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Publication - Research and analysis

Coronavirus (COVID-19): modelling the epidemic (issue no. 24)

Published
29 October 2020
From
Director-General Health and Social Care
Directorate
Constitution Directorate
Topic
Coronavirus (COVID-19) in Scotland
ISBN
9781800042766

Latest findings in modelling the COVID-19 epidemic in Scotland, both in terms of the spread of the disease through the population (epidemiological modelling) and of the demands it will place on the system, for example in terms of health care requirement.

View supporting documents Supporting documents

Coronavirus (COVID-19): modelling the epidemic in Scotland (Issue No. 24)

Background

This is a report on the Scottish Government modelling of the spread and level of Covid-19. This updates the previous publication on modelling of Covid-19 in Scotland published on 22 October 2020. The estimates in this document help the Scottish Government, the health service and the wider public sector plan and put in place what is needed to keep us safe and treat people who have the virus.

This edition of the research findings focuses on the epidemic as a whole, looking at estimates of R, growth rate and incidence as well as local measures of change in the epidemic.

Key Points

  • The reproduction rate R in Scotland is currently estimated as being between 1.0 and 1.3.
  • The number of new daily infections for Scotland is estimated as being between 130 and 337, per 100,000 people.
  • The growth rate for Scotland is estimated as being between +1% and +5%.
  • The estimated doubling time for Scotland was between 17 and 52 days.
  • There was a significant increase in contacts when the schools went back (from just below 6 to 8 per day). However, the number of contacts has fallen significantly in the last two weeks (down around 17%), and is now lower than the level immediately before the schools returned by around 15%.
  • There has also been a drop off in how many people are visiting different locations, particularly in the hospitality category. This suggests restrictions on households meeting and further restrictions on hospitality are having a noticeable effect that is likely to feed through to confirmed cases over the next week.
  • Modelled rates per 100K indicate that by the week of 8 - 14 November, 23 local authorities have at least a 75% probability of exceeding
    50 cases, 20 have at least a 75% probability of exceeding 100 cases, 12 have at least a 75% probability of exceeding 300 and 7 of those have at least a 75% probability of exceeding 500.

Overview of Scottish Government Modelling

Epidemiology is the study of how diseases spread within populations. One way we do this is using our best understanding of the way the infection is passed on and how it affects people who catch it to create mathematical simulations. Because people who catch Covid-19 have a relatively long period in which they can pass it on to others before they begin to have symptoms, and the majority of people infected with the virus will experience mild symptoms, this "epidemiological modelling" provides insights into the epidemic that cannot easily be measured through testing e.g. of those with symptoms, as it estimates the total number of new daily infections and infectious people, including those who are asymptomatic or have mild symptoms.

Modelling also allows us to make short-term forecasts of what may happen with a degree of uncertainty. These can be used in health care and other planning. The modelling in this research findings is undertaken using different types of data which going forward aims to both model the progress of the epidemic in Scotland and provide early indications of where any changes are taking place.

Modelling outputs are provided here on the current epidemic in Scotland as a whole, based on a range of methods. Because it takes a little over three weeks on average for a person who catches Covid-19 to show symptoms, become sick, and either die or recover, there is a time lag in what our model can tell us about any re-emergence of the epidemic and where in Scotland this might occur. However modelling of Covid deaths is an important measure of where Scotland lies in its epidemic as a whole. In addition the modelling groups which feed into the SAGE consensus use a range of other data along with deaths in their estimates of R and growth rate. These outputs are provided in the first part of this research findings. The type of data used in each model to estimate R is highlighted in Figure 2.

A short term forecast of the number of cases in the next two weeks is also provided, as the focus at this stage of the epidemic is the re-emergence of the virus in Scotland.

A new tranche of results are provided from the Scottish Contact Survey (SCS), to indicate how people's contacts are changing.

What the modelling tells us about the epidemic as a whole

Figure 1 shows how Rt has changed since February. Before the "stay at home" restrictions were put in place Rt was above 1, and most likely to have been between 3 and 4 before any interventions were put in place.

Figure 1: Trends in R t for Scotland, 2020
Figure 1. A graph showing the trends in the Rt value for Scotland over time, as calculated by the model. The graph shows step changes downwards at the point when each intervention was introduced. This figure shows Rt fell below 1.0 on the 23rd of March, when the “stay at home” advice was given. The Rt has been above 1 since around the time the school summer holidays ended.

