^{1}

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^{3}

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^{1}

^{4}

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The authors have declared that no competing interests exist. We received funding from Microsoft Corporation, a commercial source. E.B. is employed by a commercial company, JPMorgan Chase & Co. W.B. works for JPMorgan Chase & Co, and is employed by Kubrick Group, London. None of this does not alter our adherence to PLOS ONE policies on sharing data and materials.

We apply Bayesian inference methods to a suite of distinct compartmental models of generalised SEIR type, in which diagnosis and quarantine are included via extra compartments. We investigate the evidence for a change in lethality of COVID-19 in late autumn 2020 in the UK, using age-structured, weekly national aggregate data for cases and mortalities. Models that allow a (step-like or graded) change in infection fatality rate (IFR) have consistently higher model evidence than those without. Moreover, they all infer a close to two-fold increase in IFR. This value lies well above most previously available estimates. However, the same models consistently infer that, most probably, the increase in IFR

The alpha variant B.1.1.7 of the SARS-CoV-2 virus first emerged in the UK in September 2020. It is now well known to be more infectious than the prior UK strain and for this reason, in the following months, it not only became dominant in the UK itself, but rapidly took hold in a number of other countries (including the USA) where it soon became the dominant variant [

However, the relation between reported cases and mortalities is not always straightforward to interpret. The time between infection and potential deaths is stochastic, so that a time series of mortality data will tend to show less rapid changes than corresponding case numbers. Moreover, as the available testing capacity changes with time alongside the demand for tests, which leads to a time-dependent ascertainment rate. Hence, reported case numbers are not directly representative for the true number of cases.

In order to overcome these challenges while still focusing on (nationally) aggregated data for cases and mortalities, we choose to analyse such data in the context of well-mixed compartmented models, whose complexity is adjustable for this purpose [

Our goal is to establish the evidence for a change in the UK lethality of COVID-19 in a Bayesian fashion, using the reported age-structured data for nationwide cases and mortality. For that purpose, we compare different models that either do or do not allow for a change in the infection fatality rate (IFR). We analyse differences in their posterior probability, optimised for fixed data over a set of model parameters. Generally, one would expect that the likelihood of a model increases as more details are added. In order to judge the significance of changes in the posterior, we compare a whole set of model variants differing in the level of detail, and see whether the ones that do allow for changing IFR perform better than those that do not.

Our comparison also includes model variants that would explain perceived changes in the case fatality rate merely through changes in the ascertainment rate. However, these models require implausible assumptions about changes in the testing strategy.

The model specification, simulation, likelihood computation, and optimisation is carried out using the software package PyRoss, which we have developed during the past year [

We consider a suite of compartmented models (referred to as model variants below) all with _{i}, the intervention function _{i}(_{ij}, the total population per age group _{i}, and a factor

Every compartment (except S and E) has a quarantined version, transitions into which occur via testing. New infections are transmitted by individuals of the classes coloured red (with the stage Is2 being less infectious). Classes R and Im (mortalities) are no longer infectious.

After a presymptomatic stage (A), individuals become either symptomatically (Is1/Is2) or asymptomatically (Ia) infected, according to an age-dependent fraction _{i} of asymptomatic cases. The outcome of the infection is either recovery (R) or death (Im). Progression though all these stages is modelled though linear transition rates, matching the latent and incubation periods, and the typical time from infection until death. The latter is determined by the exit rate _{s} from both stages Is1 and Is2, and is inferred. Otherwise, we fix parameters relating to disease progression to values informed by the literature, in line with established literature (see Ref. [_{E}, stage A, _{A}, and stage Ia, _{a}. The complete constitutive equations of the model are given in part A in

We model changes in the contact behaviour and (potentially) in the infectiousness via the time dependence of _{i}(_{i}(_{i}(_{i}) and constrain the vector _{i} for the age dependence to the form [0, 0, 0, _{4}, _{5,6}, _{5,6}, _{7}], with the largest element set to 1 as a reference. The parameter

dates | type | control parameters |
---|---|---|

before 2020–03–20 | before lockdown (reference) | |

2020–03–20 to 2020–03–27 | imposition of lockdown | linear decrease of |

2020–03–27 to 2020–07–24 | easing of / increasing non-compliance with lockdown | linear increase of |

