Modelling the UK pandemic in 2021
For anyone trying to predict the aviation recovery in 2021, the most important assumption that needs to be made is how the pandemic will develop. There is a surprising lack of publicly available forecasts that I could find. Imperial University has one on its website, but it is hard to work out what assumptions lie behind it. So after taking an online course from Imperial on “Infectious Disease Modelling” (available via Coursera), I decided to build my own model for the UK and generate some forecasts.
I will apologise now for the fact that this post is rather longer than usual, as it is quite a complicated subject. Hopefully you will find it interesting and I look forward to getting feedback on areas I could improve, either in the model or in the assumptions.
The model
Imperial’s course uses a language called R to build the model and therefore I did the same initially. At the risk of offending any R-lovers out there, I have to say that R is quite an annoying language and so after a bit I rewrote the model in Python.
I subdivided the population into two groups, those aged 70 years or older and the rest of the population. That is because the hospitalisation and fatality risk is much higher for the older group. They have a five times higher hospitalisation rate and a 17 times higher fatality rate. They are also less likely to catch and spread the disease. The data from the UK’s Office for National Statistics (ONS) suggest an infection rate of only 40% compared to the average. Because of their higher risk, this group is getting priority access to the vaccine.
The population is then broken down further between different “COVID categories”:
Susceptible: no-immunity to the virus
Vaccinated: people with (possibly imperfect) immunity through having received a vaccination
Infectious: those who have caught the virus and are infectious
Hospitalised: people with sufficiently severe symptoms to become hospitalised
Recovered: those who have had the disease and recovered, assumed to be immune
Dead: people who have sadly died from the disease
Transitions between the states are governed by assumptions about the vaccination programme, the infectiousness of the disease, how long people take to recover etc. There are a lot of assumptions, but I’ve tried to cover the most important in the next few sections.
The R value
I doubt there is anyone who hasn’t heard about the “R value” by now. There are two important concepts, R0 and the effective R.
R0 measures the average number of people infected by an infectious individual in a scenario where the entire population is susceptible. I believe that the “classic” strain of the COVID virus has an R0 of 2.65 in the absence of restrictions such as lockdowns. However, in the UK, it is the new “Kent variant” of the virus that has become the dominant strain and that has a transmissibility which is 54% higher, according to Public Health England. So I’ve used an R0 of 4.
The effective R is lower than R0 due to immunity in the population and also due to the effect of public behaviour change and government restrictions, such as lockdowns. The UK governments’ advisory group SAGE estimated that the UK’s lockdown earlier in 2020 reduced R by 75%. Whether the current lockdown in the UK will be as effective is not yet clear. Restrictions are not quite as severe and compliance may not be as high this time. I used 70% as my base assumption but also looked at 65% as a sensitivity.
One of the key drivers of the effective R is the existing level of immunity in the population. There have been just under 3 million confirmed cases in the UK, but everybody knows that this is understated. The low level of testing during the first wave meant most cases were not recorded. Back in July, it was estimated that 7.1% of the UK population had antibodies against COVID, which would be 4.8m people. At that time, the cumulative confirmed case count was only about 250,000. Although testing has increased massively, the under-reporting of cases is still an issue. Presumably the missing cases are either asymptomatic or for people with less severe symptoms, but I think the gap is quite large. The Office for National Statistics conducts a regular random sample test on the UK population, which suggests that 1 in 50 have the virus today, which is 1.3 million people. I think that once people catch the virus, they will test positive for about a 10 day period on average. With reported new cases having been running at about 55,000 people recently, that should mean that only 550,000 would test positive at any one time. The ONS data point of 1.3 million therefore suggests that the real number of new cases is still around 2.3 times higher than the official statistics.
Adjusting for all those missing cases and allowing for the 1.5 million people who have already been vaccinated, I estimate that 81% of the UK population is still susceptible to the virus. If the current lock-down rules are almost as effective as the first one, with a suppression factor of 70% that gives me an effective R of marginally under one (R0 of 4, times 81% susceptible * 30% to allow for suppression). If suppression is only 65%, the effective R at the moment is 1.1, with the pandemic continuing to grow exponentially.
That means that the job of bringing the infection rate down is fully dependent on the pace of the vaccination rollout. Let us look at that next.
Vaccinations
The UK has now approved three vaccines, but most of heavy lifting will have to be done by the Oxford / AstraZeneca one, since that is the vaccine that is both easiest from a logistical point of view and where the UK has secured the most supply over the critical next couple of months.
The government has stated a target of delivering 2 million vaccinations a week, and have also said they plan to have offered vaccination slots to the 13 million people in the top priority groups by 15th February. They only managed to do 333,234 in the last week, so there is clearly going to be a period of ramp up. I’ve assumed that going forward they will manage 0.5m a week, ramping smoothly up to 2m a week within a month. That implies they will only get to 9.2 million vaccinations by the 15th February. I find with this government, it is best to assume that they will overpromise and underdeliver.
Somewhat controversially, the UK has decided to prioritise “first doses” and come back to doing second doses up to 12 weeks later. I haven’t allowed for second dose volumes in my model, so I am implicitly assuming that the rate of vaccination will actually be higher than 2 million a week from April onwards to sustain “first dose” volume levels.
I have made an allowance for the vaccines not being fully effective. I’ve assumed 70%, which is reported to be the effectiveness of the Oxford / AstraZeneca vaccine after one dose. I have assumed that as well as being effective in preventing symptoms, the vaccine is also effective at preventing people becoming infectious. That has not been proven as yet, but most experts believe it is very likely.
I’ve allocated the vaccinations as a first priority to my 70+ age category, with the other category only being vaccinated once that group has been done (around the middle of February). That isn’t quite right, as health care workers and other high risk people are also being prioritised, but it is close enough for me.
