICU would be a really good thing.

Test 3: Immunisations

Assuming that we’re looking for a Divoc-91 vaccine at some point in the future, how much can we relax in our lockdown processes because at some point in the future we will have a vaccine.

Test 3A: Vaccination dates and spread rates

I’ll assume that we get a vaccine 6 months, 12 months, 18 months and 24 months after the infection starts, with Ro values of 3, 2 and 1.5. I’ll set the fraction of infected people that end up in ICU as 2%, and the mortality in ICU as 50%. At the timepoint vaccination occurs I’ll move half of the remaining S (susceptible) population to the R (resistance) population.


For Ro = 3 it doesn’t make much difference: the epidemic has peaked and is largely over  over after 6 months. For Ro =2 we have more time, but we’d need the vaccine at around the 12 month point to make any difference.  Ro of 1.5 is really what we need to be aiming at.

Test 3B: Vaccination coverage and dates

I’ll assume that we get a vaccine 6 months, 12 months or 18 months, and that the coverage of the vaccine changes from 50% to 75% and 90% (say for example we only get limited vaccine quantities to start off with). I’ll keep the same values as in Test 3A, and use an Ro value of 2. At the vaccination timepoint I’ll move the required amount of the remaining S (susceptible) population to the R (resistance) population.


Vaccine timing seems to be key here: although all the timings will be relative to the Ro values of the virus moving through the population.

It might be worth saying at an Ro of 3 we are already between 6-7 months into the infection, at an Ro of 2 we are already between 9-11 months into the infection, and at an Ro of 1.5 we are already 20-21 months into the infection (I’m assuming that 1 million people are infected at the current time from 100,000 confirmed infections with a 1:10 diagnosis:infection ratio). I should have done this analysis with high I and R numbers.

Summary thoughts

In the next post, I’ll try and sum up some personal thoughts on what this has taught me.


Categories: Tags: , ,

1 Comment

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s