# Long-term forecasting of the COVID-19 epidemic

### Dynamic Causal Modelling, UCL, UK

**The real-time estimate of the R-number is 1.00**(credible interval from 0.79 to 1.20) on 25 July 2021.**Daily deaths are predicted to peak — at about 94 deaths per day — in the next 2 weeks.****These predictions rest upon a prevalence–dependent unlocking**. A final lifting of restrictions is anticipated on**11 Sept 2021**.**The basic reproduction number R**This corresponds to a 17% increase in transmission risk, relative to the average since 1 Feb 2020. This estimate includes seasonality effects and may rise to about 6 as winter approaches._{0}is currently estimated to be 4.8.**Current estimates of the vaccination efficacy are:**- preventing exposure to infection: 19.2% (CI 15.4 to 22.8)
- preventing transmission following infection: 84.7% (CI 83.5 to 85.7)
- preventing serious illness when symptomatic (age 15-34): 74.0% (CI 73.3 to 74.8)
- preventing serious illness when symptomatic (age 35-70): 49.3% (CI 47.8 to 50.8)
- preventing fatality when seriously ill: 92.9% (CI 92.3 to 93.4)

**The corresponding cumulative (vaccinated vs. unvaccinated) risks are:**- relative risk of infection: 80.8%
- relative risk of mild illness: 42.5%
- relative risk of severe illness: 15.5%
- relative risk of fatality: 1.1%

These headlines furnish a national picture of the epidemic and obscure regional variations. A more detailed picture — at the level of lower tier local authorities — can be found in the accompanying local dashboard. Please see the national dashboard for the data fits upon which the long-term forecasts are based.

**Disclaimer:** *the modelling and accompanying estimates are reported in these pages for purely academic (open science) purposes. This modelling has not been commissioned. In particular, dynamic causal modelling is not commissioned by the Independent SAGE (on which Prof Friston serves as a panellist). The independent SAGE does not commit to — or engage in — any particular modelling initiative.*

These long-term forecasts are based upon a dynamic causal model (DCM) of viral transmission and mitigated responses. This particular (age-stratified) model is equipped with a vaccination state that affords immunity of various kinds (i.e., reduces the risk of infection, transmission, developing serious illness and dying when seriously ill).

The DCM generates various data that quantify the progression of the epidemic, including the number of reported (first) vaccinations. These data are used to estimate model parameters controlling contact rates, transmission risk and periods of infectiousness using standard variational procedures. Crucially, these variables are themselves time-dependent and depend upon mitigating responses, modelled as the (prevalence-dependent) probability of moving from low to high contact rate locations (e.g., from home to the workplace). The data include daily positive tests, reported deaths within 28 days of a positive PCR test, certified deaths disaggregated by age and place of death, hospital admissions, contact rate proxies, such as car use and Google mobility data, and so on.

Much like weather forecasts, the ensuing predictions should not be overinterpreted because there is an inherent (although quantified) uncertainty about underlying epidemiological and sociobehavioural variables. These reports will be released on a weekly basis so that people can see how the predictions change — and uncertainty resolves — as time progresses and more data are assimilated.

Because the DCM combines an epidemiological model with an agent-based behavioural (and testing) model, it can predict mitigating responses to changes in community transmission. These predictions are the **most likely outcomes** given population responses to date. In other words, DCM does not rely on information about future scenarios (e.g., roadmaps for unlocking). This complements — and contrasts with — SPI-M projections of **reasonable worst-case scenarios** that assume a particular schedule of interventions. Generally, the most likely predictions of mitigated responses — i.e., what is likely to happen — are more optimistic than worst-case projections of unmitigated responses—i.e., what could happen.

#### Summary graphs

In the summary graphs below, the lines and shaded intervals correspond to predictions and 90% credible intervals, while the black dots are (smoothed) data upon which the estimates are based.

**Prevalence and reproduction ratio:** this panel provides a forecast of (i) the **prevalence of infection** and (ii) the reproduction ratio or **R-number** (blue and orange lines, respectively) with their accompanying confidence intervals (shaded areas). These forecasts are based upon parameters estimated from data up until the reporting date (the vertical line). The data include GOV.UK estimates of the R-number, which are shown for comparison with the DCM estimates. The **basic reproduction ratio** (R0 – yellow line) can be read as the R-number in the absence of any mitigating reductions in contact rates (under the simplifying assumption that the mean period of infectivity is constant). This reflects fluctuations in transmission risk due to seasonality effects and viral mutations. The ‘alpha’ and ‘delta’ indicate when the alpha and delta variants were introduced to the UK.

The DCM estimate of the R-number is based upon a generative model (i.e., a real-time estimate using data assimilation). The corresponding consensus estimates from the SPI-M and UKHSA are based upon retrospective (e.g., Bayesian regression) analysis of recent data and are therefore treated as a lagged estimator. The black dots correspond to the GOV.UK estimates moved backwards in time by 16 days from their date of reporting.

**Attack rate and immunity:** this panel shows long-term forecasts of attack rate, population or herd immunity and the percentage of people who have seroconverted following vaccination (in blue, red and purple, respectively). In addition, an estimate of the effective immunity threshold is provided under simplifying (SIR) assumptions (yellow line). These simplifying assumptions ignore heterogeneity in susceptibility or vaccine uptake and fluctuating contact rates (e.g., superspreader events).