2020–07–24 to 2020–11–06 | lockdown lifted | new values of |

inferred | increase of contacts / infectiousness in autumn | tanh-shaped increase of |

inferred | several local interventions, summarised as a single one at time to be inferred | new values for |

2020–11–06 to 2020–12–04 | national lockdown (England) | new values for |

2020–12–04 to 2021–01–08 | tiered lockdown | new values for |

after 2021–01–08 | national lockdown | new values for |

Dates are always rounded to the closest Friday.

Testing is modelled as the transfer of individuals from the undiagnosed version to the diagnosed (or “quarantined”) version of a compartment. For a given overall rate of testing _{X} = 0 for X being S, E, or R, otherwise the true positive rate _{X} = 1. The factors _{X} encode testing priorities for the various compartments. We set _{Is1} = _{Is2} = 1 as a reference, and _{Im} = 20 to ensure that mortalities get reliably detected (For the early weeks, when tests were scarce, we set _{Is2} = 5 and _{Im} = 100.). The only fit parameter we infer is _{a}, the priority for testing individuals that are not symptomatically infected (i.e. of classes S, E, A, Ia, R, setting _{S} = _{E} = _{A} = _{Ia} = _{R} ≡ _{a}). It interpolates between random testing for _{a} = 1 and very targeted testing for _{a} = 0. The progression through stages in the quarantined compartments is the same as in the non-quarantined ones, but quarantined individuals cause no further infections (effectively assuming perfect self-isolation). Once tested positive, individuals remain in the quarantined compartments. The recovered class RQ therefore includes individuals that have actually left quarantine, but we keep this class separate for the purpose of counting previously diagnosed cases.

In summary, our model implicitly accounts for changes in the ascertainment rate, in a way that is informed by the supply (the total number of tests) and the demand (the yet untested symptomatic cases).

The lethality of COVID-19 is encoded in the infection fatality rate (IFR), i.e., the probability of any infected to die eventually. As an auxiliary quantity for the specification of our model, we define a

We choose both transition rates from Is1 to Is2 and from Is2 to Im as _{i}; recoveries (to class R) happen from both stages at rate _{i} of asymptomatic cases, the IFR follows readily as

All transitions are modeled as Markov rates, i.e. they are inherently stochastic. In order to account for additional sources of noise or variation that are not present in the well-mixed model, we infer overdispersion parameters that scale up the fluctuations in transitions related to infections, testing, and deaths.

We consider the reported cumulative case numbers as the sum of all quarantined classes for each age cohort. The reported cumulative mortalities are identified with the numbers of class ImQ (where the age groups 15–29 and 30–44 to match the available data). We ensure that no deaths remain unnoticed, by formally assuming that individuals in class Im continue to get tested at high priority. The numbers in all other compartments are considered as hidden, and are implicitly reconstructed by the inference procedure.

We use weekly data from the week beginning 2020–03-07 to the week ending 2021–01-15. For cases, we use the daily numbers reported on the UK government webpage [

Death numbers by week of reporting have been obtained from the UK Office for National Statistics (ONS) webpage [

The daily number of PCR tests performed is available from the government webpage [

For France, we use data for deaths in hospitals [

For Germany, we use data provided by the Robert Koch-Institute for cases and deaths [

We consider several variants of the basic model outlined above, labeled by A0, A1, B0, etc. Some of the variants have an IFR that is constant in time, as indicated by the Type number 0. Type number 1 indicate time-dependent changes of the IFR; Type 2 also indicates this but via a mechanism involving slowed recovery rather than higher death rate (which also ultimately results in more deaths). The Type letters refer to other details of the model variant,

The model outlined above, without any further additions.