Source: Scottish Government modelled estimates using Imperial College model code; actual data from https://www.nrscotland.gov.uk/statistics-and-data/statistics/statistics-by-theme/vital-events/general-publications/weekly-and-monthly-data-on-births-and-deaths/deaths-involving-coronavirus-covid-19-in-scotland

The various groups which report to the Scientific Pandemic Influenza Group on Modelling (SPI-M) use different sources of data in their models (i.e. deaths, hospital admissions, cases) so their estimates of R are also based on these different methods. SAGE's consensus view across these methods, as of 28 October, was that the value of Rt in Scotland was above 1, between 1.0 and 1.3, meaning that the epidemic is growing exponentially. The R value estimated by the Scottish Government is higher this week, this is due to deaths, which are used in this model, lagging cases (Figure 2).

Figure 2. Estimates of R t for Scotland, as of 28 October, including 90% confidence intervals, produced by SAGE. The blue bars are death-based models, purple use multiple sources of data and cyan use Covid‑19 test results. The estimate produced by the Scottish Government (a semi-mechanistic model) is the 3rd from left (yellow), while the SAGE consensus range is the right-most (red).
Figure 2. A graph showing the range of values which each of the academic groups reporting an R value to SAGE are likely to lie within, as of 28 October. The blue bars (first and second from left) are death-based models, purple (4th to 10th from the left) use multiple sources of data and cyan (11th from the left) use Covid-19 test results. The estimate produced by the Scottish Government (a deaths-based model) is the 3rd from left (yellow). The R value estimated by the Scottish Government is similar to the estimates of other groups using models which draw upon numbers of deaths. The SAGE consensus, shown at the right hand side of the plot, is that the most likely “true” range is between 1.0 and 1.3.

Source: Scientific Advisory Group for Emergencies (SAGE).

On 26 October, Public Health Scotland recorded 1,327[1] positive new cases, with 8,153 positive new cases over the week of 20 – 26 October.

The various groups which report to the Scientific Pandemic Influenza Group on Modelling (SPI-M) use different sources of data in their models to produce estimates of incidence. SPI-M's consensus view across these methods, as of 28 October, was that the incidence of new daily infections in Scotland was between 130 and 337 new infections per 100,000. This equates to between 7,100 and 18,400 people becoming infected each day in Scotland.

Figure 3. Estimates of incidence for Scotland, as of 28 October, including 90% confidence intervals, produced by SPI-M. The purple bars represent models which use multiple sources of data. The estimate produced by the Scottish Government (a semi-mechanistic model) is the third from left (yellow), while the SAGE consensus range is the right-most (red).
Figure 3. A graph showing the ranges the values which each of the academic groups in SPI-M are reporting for incidence (new daily infections per 100,000) are likely to lie within, as of 28 October. The blue bars are death based models (1st and 2nd from left). The purple bars (4th to 6th from the left) use multiple sources of data. The estimate produced by the Scottish Government (a deaths-based model) is the 3rd from the left (yellow). The SAGE consensus (130 to 337 new daily infections per 100,000) is shown at the right hand side of the plot.

Source: Scientific Pandemic Influenza Group on Modelling (SPI-M).

The consensus from SAGE for this week is that the growth rate in Scotland is between +1% and +5% per day. Last week the growth rate was in the range +4% to +7%.

The spread of the epidemic can be expressed in terms of the length of time it takes for numbers of new daily cases to double. Doubling times were provided by SPI-M on 28 October. The consensus estimated doubling time for Scotland was between 17 and 52 days.

Figure 4 shows the epidemiological model forecasts of daily deaths produced by the Scottish Government, given the present set of interventions. This measure of the epidemic is forecast to increase in the weeks ahead.

Figure 4. Scottish Government short-term forecast of the number of deaths from Covid-19 in Scotland, based on actual data (20 October).

Figure 4. Scottish Government short-term forecast of the number of deaths from Covid-19 in Scotland, based on actual data (20 October).
Figure 4. A bar chart showing daily numbers of deaths caused by Covid-19 in Scotland between 12th March and 20th October, 2020. Overlain on this is the “estimated deaths” result from the model, which smooths out the cyclical weekly pattern in the reported numbers, due to fewer deaths being registered over a weekend.

