You will be shown a paragraph from a biomedical research paper, followed by a question about the paper. Please extract text from the paragraph to answer the question. Do not include any text in your repsonse other than the answer.

Paragraph: The onset-to-death distribution is also Gamma distributed with a mean of 18.8 days and a
coefficient of va riation 0.45 . ifrm is population averaged over the age structure of a given country. The
infection-to-death distribution is therefore given by:

um ~ ifrm ~ (Gamma(5.1,0.86) + Gamma(18.8,0.45))

Figure 6 shows the infection-to-death distribution and the resulting survival function that integrates
to the infection fatality ratio. Figure 6: Left, infection-to-death distribution (mean 23.9 days). Right, survival probability of infected
individuals per day given the infection fatality ratio (1%) and the infection-to-death distribution on
the left. Using the probability of death distribution, the expected number of deaths dam, on a given day t, for
country, m, is given by the following discrete sum:


The number of deaths today is the sum of the past infections weighted by their probability of death,
where the probability of death depends on the number of days since infection. 8.2 Infection model

The true number of infected individuals, C, is modelled using a discrete renewal process. This approach
has been used in numerous previous studies13'16 and has a strong theoretical basis in stochastic
individual-based counting processes such as Hawkes process and the Bellman-Harris process.”18 The
renewal model is related to the Susceptible-Infected-Recovered model, except the renewal is not
expressed in differential form. To model the number ofinfections over time we need to specify a serial
interval distribution g with density g(T), (the time between when a person gets infected and when
they subsequently infect another other people), which we choose to be Gamma distributed:

g ~ Gamma (6.50.62). The serial interval distribution is shown below in Figure 7 and is assumed to be the same for all
countries. Figure 7: Serial interval distribution g with a mean of 6.5 days. Given the serial interval distribution, the number of infections Eamon a given day t, and country, m,
is given by the following discrete convolution function:

_ t—1
Cam — Ram ZT=0 Cr,mgt—‘r r
where, similarto the probability ofdeath function, the daily serial interval is discretized by

fs+0.5

1.5
gs = T=s—0.Sg(T)dT fors = 2,3, and 91 = fT=Og(T)dT. Infections today depend on the number of infections in the previous days, weighted by the discretized
serial interval distribution. This weighting is then scaled by the country-specific time-Varying
reproduction number, Ram, that models the average number of secondary infections at a given time. The functional form for the time-Varying reproduction number was chosen to be as simple as possible
to minimize the impact of strong prior assumptions: we use a piecewise constant function that scales
Ram from a baseline prior R0,m and is driven by known major non-pharmaceutical interventions
occurring in different countries and times. We included 6 interventions, one of which is constructed
from the other 5 interventions, which are timings of school and university closures (k=l), self—isolating
if ill (k=2), banning of public events (k=3), any government intervention in place (k=4), implementing
a partial or complete lockdown (k=5) and encouraging social distancing and isolation (k=6). We denote
the indicator variable for intervention k E 1,2,3,4,5,6 by IkI’m, which is 1 if intervention k is in place
in country m at time t and 0 otherwise. The covariate ”any government intervention” (k=4) indicates
if any of the other 5 interventions are in effect,i.e.14’t’m equals 1 at time t if any of the interventions
k E 1,2,3,4,5 are in effect in country m at time t and equals 0 otherwise. Covariate 4 has the
interpretation of indicating the onset of major government intervention. The effect of each
intervention is assumed to be multiplicative. Ram is therefore a function ofthe intervention indicators
Ik’t’m in place at time t in country m:

Ram : R0,m eXp(— 212:1 O(Rheum)-

The exponential form was used to ensure positivity of the reproduction number, with R0,m
constrained to be positive as it appears outside the exponential. The impact of each intervention on

Ram is characterised by a set of parameters 0(1, ...,OL6, with independent prior distributions chosen
to be

ock ~ Gamma(. 5,1). The impacts ock are shared between all m countries and therefore they are informed by all available
data. The prior distribution for R0 was chosen to be