As model A0, but with a simple step-change in IFR. The size of the change and time of change are inferred (except for the two youngest cohorts, where fatal cases are extremely rare). The change in IFR is parameterised in terms of the log-ratio of the values, with a prior that is normal distributed with mean zero and standard deviation log(3). The prior for the time of the change is normal with mean 2020–12-12 and standard deviation 2 weeks.

As model A0, but with a step-change of the recovery rate to

As model A0, but with a linear increase in

As model B0, but with a tanh-shaped change in IFR. The time around which this change is centred (which we refer to as the onset time), the width, and the amplitude are inferred. (We deem that this increased level of detail is harmonious with the already more detailed model B0.)

As model A0, but with a change in the three overdispersion parameters for infections, testing, and deaths. The change is allowed to happen on 2020–10-02, a date chosen to match a potentially new stochastic dynamics as the second wave gains momentum. The new values of the parameters are inferred independently. This reflects potential changes in the testing strategy and in the infection dynamics in the second wave, and can avoid the overestimation of case and death numbers, that is often observed as a side-effect of mismatching overdispersion parameters.

As model C0, but with a simple jump-like change in IFR.

A combination of models B0 and C0: It has a change in overdispersion parameters in the second wave and easing/non-compliance (or increasing infectiousness) during the November lockdown.

A combination of models B1 and C1: As model BC0, but with a tanh-shaped change of IFR as in B1.

When case numbers are low, effective contact tracing is possible. This could mean that more asymptomatic cases are uncovered in summer than at the height of the first and second wave. As a simple model for this effect, building on model BC0, we allow for the inference of the testing priorities _{A} and _{Ia} for pre- and asymptomatic infected individuals different from the priority _{a} of classes S and R. This change comes into effect with the beginning of large-scale contact tracing on 28th May 2020. Testing priorities remain unchanged thereafter; however, as long as the testing priorities of A and Ia remain below those of Is, the effect of contact tracing will only become relevant for large test rates and low case numbers, so that the class Is of undiagnosed individuals can be depleted.

As model TT0, but with a tanh-shaped change in IFR (as in B1).

As model BC0, but with a time-dependent change in _{a}, the only parameter entering our model for testing. The change in _{a} is tanh-shaped, with centre (the onset time), width and amplitude to be inferred. This change may reflect changes in the testing strategy, that have happened during the course of the pandemic.

Using our software package PyRoss, we can calculate the logarithmic likelihood of the observed data for each of the model variants and for any choice of the model parameters and initial conditions [

Given an informed choice of prior distributions for all parameters and initial conditions, we have determined for each model variant the parameters that maximise the posterior probability (MAP). To decide between different modelling hypotheses, a rigorous approach is to compute the model evidence (also known as the marginal likelihood). In this Section we take the simpler approach of comparing the log-posterior probabilities—to a first approximation, models with large posterior probability should be preferred. However, that approach can suffer from over-fitting: that issue can be addressed via estimates of the evidence. This point is addressed in Sec 5, below. (Anticipating the answer, we find that the conclusions of this Section—based on the log-posterior—are robust.)

Results are summarised in

Models without change in IFR in blue, models with change in IFR in orange.

Country | Model | # Params | log-Prior | log-Posterior | log-Evidence | IFR change | ||||
---|---|---|---|---|---|---|---|---|---|---|