One other observation is that the government has said it is not going to differentiate between people that have had COVID and those that haven’t when it comes to vaccinations. That seems like the wrong policy to me, because it is “wasting” scarce vaccine on people that should already have immunity. Based on my projections, around a third of the vaccines given over the next six months are likely to go to people that will probably already be immune. But I have followed the government approach in my modelling.
Enough preamble, what are the results from the model?
Case numbers
With a suppression assumption of 70%, the number of cases starts to fall immediately and the rate of decline starts to accelerate as the vaccination programme cuts in. However, it takes until early March for the proportion of the population with COVID to halve, dropping to 1 in 100, compared to 1 in 50 today.
To get to the levels of infection we were at in the summer (around 1 in 2,000) takes until the end of May. The extra transmissibility of the “Kent virus strain” makes it so much harder to bring down infection levels than it was for the first wave, despite the helping hand from the vaccines.
The number of susceptible people starts off at about 81% of the population, but drops progressively due to both vaccinations and to natural immunity as the cumulative cases continues to climb from an estimated 10 million people at the start to 18 million by the time the pandemic is brought under control.
With an R0 value of 4, herd immunity requires 75% of the population to be immune (it’s an easy calculation, 1 - 1/R0). Even if recovery from the disease confers full immunity, if the vaccine is only 70% effective, that level of immunity will still not be reached by the end of July, with only 68.5% immune. That would mean that suppression measures which deliver a 20% reduction of transmission would still be required. So I guess mask-wearing may be here to stay for much of 2021.
Of course, from April onwards, the second vaccination doses will have been delivered and the level of immunity of vaccinated people may be higher than 70% by then. If the vaccine proved to be 90% effective, we would achieve herd immunity without suppression measures by early July.
Even though the case numbers start to decline almost immediately, the high number of cases that we already have of course feeds through into additional fatalities. My projection is that daily deaths will be above 500 for most of January and the final count will reach almost 110,000, an additional 36% compared to today.
This projection assumes that mortality rates are not increased due to the new strain and also that deaths are not driven up by an overwhelmed health service. Anyone who has been watching the news in the UK recently will know that there is a real risk of that happening.
My projection is for the number of people in hospital to continue to climb for the next week before falling back. Actually, I think the lags in the system are probably bigger than accounted for in my model due to things such as reporting delays, so I’d probably guess it might take two weeks before hospital numbers start to drop. Update 10th Jan: I had been using 9 days as the average length of stay in hospital for COVID patients based on a media report. I’ve now crunched the recent NHS numbers and it looks like the real value is 11 days. Updating that assumption increases and slightly delays the peak, which I now estimate to be over 36,000, hit around the 20th January.
What if the lock-down is less effective?
If the suppression of R achieved by the latest lock-down measures is only 5% less than I've assumed (a suppression factor of 65%), then case numbers will continue to grow for the next month, before vaccinations and natural immunity start to bring the numbers down. In this scenario, it takes a month longer to get back to low levels of infections. There are an extra 5.1 million cases and 13,000 more fatalities. Hospitalisations continue to grow until mid February and hit a peak which is 14% higher. This is why the government is putting so much emphasis on compliance with the rules.
What about aviation?
The model allows me to look at what might be the impact on the pandemic from different aviation restrictions.
I’ve constructed a scenario for the number of international air arrivals by month which is deliberately at the top end of what could possibly happen in 2021. In reality, travel and other restrictions will see numbers below this, but it is designed to test the level of risk to the UK from continuing to accept air arrivals.
In November, there were about 630,000 international air arrivals into the UK, 8% of 2019 levels. I’ve assumed January and February at a similar level, with March at 25% of 2019 volumes, April at 50% and then a gradual ramp up towards 80% from July onwards. That gives a total of 79 million international arrivals in 2021.
I’ve tested a conservative scenario where 2% of these passengers are infectious and they simply add to the pool of infected people in the UK, in a sense simulating a scenario where these passengers are coming from places that are and continue to be about as bad as the UK is at the moment, and no testing or quarantine rules are in place (or they are completely ineffective).
As well as the imported cases created by air travel, there is also an effect in the opposite direction. “Exported cases”, if you like. Planes go in both directions.
What is the impact on my forecast for the UK pandemic? I get an additional million cases and 1,900 deaths. Peak hospitalisations are not impacted, as arrivals are so low during the critical January and February period.
I have written in the past about the effectiveness of a single COVID test in filtering out infected airline passengers and concluded that it is about 60% effective. Applying that 60% reduction would reduce the additional cases to 400,000 and additional fatalities to under 800. Is that still too many, even for a worst case scenario? I’m sure that people will take different positions on that.
Other considerations and risks
All of the above has been based on assumptions, and there are some downside risks if they are wrong.
Firstly, I have assumed that having COVID makes you 100% immune. Whilst confirmed cases of reinfection so far have been very rare, mutations of the virus could change that. Likewise, I have assumed that vaccine effectiveness is not similarly compromised by new strains.
Secondly, although I have allowed for the higher transmission rate of the new “Kent variant”, I have not assumed that it causes more hospitalisations or deaths. The evidence so far makes that look like a reasonable assumption, but there are one or two anecdotal suggestions that it might hit young people harder.
Thirdly, I haven’t allowed for the possibility of yet worse strains arriving in the UK. The South African variant could be one such, and reducing the risk of importing such new strains is the main reason the government has been imposing additional flight restrictions.
Feedback
Let me know if you found this useful and whether there are different assumptions or scenarios you’d like me to look at. I’ll be trying to keep the assumptions up to date as events unfold.
For the UK and for many other countries in a similar position, I think we all have a difficult three months ahead of us. But hopefully there is light at the end of the tunnel now.
