To assess the potential impacts of successive lockdown-easing measures in England, at a point in the COVID-19 pandemic when community transmission levels were relatively high.

We developed a Bayesian model to infer incident cases and reproduction number (

England.

Publicly available national incident death data for COVID-19 were examined.

Excess cumulative cases and deaths forecast at 90 days, in simulated scenarios of plausible increases in

Our model inferred an

When levels of transmission are high, even small changes in

This study provides urgently needed information about the potential impact of successive lockdown-easing measures in England when community transmission of SARS-CoV-2 is relatively high.

We use a robust Bayesian model based on Office of National Statistics-registered deaths in England, to infer incident cases and reproduction number and then forecast deaths and cases considering multiple plausible scenarios of increase in reproduction number with successive easing of lockdown in England.

Our study focuses on the impact of easing lockdown in the conservative scenario when

The excess cumulative cases are likely to be sensitive to the specified infection fatality ratio, although this is not expected to materially change the results and inferences. We have assumed a constant infection fatality rate across time, which would not account for changes in the age composition of the infected cases over time.

The model inference is dependent on reliable reported statistics on incident deaths. Underestimation of recent registered deaths would lead to more conservative

As countries around the world negotiated the first wave of the COVID-19 pandemic, governments had to make critical decisions about when and how they eased the lockdown measures instituted to control the pandemic. Given the significant risks of a resurgence of the pandemic and the consequent implications, these decisions have had important consequences on pandemic control following easing of lockdown restrictions globally.

Different countries eased lockdown in different ways and at different points in their epidemic trajectory.

Several experts, including the UK government's Scientific Advisory Group for Epidemics (SAGE), cautioned against easing lockdown in May 2020,

As the country proceeded to rapidly ease lockdown, it was vital to understand and quantify the potential impact of this so as best inform public health strategy. In June 2020, we modelled these impacts across a range of plausible scenarios over the 90-day period from 1 June to 29 August. Using an epidemiological model of COVID-19 spread with Bayesian inference, we inferred parameters of the epidemic in England using daily death data from the Office of National Statistics (ONS). We estimated the time-varying

During the manuscript review process, we were able to examine the observed data that accrued through the original forecasting period and compare it against the model predictions.

The original model inference and forecasting were carried out in June 2020 and the model development is described below. Following this, we describe the comparison of the model predictions from the original forecasts with the observed data from the forecasting period.

In order to model the impact of easing lockdown, we needed to know the levels of transmission and growth parameters of the regional epidemic. Given the limited community testing and case detection in the UK, incident case numbers at that point were likely to be substantially underestimated. We therefore based our model on the number of incident deaths by date of occurrence, which is likely to be more reliable.

As only publicly available aggregate incident death statistics were used, there was no direct patient or public involvement.

We assessed the excess cumulative predicted cases and deaths, over a 90-day period from 1 June. We assumed different scenarios of changing

Incident cases and time-varying

We modelled cases from 30 days prior to the first day that 10 cumulative deaths were observed in England, similar to previous methods.

Parameters estimated by Bayesian model

Variable | Parameter | No | Priors |

_{t} | Number of initial cases on first 6 days | 6 | Exponential(1.0/tau) |

_{0} | Baseline reproduction number | 1 | Normal(2.4,0.5) |

_{t} | Time-varying effective reproduction number | 9 | Normal(0.8,0.25) |

ϕ | Variance parameter for negative binomial distribution of deaths | 1 | Normal(0,5) |

τ | Parameter in prior of _{t} | 1 | Exponential(0.03) |

Subsequent incident case numbers would then be a function of these initial cases and estimated

For s=1,2…N, where N is the total number of intervals (each interval being 1 day) estimated. We estimated the distribution for 201 days, to align with the 111 days of data up to 29 May, plus 90 days of forecasting. Given an SI distribution, the number of infections _{t} on a given day

The incident cases on a given day

The baseline reproduction number (_{0}), and the subsequent time-varying effective reproduction number (_{t}_{t}_{t}_{t}_{t}

We assessed and compared models that allowed _{t}

Model 1: 16 March, 23 March, 13 May and 1 June.

