Coronavirus (COVID-19): modelling the epidemic (issue no. 14)

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.

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Coronavirus (COVID-19): modelling the epidemic in Scotland (Issue No. 14)


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 13 August 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, it also looks at whether certain parts of Scotland are exceeding the number of cases which we would expect at this point in the epidemic, and then goes on to introduce another improved modelling based on how many contacts people have with one another.

Key Points

  • The reproduction rate R in Scotland is currently estimated at being between 0.8 and 1.2.
  • The growth rate for Scotland is estimated as being between -6% and +2%.
  • There were 6 local authority areas which exceeded what would be expected at this stage in the epidemic in the last 7 days. These were Perth and Kinross, Dundee, North Lanarkshire, Angus, Glasgow and Aberdeen. These are now returning towards normal levels, however levels of new infections in Perth and Kinross are still exceeding what would be expected at this point in the epidemic. Levels in Aberdeenshire have now returned to background levels.

Overview of Scottish Government Modelling

Epidemiology is the study of how diseases spread within populations. One way we do this is to use 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 the relatively long period in which they can pass it on to others before they begin to have symptoms, and that 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.

Firstly, 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 this model can tell us about any re-emergence of the epidemic and where in Scotland this might occur. However modelling of Covid deaths across the epidemic is an important measure of where Scotland lies in its epidemic as a whole. In additional the modelling groups which feed into the SPI 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. This week the type of data used in each model to estimate R is highlighted in Figure 2.

Secondly, modelling is provided this week around whether the number of positive test results we are seeing exceeds what would be expected. This replaces the short term modelling of NHS capacity provided in these reports previously, as the focus at this stage of the epidemic is around identifying any possible re-emergence of the virus in Scotland rather than whether there is sufficient hospital capacity to treat large numbers of Covid cases.

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 4 and 6 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 falling below 1.0 on the 23rd of March, when the “stay at home” advice was given.

Source: Scottish Government modelled estimates using Imperial College model code; actual data from

The various groups which report to the Scientific Advisory Group on Epidemics (SAGE) use different sources of data in their models (i.e. deaths, hospital admissions, cases) so their at estimates of R are also based on these different methods. SAGE's consensus view across these methods, as of 20 August, was that the value of Rt in Scotland was between 0.8 and 1.2. The R value estimated by the Scottish Government falls within this range, and is similar to the estimates of other groups using models which draw upon numbers of deaths (Figure 2).

Figure 2. Estimates of R t for Scotland, as of 19 August, including 90% confidence intervals, produced by SAGE. The green bar represents a model driven by numbers of new cases, blue 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 deaths-based model) is the 6th 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 19th August. The green bar (at the left hand side of the plot) represents a model driven by numbers of new cases, blue (bars second to fifth from the left) are death-based models, purple (7th to 10th from the left) use multiple sources of data and cyan (11th and 12th from the left) use Covid-19 test results. The estimate produced by the Scottish Government (a deaths-based model) is the 6th from left (yellow) and is around 0.8. 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 0.8 and 1.2.

Source: Scientific Advisory Group for Emergencies (SAGE).

On the 14 August , Public Health Scotland recorded 65 positive tests, with 348 positive tests over week of 8-14 August. At this point in the epidemic, with very low numbers of deaths to inform its outputs, the results of our model have begun to underestimate prevalence and incidence. We are working with SAGE to improve our modelled incidence estimates in light of the recent local outbreaks in Aberdeen and Perth and Kinross. However for this week it should be noted that the number of infectious people in Scotland will be higher than the number of positive tests published by Public Health Scotland as outlined above.

The consensus from SAGE for this week is that the growth rate in Scotland is between -6% and +2% per day. This is similar to last week where growth rate was in the range -7% to +4%.

Figure 3 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 remain near zero in the weeks ahead.

Figure 3. Scottish Government short-term forecast of the number of deaths from Covid-19 in Scotland, based on actual data (11 August).
Figure 3. A bar chart showing daily numbers of deaths caused by Covid-19 in Scotland between 12th March and 11th August, 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

What the modelling tells us about whether Covid infections exceeded what would be expected at this stage in the epidemic

While metrics such as the reproductive rate - the R number - have been useful in guiding our response to Coronavirus so far, the models we use rely on numbers of deaths to track the epidemic. These have fallen to low levels, which means we need to find new ways to monitor trends in the epidemic. One of the ways we can do this is to calculate whether the number of confirmed infections (based on testing) in each area exceeds the number that was expected, given the number recorded across the country - this is called "exceedance". An analysis of trends across Local Authorities in Scotland has been developed by modellers at the University of Warwick on behalf of the Scottish Government.

Numbers of positive tests recorded each day, adjusted for population of each local authority and number of cases seen in preceding weeks, should fall within a certain distribution of values, which will rise and fall depending on the number of cases being seen nationally. Areas where the number of positive test results fall beyond the upper 95th percentile of this distribution may be at risk of seeing increased local transmission of Covid and heightened vigilance may be required. This happens when the cumulative exceedance is higher than 6.0.

In the week preceding 17August, across Scotland the epidemic in Scotland was driven by the outbreak in Perth and Kinross (Figure 5). Recent cumulative exceedance highlighted Perth and Kinross (exceedance = 12.7), Dundee (7.7), North Lanarkshire (5.1), Angus (4.3), Glasgow (3.9) and Aberdeen (3.6) as areas of higher risk of transmission (Figure 6). In most cases these are a reflection of a small number of cases, however the ongoing nature of the outbreak in Perth and Kinross may be a cause for concern. Levels in Aberdeenshire are falling back towards background levels.

Figure 4. Map of cumulative exceedance, 17 August, for Scottish Local Authorities.
Figure 4. A choropleth map of Scotland, divided into Local Authority areas. The shading colour of each Local Authority area is determined by the recent cumulative exceedance at 17th August. Perth and Kinross is shown to have the highest exceedance of any Local Authority.
Figure 5. Graphs of daily and cumulative exceedance for the local authorities deemed as higher risk over the period 11 – 18 August.
Figure 5. Graphs showing the daily and cumulative exceedance for the local authorities of Perth and Kinross, Dundee City, North Lanarkshire, Angus, Glasgow City and Aberdeen City from 26th July 2020 to 18th August 2020.

What next?

Most models are based on the number of daily deaths, however as the numbers of deaths in Scotland are now at low levels, and as deaths happen sometime after people become ill (around three weeks), it is important to have models that pick up immediate upturns in the post-peak phase of the epidemic.

The Scottish Government has also been working with a number of academic modelling groups to develop other estimates of the epidemic in Scotland. Recently, we introduced the Exceedance method developed by the University of Warwick, and next week we hope to introduce a new way of measuring the epidemic which is based on a survey of people's behaviours, rather than numbers of cases or deaths.

The London School of Hygiene and Tropical Medicine have developed an online survey called "CoMix" which collects weekly data on how many social contacts people have made on a given day as well as information on behaviours and attitudes related to transmission. Researchers use the survey data to estimate the changes in social contact patterns and the state of the epidemic (the basic reproduction number at that point in time). Until now Scotland has been included in their modelling but represented by around 10% of respondents (under 150 people) leading to a high level of uncertainty in the results. Scottish Government are now running a Scottish survey to boost these numbers to over 3,000 people, the "COVID-19 online research panel". The outputs of the model will provide alternative measures of the epidemic at the time (e.g. R number) and will provide an early indication of a resurgence in the virus. Having a range of different types of models for Scotland is the best chance of spotting an upturn in cases and help us plan what to do if this happens.

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



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