abs | Δ | abs | Δ | abs | Δ | Onset | Factor | |||

UK | A0 | 67 | −354 | −4281 | −4149 | |||||

A1 | 69 | −20 | +55 | +50 | 27 Oct (±2 d) | 1.97(±0.11) | ||||

A2 | 69 | −9 | +49 | +44 | 24 Nov (±8 d) | |||||

UK | B0 | 69 | −350 | −4269 | −4147 | |||||

B1 | 72 | −21 | +61 | +64 | 8 Nov (±7 d) | 2.14(±0.17) | ||||

UK | C0 | 70 | −365 | −4282 | −4150 | |||||

C1 | 72 | −35 | +52 | +63 | 28 Oct (±2 d) | 2.02(±0.12) | ||||

UK | BC0 | 72 | −400 | −4272 | −4139 | |||||

BC1 | 75 | −26 | +56 | +56 | 9 Nov (±7 d) | 2.20(±0.18) | ||||

P0 | 75 | −22 | +59 | +69 | 29 Oct (±5 d)^{a} |
[3.000]^{a}^{,}^{b} |
||||

UK | TT0 | 73 | −368 | −4270 | −4153 | |||||

TT1 | 76 | −25 | +56 | +57 | 9 Nov (±7 d) | 2.20(±0.19) | ||||

GER | C0 | 59 | −268 | −3438 | −3376 | |||||

C1 | 61 | +3 | +79 | +77 | 26 Nov (±2 d) | 1.80(±0.08) | ||||

FRA | C0 | 75 | −254 | −4137 | −4121 | |||||

C1 | 77 | 0 | +55 | +58 | 3 Nov (±1 d) | 1.37(±0.11) |

We list the country considered along with the model variant, the number of inferred parameters and initial conditions, the logarithmic prior, posterior, and model evidence, and, if applicable, the inferred onset time and factor of a change in IFR. For Type 0 models without any IFR change (printed in bold), absolute values of log-prior, log-posterior (non-normalised), and the logarithmic model evidence are shown, indicated by “abs”. For other models, we show values relative to the corresponding base model, indicated by Δ. The indicated uncertainties in the inferred parameters for the IFR change correspond to a single standard deviation in a Gaussian approximation of the posterior.

^{a} Changes in _{a}

^{b} MAP value has attained an upper bound set by prior

It is remarkable that even though the models A0, B0, C0, and BC0 have different numbers of fit parameters, their variation in the log-posterior is nowhere near as big as the difference to the variants with a changing IFR. This observation already suggests that the increase in likelihood for the variants with changing IFR is not a mere consequence of overfitting due to the additional parameters.

For each model and set of MAP parameters, we can plot a deterministic solution. This is the most likely trajectory, conditional on the inferred initial condition, and the mean of the multivariate Gaussian approximation for all compartment values at all times. Fully detailed plots of the MAP trajectories for each of the model variants are shown in part C in

Unsurprisingly, the results for the most detailed model variants BC0 and BC1 produce mean trajectories for cases and mortalities that fit the data best, as shown in

Deterministic trajectories for the MAP parameters of models BC0 (dashed blue) and BC1 (solid orange), along with data (black).

In

Deterministic trajectories for the MAP parameters of the various model variants, along with data (black). Models without change in IFR are shown as dashed, models with change in IFR solid.

Some differences between models A/B and C are due to a generic feature of our computation of the likelihood. When the estimated overdispersion parameters are too low to account for the observed noise in the data, the optimiser tends to overestimate the expected case and deaths numbers, thereby increasing the variance of weekly changes. Considering the resulting MAP trajectories for models A and B, it seems that the inferred overdispersion parameters, which serve well to fit the first wave, are too small to match the level of noise in the second wave. This leads to an overestimation of both expected case and death numbers. Differences between cases and deaths in this overestimation could negate the perceived change in IFR. This has prompted us to analyse model C, allowing for changes in the overdispersion parameters for the second wave. It is consistent with the results of models A and B, which rules out that the observed changes in the IFR stem from temporal changes in the overdispersion.

The models C0/C1 reproduce the observed height of the second wave, but not the dip between the second and third wave. Note that model C (just like model A) has the November lockdown fixed without easing, leading to a larger reduction in cases and deaths than in reality. Model BC (easing and change of overdispersion) reproduces the short-livedness of this reduction better.

The sudden drop in mortalities (individuals with COVID-19 mentioned on the death certificate) in late August / early September is not reproduced by any of our models. This might be related to changes in the legal definition of such deaths. We do not model this here, but note that adjusting the data for a lasting change in the definition from this time onwards would likely lead to an even larger increase in the IFR than the MAP estimates 1.9–2.2 reported above.