The attack rate corresponds to the number of people who have been infected since the onset of the outbreak (blue line). This can be supplemented with a small proportion of the population that have pre-existing immunity (e.g., mucosal immunity or cross immunoreactivity with other SARS viruses), shown in red. The combination can be read as the herd or population immunity

The effective immunity threshold is based upon the effective reproduction ratio under pre-pandemic contact rates and the efficacy of vaccination in precluding transmission. The reproduction ratio corresponds to the product of the contact rate, transmission risk and mean infectious period. Note that the effective immunity threshold fluctuates. This reflects the fact that transmission risk changes with time. In this model, transmission risk is modelled as a seasonal fluctuation multiplied by smooth (increasing) function of time. A fluctuating transmission risk accommodates changes in transmissibility (e.g., due to viral evolution) that is contextualised by seasonal variations in transmission (e.g., due to changes in temperature, humidity, socialising outdoors and the propensity for aerosol transmission).

Based upon the changes in testing, death rate and other data, one can estimate the efficacy of vaccination at several points in the chain from infection to death. This DCM is equipped with two infection states: infected (i.e., exposed) versus infectious (i.e., contagious) and two clinical states: mildly symptomatic versus systemic illness. The efficacy of vaccination is modelled in terms of the relative risk of (i) becoming infected, (ii) becoming infectious after exposure to the virus (i.e., being able to transmit the virus), (iii) becoming seriously ill when mildly symptomatic, and (iv) dying when seriously ill. The relative risk of becoming infected subsumes contact rates and transmission risk. This means that being vaccinated can, in principle, increase contact rates (e.g., due to sociobehavioural changes), leading to a negative efficacy for preventing infection. These relative risks can be expressed in terms of conditional efficacies or composed to evaluate the overall risk of morbidity and mortality, relative to not being vaccinated.

**Daily deaths:** this panel shows fatality rates as assessed by patients who died within 28 days of positive PCR test and people who died from certified COVID-19. The former represents an underestimate of COVID-related mortality, where the degree of underestimation depends upon testing rates. The discrepancy is modelled by evaluating the probability of succumbing to COVID-19 and having had a positive PCR test within 28 days.

**Mobility and lockdown:** a long-term forecast of locking and unlocking, based upon car use as quantified by the Department of Transport and - in this graph - Google mobility data. These measures of mobility are expressed in terms of the percentage of pre-pandemic levels. The expected mobility has been thresholded at three levels to illustrate different levels of lockdown. The dates on the lower (graded pink) bar annotate a transition from a more restrictive level of mobility to a less restrictive level. Forecasts of mobility are based upon underlying contact rates that depend upon the prevalence of infection, which are then modulated with a smooth function of time (parameterised with Fourier coefficients).

This dynamic causal model includes age-stratification into four groups (below the age of 15, between 15 and 35, between 35 and 70, and over 70 years of age). The contact rates within and between the groups (for high and low contact rate locations) are estimated from the data, under mildly informative lognormal shrinkage priors. Please see the following technical report for further technical details.

**Changes since last report:**

- Age-specific vaccination effects on pathogenicity added
- Age-specific (cumulative) hospital admissions and now included in the parameter estimation
- Age-specific duration of serious illness (requiring hospitalisation) has been included to provide a simpler model of hospital admission and occupancy data
- A prediction of the incidence of long COVID has been included (assuming one in 400 people will develop long COVID following symptomatic disease and one in two people will develop post COVID syndrome following severe disease)

**Software note:** The figures in this report can be reproduced using annotated (MATLAB) code available as part of the free and open source academic software SPM (https://www.fil.ion.ucl.ac.uk/spm/), released under the terms of the GNU General Public License version 2 or later. The routines are called by a demonstration script that can be invoked by typing `>> DEM_COVID_UK4`

at the MATLAB prompt.

**Data sources:** (also available as CSV files)

https://coronavirus.data.gov.uk

https://www.ons.gov.uk/peoplepopulationandcommunity/healthandsocialcare/conditionsanddiseases/datasets/coronaviruscovid19infectionsurveydata

https://covid.joinzoe.com/data#levels-over-time

https://www.gov.uk/guidance/the-r-number-in-the-uk#contents

https://www.gov.uk/government/statistics/transport-use-during-the-coronavirus-covid-19-pandemic

https://www.google.com/covid19/mobility/

**Peer-reviewed references** [1-3] and **archival papers** [4-11]

- Friston, K.J., et al., Second waves, social distancing, and the spread of COVID-19 across America. Wellcome Open Research, 2020. 5(103): p. 103.
- Friston, K.J., et al., Dynamic causal modelling of COVID-19. Wellcome Open Research, 2020. 5(89): p. 89.
- Friston, K.J., A. Costello, and D. Pillay, ‘Dark matter’, second waves and epidemiological modelling. BMJ Global Health, 2020. 5(12): p. e003978.
- Moran, R.J., et al., Using the LIST model to Estimate the Effects of Contact Tracing on COVID-19 Endemic Equilibria in England and its Regions. medRxiv, 2020.
- Friston, K.J., G. Flandin, and A. Razi Dynamic causal modelling of mitigated epidemiological outcomes. 2020. arXiv:2011.12400.
- Daunizeau, J., et al., On the reliability of model-based predictions in the context of the current COVID epidemic event: impact of outbreak peak phase and data paucity. medRxiv, 2020.
- Daunizeau, J., et al., Modelling lockdown-induced secondary COVID waves in France. medRxiv, 2020.
- Gandolfi, D., et al., Dynamic causal modeling of the COVID-19 pandemic in northern Italy predicts possible scenarios for the second wave. medRxiv, 2020: p. 2020.08.20.20178798.
- Moran, R.J., et al., Estimating required ‘lockdown’ cycles before immunity to SARS-CoV-2: Model-based analyses of susceptible population sizes, ‘S0’, in seven European countries including the UK and Ireland. medRxiv, 2020: p. 2020.04.10.20060426.
- Friston, K.J., et al., Viral mutation, contact rates and testing: a DCM study of fluctuations. medRxiv, 2021: p. 2021.01.10.21249520.
- Friston, K.J., et al., Tracking and tracing in the UK: a dynamic causal modelling study.

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