Model 2: every week from the beginning of the modelling period, including on 16 March, 23 March, 13 May and 1 June.

Model 3: 16 March, 23 March and 13 May, and every week between 23 March and 13 May, that is, during lockdown.

For each model, we used the R package

In addition, we also compared model 1 (four change points) with models where each of the change points was left out in turn, as done by Dehning _{t}.

Incident deaths from COVID-19 are a function of the infection fatality rate (IFR), the proportion of infections that result in death and incident cases that have occurred over the past 2–3 weeks. For observed daily deaths (_{t}_{t}

_{t}=E(_{t})

As described in Flaxman _{t} as following a negative binomial distribution with mean _{t} and variance

where ψ ∼ ^{+} (0,5).

Similar to estimation of incident cases, deaths at time point _{t}

IFR was assumed to be 1.1%, based on the most recent estimates from the University of Cambridge MRC nowcasting and forecasting model.

To discretise the time to death distribution, we estimated the probability of death within each discrete time interval (1 day), conditional on surviving previous intervals. First, we calculate the hazard (_{t}

As the denominator excludes individuals who have died, this ensures that _{t}

The cumulative survival probability of surviving up to the interval

where _{t}_{t}_{t}

Given this, we now estimate the probability of death within interval

Here,

Here, the number of deaths within interval

We estimated the set of model parameters θ={c_{1−6}, R_{0}, R_{t}_{t}_{0}_{0}_{t}_{t}

where

Parameters were estimated using the Stan package in R with MCMC algorithms used to approximate a posterior distribution of parameters by randomly sampling the parameter space. We used 4 chains with 1000 warm up samples (which were discarded), and 3000 subsequent samples in each chain (12 000 samples in total) to approximate a posterior distribution using the Gibbs sampling algorithm. From these, we obtained the best-fit values and the 95% credible intervals (CIs) for all parameters. We used these parameters to estimate the number of incident cases and deaths in England. We examined the fit-of-the-model predicted deaths to the observed daily deaths from the ONS, and also the consistency of the model parameters with known values in the literature, estimated from global data. We assessed the distribution of

We carried out sensitivity analyses using uninformative priors for _{0}_{t},_{t}

All forecasts were carried out up to 90 days (29 August 2020) after 1 June. We considered a set of scenarios in which _{t}_{t}_{t}_{t}_{t}

For each of these scenarios, we predicted the number of incident cases and incident deaths, using the functions from the inference model above. Briefly, cases are a function of _{t}

Deaths are a function of incident cases over previous weeks, and the distribution of onset of infection to death times:

All scenarios were compared with a baseline scenario of no change in _{t}

The observed death data for daily deaths in England up to 28 August as obtained from the ONS (from data up to 11 September) were plotted against the original model predictions from June, and the RMSE was calculated between the observed data and the predicted deaths in the different modelled scenarios. The model was rerun with these data, to infer values of _{t}_{t}

Model 3, which allowed weekly changes in _{t}

We inferred _{0}_{t}_{t}_{t}_{t}_{t}_{t}_{t}_{t}

Estimated time-varying reproduction number (_{t}_{t}_{t}_{t}

The model showed a good fit to the observed distribution of deaths up to 12 June (

Model fit to observed death data. Daily deaths predicted by model 3 (blue) with 95% credible intervals (grey) show a good fit to the observed deaths from the ONS (red). ONS, Office of National Statistics.

In the baseline forecasting scenario where _{t}_{test}=0.75) through the 90-day forecasting period (1 June–29 August 2020), the model predicted 48 501 (46 170–50 989) cumulative deaths in England (

In the scenarios where _{t}_{t}

Predicted deaths with _{t}_{t}_{t}_{t}_{t}_{t}, time-varying reproduction number.