We note that for the exit rate from the two stages of symptomatic infection, the value _{Is} ≈ 0.43 (per week) is inferred for model BC1 (and similar values for the other UK model variants). This corresponds to a mean time from onset of symptoms to death of approximately 33 days, which is considerably more than the 18 days reported in the cohort based study of Ref. [

We also did the inference procedure for model C0/C1 with data for France and Germany, using appropriate forms of the intervention function, as detailed in part B in

In

Mean fraction of people infected (symptomatic and asymptomatic) in the total population, conditional on the observation of cases and deaths in models BC0 (blue), BC1 (orange) and P0 (grey). For comparison, we also show the prevalence of infections reported in the ONS infection survey.

We note that differences in the conditional numbers of infections between models BC0 and BC1 mainly show up in the first wave. For that time, data from the infection survey is not available, and data on testing may be incomplete. Nonetheless, it is remarkable that the inferred numbers of infection largely agree between models BC0 and BC1 from June onwards, encompassing the inferred time of change in IFR. Hence, we can rule out that the change in IFR in model BC1 is merely due to changes in the inferred true number of cases. It is rather that the observed timeline of deaths is more likely in model BC1 than in model BC0, for similar estimated total numbers of infections.

Based on the remaining discrepancies between the our inferred infection numbers and the ONS survey, one could still argue that we overestimate the true case numbers early on and/or underestimate them later, leading to an apparent increase in the IFR. The model variants TT0/TT1 and P0 serve to address this possibility.

The goal of contact tracing is to detect and isolate asymptomatic cases of COVID-19 and ideally to also detect cases early on in the presymptomatic stage. However, due to limited capacity, the test and trace system is only effective when case numbers are low and there is sufficient testing capacity, such as in the summer months. This could mean that with increasing testing numbers and the large-scale test and trace system being put in place in late May, the reported cases after the first wave are closer to the true cases than expected by the model variants considered so far. The results for the pair of variants TT0 and TT1, with their (albeit rudimentary) realisation of contact tracing, give no indication that this might explain an apparent change in the IFR. Quite to the contrary, TT1 infers a somewhat larger change in IFR than the Type 1 models already considered. The posterior of models TT0 and TT1 change only marginally compared to BC0 and BC1, respectively.

In principle, it is possible that the inferred factor-two change in IFR in early November could instead be explained by large changes in the numbers of undiagnosed cases. We illustrate this fact using model P0. It allows for a time-dependent change of the testing priority _{a}, and we deliberately set a loose prior on the timing and amplitude of this change. This model attains a posterior probability that is comparable to that of the models with a change in the IFR. (Coincidentally, prior and posterior are almost identical to those of TT1, therefore P0 is not shown in _{a} as early as July, yet it is inferred as late November (four weeks _{a} is inferred as a factor 3, unexpectedly saturating an upper bound we set on this parameter. This would mean that, to explain the data in terms of a changed testing regime rather than an actual IFR change, at the height of the second wave, tests must have become at least three times

This section refines the model comparison by using an estimate of the Bayesian model evidence (also known as the marginal likelihood). The core of our analysis is the Bayes theorem

For a complete analysis of the model evidence, a sampling of the posterior over the parameter space would be necessary, e.g. using Markov Chain Monte Carlo (MCMC) techniques [

Data for model variants BC1 (blue) and C1 (orange) are shown. The computed posterior is shown as points, the Gaussian approximation of

The logarithmic model evidence follows as

The local Gaussian approximation of the posterior can also be used to estimate uncertainties in the inferred parameters. The inverse ^{−1} is the covariance matrix of the inferred parameters in the posterior distribution. The square root of its diagonal elements yields a single standard deviation (or 68% credible interval) for each individual parameter, such as the ones quoted in

Moreover, we can explore the posterior distribution of the IFR before and after the change by drawing samples from the multivariate Gaussian distribution of the parameters describing the IFR. The comparison between models BC0 and BC1 in

We show data for model variants BC0 (blue) and BC1 (orange, time-dependent), for four different age groups (indicated in square brackets). The thick lines represent the IFR determined from the MAP parameters, the thin lines correspond to 100 samples of the parameters for the IFR drawn from the Gaussian approximation of the posterior.