Source: Scottish Government modelled estimates using Imperial College model code; actual data from https://www.nrscotland.gov.uk/statistics-and-data/statistics/statistics-by-theme/vital-events/general-publications/weekly-and-monthly-data-on-births-and-deaths/deaths-involving-coronavirus-covid-19-in-scotland

The logistical model developed by Scottish Government to assess implications for health care demand (see previous Research Findings) has been adapted to produce a short/medium-term predictions of infections.

The following two week prediction uses this model to extend the estimated number of infections from the Imperial College model, in a manner that fits with the estimated number of actual infections, adjusting people who have tested positive to account for asymptomatic and undetected infections.

Figure 5 shows a "better scenario", which assumes the current Rt value was reduced significantly as a result of measures announced on 7 October, and a "worse scenario", which assumes that transmission decreased by a smaller amount. In comparison to this chart in previous versions of this paper, a calculation has been applied to data on people who have tested positive to correct for weekend effects in testing, in order to better estimate actual new infections.

Figure 5. Short term forecast of modelled total new infections, adjusting positive tests to account for asymptomatic and undetected infections, from Scottish Government modelling, positive test data up to 24 October.
Figure 5. A line graph showing the two week ahead better and worse scenario predictions using the logistics model to extend the estimated number of infections from the Imperial College model in a manner that fits with the number of actual cases. The worse scenario indicates over 15,000 cases, whereas the better scenario indicates up to around 4,800 cases in two weeks.

What the modelling tells us about Hospital bed and ICU bed demand

Figure 6 shows the impact of the better and worse scenarios on the number of people in hospital. As figure 5 shows, infections appear to be following a path between the worse and better scenario, and the trajectory of the hospital bed demand is likely to follow suit.

Figure 6. Short term forecast of modelled hospital bed demand, from Scottish Government modelling.
Figure 6. A line graph showing the two week ahead better and worse scenario predictions of hospital bed demand. The worse scenario indicates up to 2,500 beds, whereas the better scenario indicates over 1,500 hospital beds required in two weeks.

Figure 7 shows the impact of the better and worse scenarios on ICU bed demand. As things stand, the bed demand estimates lie consistently above the actual number of people in ICU, though it is well within the prediction interval.

Figure 7. Short term forecast of modelled ICU bed demand, from Scottish Government modelling.
Figure 7. A line graph showing a short term forecast of modelled ICU bed demand, from Scottish Government modelling. The worse scenario indicates up to 350 ICU required, whereas the better scenario indicates up to around 250 ICU required in two weeks.

What we know about how people's contact patterns have changed

It is now possible for us to estimate how much contact people in Scotland have with each other, with a good degree of accuracy. This provides an update of the modelled results presented in issue 22 using methodologies developed by the London School of Hygiene and Tropical Medicine. The modelling is based on a survey asking about where respondents have been and how many contacts they met in a given week.

The average number of contacts per day are approximately two thirds higher than the level at the beginning of Stay-at-home-advice, and around half the level pre-Stay-at-home-advice (UK comparison 10.8). There was a significant increase in contacts when the schools went back (from just below 6 to 8 per day). However, the number of contacts has fallen significantly in the last two weeks (down around 17%), and is now lower than the level immediately before the schools returned (Figure 8) by around 15%. There has also been a drop off in how many people are visiting different locations, particularly in the hospitality category. This suggests restrictions on households meeting and further restrictions on hospitality are having a noticeable effect that is likely to feed through to confirmed cases over the next week.

From the results older people generally have fewer contacts than younger people, but the difference is largely from work and school contacts, rather than in the home or in other settings.

Although the mean number of contacts per person in the latest wave (15 to 21 October) of the SCS has reduced as shown in Figure 8, it remains significantly higher than the 2.8 contacts per person for the UK as a whole at the beginning of Stay-at-home-advice.

Figure 8: Mean Adult Contacts outside of household (truncated at 100) from SCS.
Figure 8. A barchart showing mean adult contacts outside household with non-household members by age group from 6 Aug to 21 Oct.

