Coronavirus (COVID-19): modelling the epidemic (issue no. 17)

Coronavirus (COVID-19): modelling the epidemic in Scotland (Issue No. 17)

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 3 September 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 improved modelling based on how many contacts people have with one another.

Key Points

• The reproduction rate R in Scotland is currently estimated as being between 1.1 and 1.5.
• The growth rate for Scotland is estimated as being between +3% and +9%.
• The number of new daily infections for Scotland is estimated as being between 0 and 13 per 100,000 people.
• There has been a sustained rise in positive new cases in most local authority areas rather than a small number exceeding expected levels in the week leading up to 7 September.

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.

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 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. This week the type of data used in each model to estimate R is highlighted in Figure 2.

Modelling is provided this week around the number of positive test results we are seeing, and on a short-term forecast of cases. The focus at this stage of the epidemic is the 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 R_t has changed since February. Before the "stay at home" restrictions were put in place R_t was above 1, and most likely to have been between 2 and 4 before any interventions were put in place.

A graph showing the trends in the R_t 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 R_t fell below 1.0 on the 23^rd of March, when the “stay at home” advice was given. It 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 9 September, was that the value of R_t in Scotland was above 1, between 1.1 and 1.5, meaning that the epidemic is growing exponentially. The R value estimated by the Scottish Government falls within this range, and is similar to the estimates of other modelling groups (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 9^th September. The blue bar (first from left, second from right) are death-based models, purple (3^rd to 6^th from the left) use multiple sources of data and cyan (7^th from the left) use Covid-19 test results. The estimate produced by the Scottish Government (a deaths-based model) is the 2^nd 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 1.1 and 1.5.

Source: Scientific Advisory Group for Emergencies (SAGE).

On 8 September, Public Health Scotland recorded 176^[1] positive new cases, with 755 positive new cases over the week of 1 – 7 September.

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 9 September, was that the incidence of new daily infections in Scotland was between 0 and 13 new infections per 100,000. The incidence value estimated by the Scottish Government falls within this range. This equates to between 0 – 700 people becoming infected each day in Scotland. Given the level of asymptomatic infected people, 176 positive confirmed tests remains within this range.

A graph showing the ranges the values which each of the academic groups in SPI-M are reporting for incidence (new daily cases per 100,000) are likely to lie within, as of 9th September. The blue bar (leftmost) is a death-based model, purple (3^rd and 4^th from the left) use multiple sources of data. The estimate produced by the Scottish Government (a deaths-based model) is the 2^nd from left (yellow). The incidence value estimated by the Scottish Government is within the range of the estimates of other groups using models which draw upon numbers of deaths. The SAGE consensus 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 +3% and +9% per day. This is an increase, compared to last week, where growth rate was in the range -1% to +8%.

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 remain near zero in the weeks ahead.

A bar chart showing daily numbers of deaths caused by Covid-19 in Scotland between 12^th March and the beginning of September, 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 analysts to assess implications for health care demand (see previous Research Findings) has been adapted to produce a short-term forecast of cases.

The following two week ahead predictions use this model to extend the estimated number of infections from the Imperial College model, in a manner that fits with the number of actual cases, adjusted to account for asymptomatic and undetected infections. The extension begins in mid-July, and assumes an R₀ value of around 1.3. Future R₀ values are based on judgement.

Figure 5 shows two predictions from this model - a "better scenario", which assumes the current R₀ value remains constant over the next two weeks, and a "worse scenario", which assumes that transmission increases in mid-September.

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.

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 fell to low levels, which prompted us 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 95^th 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.

Unlike previous weeks which have been characterised by local outbreaks in places like Aberdeen and Perthshire, in the week up to 7 September, the picture across Scotland was one of a general and widespread increase in the number of new confirmed cases. As a result, no local authority areas recorded "significant" (>6.0, p > 0.05) levels of cumulative exceedance (Figure 6). The analysis identified Dumfries and Galloway (5.4), Renfrewshire (3.7), West Lothian (3.3) and Glasgow City (3.1) as areas of higher risk of transmission, however given the rising background of positive new cases across the country, these were only slightly above expected levels (Figure 7).

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 7^th September. Dumfries and Galloway is shown to have the highest exceedance of any Local Authority.

Graphs showing the daily and cumulative exceedance for the local authorities of Dumfries and Galloway, Renfrewshire, Glasgow City and West Lothian from 1^st to 7^th September 2020.

What next?

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 a modelling technique called "CoMix" which is based on how may contact people have each week. The contacts people have are collected via an online survey. 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. R_t and growth rate will also be provided. Further information can be found at https://www.gov.scot/coronavirus-covid-19.