As a final note, we consider whether the observed evidence for a change in the IFR could be affected by confirmation bias. We had been motivated by previous reports, based on a non-Bayesian analysis [

In this paper, we have reported evidence for an increase in lethality of COVID-19 in the UK in late autumn 2020. Bayesian inference provides clear and consistent evidence for such an increase, across the suite of models we considered. This finding complements similar conclusions based on the visual inspection of nationally aggregated case and mortality data [

Our approach uses the same kind of data, but we use models that differentiate between reported cases and the true number of cases. They are informed by the reported number of tests performed, implicitly accounting for a time-dependent ascertainment rate. Estimates of the absolute value of the IFR are always beset with uncertainty, due to an unconstrained dark figure of less severe or asymptomatic infections [

We would not generally expect aggregated case and mortality data, analysed with a well mixed compartmental model (or suite thereof), to identify clear or definitive causes for an increase of this kind. So our conclusion is (in common with [

It is natural to speculate that the increase is related to the emergence of one or more new virus strains, whose potentially increased lethality has been the subject of several cohort-based studies summarised previously in a UK Government publication [

This suggests that factors such as seasonality and/or pressure on health services may have contributed to the change. Of these, seasonality more credibly would have had similar effects in the UK and Germany (and perhaps weaker effects in France, but this is far from clear). This conclusion may also be supported by a study from Israel addressing changes of in-hospital mortality rates [

Such arguments remain, for now, speculative. To further investigate potential connections between new variants and the observed change in IFR, one should explicitly represent the dominant mutant strain though additional model compartments, with increased infectiousness and possibly increased lethality. Such a model could be calibrated using data for the prevalence of mutant strains.

The increase in lethality might also be generic for the peaks of waves of infection, when hospitals are under severe strain. We have not yet analysed the possibility for

It will be interesting to see whether the evidence for a change in IFR persists as models become calibrated with more recent data and additional types of observations. This might include data for hospital admissions, antibody testing, or random asymptomatic testing (as already considered

As an additional caveat, we emphasise that these well-mixed models cannot describe regional variability, such as situations where the epidemic is shrinking in one region and growing in another. Moreover, the evidence for the UK IFR change is partly based on the relatively poor fits achieved by models with fixed IFR. It follows that if substantially improved fits were obtained for models that include regional variability (with fixed IFR), our conclusions might have to be re-evaluated. On the other hand, we are not aware of a specific mechanism by which regional variability would generate the discrepancies observed here between the models and the data. These possibilities might be also tested by future work.

Finally, we would like to advocate the advantages of creating a suite of models within a single platform to which consistently identical inference procedures can be applied. It is of course valuable to have independent modelling teams doing their own preferred type of data analysis and then comparing the results of these studies to see if a consensus emerges. However, the integrated model-building and inference machinery gathered within the PyRoss platform has in this case allowed rapid implementation of a number of purpose-built models and their comparison, in a fully Bayesian way, in the month immediately following the appearance of [

The PyRoss platform [

Part A: Constitutive equations for the basic model. Part B: Intervention functions for Germany and France. Part C: Detailed plots of the MAP trajectories.

(PDF)

Sheet 1: Overview of key parameters and results. Sheet 2 (3, 4): Detailed information on parameter prior and posterior values and uncertainties for all UK (GER, FRA) model variants.

(XLSX)

We thank Graeme Ackland, Daan Frenkel, and Julia Gog for helpful discussions. We also thank William Peak and Andrew Ng from JPMorgan Chase & Co. and all PyRoss contributors (see [

PONE-D-21-08131

Bayesian inference across multiple models suggests a strong increase in lethality of COVID-19 in late 2020 in the UK

PLOS ONE

Dear Dr. Pietzonka,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please respond to the reviewer comments on a point-by-point basis and revise the manuscript accordingly.