As shown in Figure 9, contacts are fewer in older age groups, with the oldest age group having similar levels of contact to the UK during Stay-at-home-advice (2.0 from CoMix), and the youngest age group having similar levels of contact to the UK average prior to Stay-at-home-advice (12.1 from POLYMOD[2]) until mid-September, although this fell significantly at the start of October and remains the same for the most recent survey.

Figure 9: Average (mean) contacts per day by setting for adults in Scotland, truncated to 100 contacts per participant (from SCS).
Figure 9. A series of bar charts showing locations visited, by age group, by participants from 6 Aug to 21 Oct.

The main difference between age groups is that younger people are having more contacts at work and at school however, the latest reporting week shows a significant drop in work contacts for the 30-39 age group. There is little difference in between age groups in the number of contacts reported in "other" settings at around 1.5 contacts per day, and the youngest and oldest age groups also report similar contacts in the home (the middle age groups report higher contacts in the home than the youngest and oldest) as set out in Figure 9.

As Figure 10 shows, there has been a reduction in the proportion of people that have visited other people's homes since the first wave of the survey until the start of October, however this has levelled out in the past two weeks. This is also reflected in the mean contacts in the home in Figure 9. The most significant decrease can be seen in the number of people visiting pubs which will be largely due to the restrictions placed on hospitality, introduced on 9 October.

There had been a slight increase in the number of people travelling to work and to gyms and sports clubs, but this has reduced again in the most recent wave (after the introduction of the rule of six and restrictions on meeting in other people's households).

Figure 10: Locations visited by participants at least once between 6 August and 21 October (from SCS).
Figure 10. A bar chart showing the proportion of respondents who visited certain locations (including shops, outside and someone else’s home) at least once from 6 Aug to 21 Oct.

What we know about which regions are experiencing high levels of Covid

We use modelling based on Covid cases and deaths, conducted by Imperial College London, to give us an indication of whether a local authority is experiencing high levels of Covid. The model projects the epidemic two weeks into the future and computes probabilities of areas being hotspots at certain thresholds, where an area is defined as a hotspot if the cases (positive tests) per 100K population exceed a threshold, e.g. 500 weekly cases.

Modelled cases per 100K (Figure 11) indicate that by the week of 8 - 14 November, 23 local authorities are projected to have at least a 75% probability of exceeding 50 cases, 20 of those have at least a 75% probability of exceeding 100 cases, 12 of those have at least a 75% probability of exceeding 300 cases and 7 of those have at least a 75% probability of exceeding 500.

Figure 11. Probability of local authority areas having more than 50, 100, 200 or 500 cases per 100K (8 - 14 Nov 2020). Data updated on 27 October [3].
Figure 11. A series of four maps showing the probability of Scottish local authorities having more than 50, 100, 200 or 500 cases per 100,000 population, corresponding to data for 8 – 14 November 2020.

Note: Hotspot 200 is included, rather than Hotspot 300 which will be in future research findings.

What next?

The Scottish Government continues to work with a number of academic modelling groups to develop other estimates of the epidemic in Scotland. This includes updates to the forecast of the number of hospitalisations and the pattern of positive tests in different age and clinical risk groups which were reported in issue 23. This has not been included in this issue as the proportions have changed very little this week. We will provide an update when we see a significant change.

The modelled estimates of the numbers of new cases and infectious people will continue to be provided as measures of the epidemic as a whole, along with measures of the current point in the epidemic such as exceedance. Rt and growth rate will also be provided. Further information can be found at https://www.gov.scot/coronavirus-covid-19.

Technical Annex

The Imperial College London results were computed using Epidemia, which extends the Bayesian semi-mechanistic model proposed in Flaxman, S., Mishra, S., Gandy, A. et al[4].

The model is based on a self-renewal equation which uses a time-varying reproduction number R to calculate the infections. However, due to a lot of uncertainty around reported cases in the early part of epidemics, they use reported deaths to back-calculate the infections as a latent variable. Then the model utilises these latent infections together with probabilistic lags related to SARS-CoV-2 to calibrate against the reported deaths and the reported cases since the beginning of June 2020. A detailed mathematical description of the original model is available[5]. The model used is an evolution of this original model, incorporating randomness in the infections to account for areas with small numbers of infections.