Predicted deaths in scenarios of _{t}_{t}_{t}_{t}_{t}, time-varying reproduction number.

Predicted deaths in scenarios of _{t}_{t}_{t}_{t}_{t}, time-varying reproduction number.

In scenarios where _{t}_{t}_{t}_{t}_{t}_{t} rising up to between 1 and 1.2, predicting a second wave of the epidemic within England (

Even in the conservative scenario where _{t}_{t}_{t}

Predicted cases in scenarios of _{t}_{t}_{t}_{t}_{t}_{t}, time-varying reproduction number.

Predicted cases in scenarios of _{t}_{t}_{t}_{t}_{t}, time-varying reproduction number.

Predicted cases in scenarios of _{t}_{t}_{t}_{t}_{t}, time-varying reproduction number.

Compared with the baseline scenario of _{t}_{t}

Using uninformative (no prior specified) priors for _{t}_{t}

Using a longer SI leads to an increase in the estimated _{0}_{t} <1.1 are higher than in the primary model with shorter SI (

The observed cases and deaths are plotted against the modelled scenarios in _{t}_{t}

Predicted deaths in different scenarios of _{t}_{t}_{t}_{t}_{t}_{t}, time-varying reproduction number.

Estimated time-varying reproduction number (_{t}_{t}_{t}_{t}

In this paper, we describe a Bayesian model for inferring incident cases and reproduction numbers from daily death data, and for forecasting the impact of future changes in _{t}

Our model inferences are robust to modelling assumptions of specified priors for _{t}_{t}_{t}_{t}

Countries like Denmark and Germany started easing lockdown when community transmission was low and this likely mitigated increases in _{t}

In September 2020, the UK is at point where community transmission is once again high and it is clear that we have entered the second wave of the pandemic. Schools reopened in the second week of September, a move that is vitally important to children’s health and development, but one that can potentially increase community transmission. Cases and hospitalisations have been increasing exponentially, which has recently translated into an increase in weekly deaths. Using the best available confirmed COVID-19 case data in England published by the UK government on 21 September (which is likely an underestimate), we modelled the potential impact of increases in transmission on daily cases and deaths over the next 2 months, assessing different scenarios of increase in _{t}_{t}_{t}

Predicted cases and deaths at different _{t}_{t}_{t}. We note that case numbers are likely underestimates, as the testing system within England is currently running at capacity, and not everyone with symptoms is able to access tests. IFR, infection fatality rate; _{t}, time-varying reproduction number.

We acknowledge some important limitations of our model. The first is that it is based on a back calculation of cases based on incident deaths, which are likely to be underestimated due to reporting delays and under-reporting. Second, our model is reliant on inferring cases and reproduction numbers, which depend on the assumed distributions of the SI and the time of onset to death distributions. Though we based our assumptions on the literature, misspecification of these would influence our estimates. While we have evaluated this, greater deviations from true estimates would make our forecasting less reliable. Third, similar to Flaxman _{t}_{t}_{t}_{t}

In summary, we show that increases in _{t}_{t}_{t}

We would like to acknowledge and thank Flaxman

DG conceived the study and designed the model with NS. DG programmed the model and made the figures. HZ and NS consulted on the model design. All authors interpreted the results, contributed to writing the article and approved the final version for submission.

HZ is partly funded by the Bernard Wolfe Health Neuroscience Fund at the University of Cambridge. NS is funded by a Strategic Award from the Wellcome Trust (208363/Z/17/Z). DG is funded by the UKRI/Rutherford HDR-UK fellowship programme (reference MR/S003711/2), and the NIHR AIM Development award (reference NIHR202646).

None declared.

Not commissioned; externally peer reviewed.

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Data are available in a public, open access repository. All data on daily deaths used in this study were taken from the Office of National Statistics website (

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