I'd also like the authors to explore whether fixing most of the \\pi parameters, e.g. \\pi_{Is}, affects the results. This seems to assume that testing rates among the symptomatic didn't change over time, or merely that those changes will all be handled by the \\pi_a term (which really appears intended to represent tracing and capture of asymptomatics). Is this assumption a problem for constraining fluctuating testing capacity and demand for testing rates over time? Evidence suggests that ascertainment rates increased over time, which would work against the rising IFR reported here. What do the raw case fatality rates show (when convolving cases forward to deaths)? Do they rise as well? How about the hospitalization fatality rates? Do they rise or fall in the study period? This can be pulled from observations and provide an indication as to whether clinical treatment has improved. Is your main finding the result of a poorly constrained total number of infection (i.e. the ascertainment rate)?

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Reviewers' comments:

Reviewer's Responses to Questions

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: Partly

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2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: No

**********

3. Have the authors made all data underlying the findings in their manuscript fully available?

The

Reviewer #1: Yes

Reviewer #2: Yes

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4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

**********

5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The paper was a pleasure to read. The summary is excellent. The clear way in which you present your models and your results gives me total confidence in your work. I recommend that it should be published. It is valuable not only for its results but also because it provides a nice demonstration of your methods and of your PyRoss software package and will help others to build on this. I only have two suggestions 1) please make the code available online and 2) it would be nice if you could give the reader access to the MAP values for all inferred parameters. I would have liked to check whether they look believable, but I also think I would have learned something from them.

Reviewer #2: Dear editor,

the manuscript analyses public COVID-19 data in the UK, France and

Germany using a SEIR type compartmental model. The text is clearly

written and the proposed models interesting and well thought-out.

The authors conclude that the IFR in the UK and Germany was higher

in the end of 2020 by a factor of around 2 when compared to the

first semester. They also conclude that this increase precedes the

widespread circulation of the new major SARS-CoV-2 variant B.1.1.7.

I consider this a very strong claim which is not convincingly

backed by the methods and analyzed data presented in the current

manuscript. I cannot recommend publication unless the authors are

able to provide further details and/or revise their analysis in

order to better substantiate their main conclusion.

I explain below both my major and minor concerns in detail.

MAJOR POINTS

1 - The authors make use of the PyRoss package in order to evaluate

their posterior and quote maximum posterior values for their

parameters. Although I understand this package was already

discussed in more detail in a previous manuscript, the authors

should explain here in more detail the methods involved in their

analysis. For instance, are the quoted values for the IFR change

marginalized over all other model parameters? Was the posterior

sampled with MCMC methods or is it the maxima found with a simpler

scheme? If the latter, what is the justification and how accurate

is it?

2 - Why do the authors refrain from quoting confidence intervals CI

(ideally highest density intervals) in most of their results? The

claim that "the calculation is tedious" seems unjustifiable, as the

discussed method based on the Hessian seems to be the very well

established method of using the Fisher information Matrix. In

particular the Li et al. 2020 paper which describes the package

PyRoss discusses the implementation of both MCMC and Fisher

information Matrix methods. In order to help make sense of the main

results for the IFR I consider estimations of the marginalized CI

for all main results imperative. Ideally one should show plots of

the posterior for the IFR (before, after and mean IFR for the

models without a change) for at least some models. Also, in the 2

cases for which uncertainties are shown, they seem quite narrow,

specially for the German IFR factor. Is that expected after

marginalization over dozens of parameters?

3 - The authors choose not to make a full model-comparison using

the Bayesian evidence and instead rely on a simple analysis of the

maxima of the Posterior. I think that if the full evidence cannot

be computed due to computational complexity, at least some better

quantitative proxy for it should be employed. Alternatives are the

Akaike or Bayesian Information Criteria, which are fast to compute.

Or at the very least a Goodness of Fit test.