R for each local authority is parameterized as a linear function of the R for its region as a whole (which it is fitted to), and a random effect specific to the local authority for each week over the course of the epidemic. The weekly random effects are encoded as a random walk, where at each successive step the random effect has an equal chance of moving upward or downward. Only a region-specific weekly random walk is used. For more information, and limitations of the projections, consult the Imperial College London web page[6].

Table 1 provides the underlying data used in the section above on "What we know about which regions are experiencing high levels of Covid". It is provided by Imperial College London.

Table 1. Probability of local authority areas having more than 50, 100, 300 or 500 cases per 100K (8 - 14 Nov 2020).
LA P(Cases>500) P(Cases>300) P(Cases>100) P(Cases>50)
Aberdeen City 0% 0% 2% 11%
Aberdeenshire 0% 1% 23% 67%
Angus 0% 4% 73% 96%
Argyll and Bute 0% 2% 53% 85%
City of Edinburgh 0% 0% 2% 28%
Clackmannanshire 71% 95% 100% 100%
Dumfries and Galloway 1% 27% 88% 97%
Dundee City 56% 92% 100% 100%
East Ayrshire 80% 95% 100% 100%
East Dunbartonshire 89% 99% 100% 100%
East Lothian 0% 16% 88% 98%
East Renfrewshire 82% 99% 100% 100%
Falkirk 2% 39% 97% 100%
Fife 0% 4% 93% 100%
Glasgow City 19% 85% 100% 100%
Highland 0% 0% 1% 17%
Inverclyde 1% 18% 91% 99%
Midlothian 4% 47% 98% 100%
Moray 0% 0% 12% 44%
Na h-Eileanan Siar 2% 7% 40% 59%
North Ayrshire 98% 100% 100% 100%
North Lanarkshire 88% 99% 100% 100%
Orkney Islands 3% 9% 40% 58%
Perth and Kinross 1% 6% 57% 87%
Renfrewshire 47% 81% 99% 100%
Scottish Borders 0% 0% 12% 49%
Shetland Islands 1% 4% 28% 47%
South Ayrshire 52% 93% 100% 100%
South Lanarkshire 81% 98% 100% 100%
Stirling 2% 20% 89% 98%
West Dunbartonshire 21% 71% 99% 100%
West Lothian 79% 98% 100% 100%

Tables 2 and 3 provide the underlying data used in the section above on "What the modelling tells us about Hospital bed and ICU bed demand". They are based on modelling undertaken by Scottish Government (for more information see research findings issue 1).

Table 2 Estimated demand for ICU beds - Worse Scenario
Area 02/11/2020 09/11/2020 16/11/2020 23/11/2020 30/11/2020 07/12/2020
Ayrshire & Arran 18 21 25 32 42 55
Borders 1 1 1 1 2 2
Dumfries & Galloway 2 3 3 4 6 8
Fife 5 5 6 8 11 14
Forth Valley 7 9 10 13 17 22
Grampian 3 3 4 5 6 8
Greater Glasgow & Clyde 50 60 70 90 118 155
Highland 2 2 2 3 4 5
Lanarkshire 50 60 69 89 117 154
Lothian 17 21 24 31 41 54
Orkney 0 0 0 0 0 0
Shetland 0 0 0 0 0 0
Tayside 9 10 12 15 20 27
Western Isles 0 0 0 0 1 1
Table 3. Estimated demand for hospital beds - Worse Scenario
Area 02/11/2020 09/11/2020 16/11/2020 23/11/2020 30/11/2020 07/12/2020
Ayrshire & Arran 130 157 184 241 318 419
Borders 6 7 8 10 14 18
Dumfries & Galloway 18 22 26 34 44 58
Fife 34 40 48 62 82 108
Forth Valley 52 63 74 97 128 168
Grampian 19 23 27 35 46 61
Greater Glasgow & Clyde 367 442 520 678 898 1181
Highland 11 13 16 21 27 36
Lanarkshire 364 439 516 674 892 1173
Lothian 127 154 180 236 312 410
Orkney 1 1 1 1 2 2
Shetland 0 1 1 1 1 2
Tayside 63 76 89 117 154 203
Western Isles 2 2 3 3 4 6

Contact

Email: modellingcoronavirus@gov.scot

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