4 - The main conclusion is that the IFR seems to have increased by

a factor of around 2 in the end of 2020. Nowhere in the text

however are the actual inferred values for the IFR (before and

after) written down. Nor is there a discussion on how these

estimates compare with other IFR estimates in the literature (for

the UK or other countries), specially with those which do not rely

on similar modelling.

5 - As the authors claim, for IFR estimates one needs accurate

estimates of the total number of cases, including undiagnosed ones.

As the authors point our a bias in the estimated number of cases

indifferent months can affect their inferred IFR increase. This

point deserves a more careful discussion as any IFR estimate hinges

strongly on the estimation of the total number of cases. I would

like the authors to discuss in more detail how the ONS random

asymptomatic testing was conducted and how reliable are their

estimates. How does it compare with other random seroprevalence

surveys (in the UK or elsewhere)?

MINOR POINTS

1 - There have been estimates in the literature of the time lag

between contagion and mortality and between development of symptoms

and mortality. How do these estimates compare with the values in

the models used?

2 - Some variables are not clearly defined in the text, which may

hinder understanding readers less familiar with SEIR models. For

instance: phi_X, gamma_A, gamma_E, a(t), s(t).

3 - For the models with allow a step-change in IFR values the

window for change has a narrow 2-week sigma. Is it possible that by

using a much broader window the models could have preferred a much

different change time? In other words, since there has already been

claims of a change of IFR in the period considered, could this "a

posteriori information" be biasing the models?

4 - The full list of parameters for the models (at least A0) should

be more explicitly written down in an appendix.

**********

6. PLOS authors have the option to publish the peer review history of their article (

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Reviewer #1: No

Reviewer #2: No

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Please see attached file reply.pdf

Submitted filename:

Bayesian inference across multiple models suggests a strong increase in lethality of COVID-19 in late 2020 in the UK

PONE-D-21-08131R1

Dear Dr. Pietzonka,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

An invoice for payment will follow shortly after the formal acceptance. To ensure an efficient process, please log into Editorial Manager at

If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact

Kind regards,

Jeffrey Shaman

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments:

Reviewer's Responses to Questions

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #2: All comments have been addressed

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #2: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #2: Yes

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The

Reviewer #2: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #2: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #2: The authors carried out a large revision which addressed the majority of my criticisms. They also provided detailed explanations in their response letter. I thus consider that the revised manuscript can now be accepted for publication.

I leave final minor comments that the authors may consider implementing in the final, published version.

1 - Even though it is well cited, I'm not sure it makes sense to cite the very controversial Ioannidis study without taking the time to properly assess its very important limitations. I suggest the authors consider alternative studies such as Meyerowitz-Katz et al IJID 2020, Hauser et al PLOS Medicine 2020, Roques et al Biology 2020, Salje et al, Science 2020, O’Driscoll et al Nature 2021, Marra et al IJID 2021.

2 - I understand IFR is highly dependent on age, but I still consider it would enhance the manuscript if some values were included in the main text. One could either perform a weighted average over the 6 age bins using estimates of age demographics or simply quote one or two age bins. Of course, the caveats regarding the absolute estimation of the IFR with the used data would need to be highlighted in these numbers.

3 - Maybe the authors should consider including in the main text a sentence summarizing heir response as to why a narrow 2-week window is considered not to affect their results.

4 - In the new Section 5 the authors should consider briefly mentioning that the diagonal terms of the inverse Hessian already provides the credible intervals which include marginalization over all other parameters.

**********

7. PLOS authors have the option to publish the peer review history of their article (

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Reviewer #2: No

PONE-D-21-08131R1

Bayesian inference across multiple models suggests a strong increase in lethality of COVID-19 in late 2020 in the UK

Dear Dr. Pietzonka:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact

If we can help with anything else, please email us at

Thank you for submitting your work to PLOS ONE and supporting open access.

Kind regards,

PLOS ONE Editorial Office Staff

on behalf of

Prof. Jeffrey Shaman

Academic Editor

PLOS ONE