R0,m ~ Normal(2.4, IKI) with K ~ Normal(0,0.5),
Once again, K is the same among all countries to share information. We assume that seeding of new infections begins 30 days before the day after a country has
cumulatively observed 10 deaths. From this date, we seed our model with 6 sequential days of
infections drawn from cl’m,...,66’m~EXponential(T), where T~Exponential(0.03). These seed
infections are inferred in our Bayesian posterior distribution. We estimated parameters jointly for all 11 countries in a single hierarchical model. Fitting was done
in the probabilistic programming language Stan,19 using an adaptive Hamiltonian Monte Carlo (HMC)
sampler. We ran 8 chains for 4000 iterations with 2000 iterations of warmup and a thinning factor 4
to obtain 2000 posterior samples. Posterior convergence was assessed using the Rhat statistic and by
diagnosing divergent transitions of the HMC sampler. Prior-posterior calibrations were also performed
(see below). 8.3 Validation

We validate accuracy of point estimates of our model using cross-Validation. In our cross-validation
scheme, we leave out 3 days of known death data (non-cumulative) and fit our model. We forecast
what the model predicts for these three days. We present the individual forecasts for each day, as
well as the average forecast for those three days. The cross-validation results are shown in the Figure
8. Figure 8: Cross-Validation results for 3-day and 3-day aggregatedforecasts

Figure 8 provides strong empirical justification for our model specification and mechanism. Our
accurate forecast over a three-day time horizon suggests that our fitted estimates for Rt are
appropriate and plausible. Along with from point estimates we all evaluate our posterior credible intervals using the Rhat
statistic. The Rhat statistic measures whether our Markov Chain Monte Carlo (MCMC) chains have

converged to the equilibrium distribution (the correct posterior distribution). Figure 9 shows the Rhat
statistics for all of our parameters


Figure 9: Rhat statistics - values close to 1 indicate MCMC convergence. Figure 9 indicates that our MCMC have converged. In fitting we also ensured that the MCMC sampler
experienced no divergent transitions - suggesting non pathological posterior topologies. 8.4 SensitivityAnalysis

8.4.1 Forecasting on log-linear scale to assess signal in the data

As we have highlighted throughout in this report, the lag between deaths and infections means that
it ta kes time for information to propagate backwa rds from deaths to infections, and ultimately to Rt. A conclusion of this report is the prediction of a slowing of Rt in response to major interventions. To
gain intuition that this is data driven and not simply a consequence of highly constrained model
assumptions, we show death forecasts on a log-linear scale. On this scale a line which curves below a
linear trend is indicative of slowing in the growth of the epidemic. Figure 10 to Figure 12 show these
forecasts for Italy, Spain and the UK. They show this slowing down in the daily number of deaths. Our
model suggests that Italy, a country that has the highest death toll of COVID-19, will see a slowing in
the increase in daily deaths over the coming week compared to the early stages of the epidemic. We investigated the sensitivity of our estimates of starting and final Rt to our assumed serial interval
distribution. For this we considered several scenarios, in which we changed the serial interval
distribution mean, from a value of 6.5 days, to have values of 5, 6, 7 and 8 days. In Figure 13, we show our estimates of R0, the starting reproduction number before interventions, for
each of these scenarios. The relative ordering of the Rt=0 in the countries is consistent in all settings. However, as expected, the scale of Rt=0 is considerably affected by this change — a longer serial
interval results in a higher estimated Rt=0. This is because to reach the currently observed size of the
epidemics, a longer assumed serial interval is compensated by a higher estimated R0. Additionally, in Figure 14, we show our estimates of Rt at the most recent model time point, again for
each ofthese scenarios. The serial interval mean can influence Rt substantially, however, the posterior
credible intervals of Rt are broadly overlapping. Figure 13: Initial reproduction number R0 for different serial interval (SI) distributions (means
between 5 and 8 days). We use 6.5 days in our main analysis. Figure 14: Rt on 28 March 2020 estimated for all countries, with serial interval (SI) distribution means
between 5 and 8 days. We use 6.5 days in our main analysis. 8.4.3 Uninformative prior sensitivity on or

We ran our model using implausible uninformative prior distributions on the intervention effects,
allowing the effect of an intervention to increase or decrease Rt. To avoid collinearity, we ran 6
separate models, with effects summarized below (compare with the main analysis in Figure 4). In this
series of univariate analyses, we find (Figure 15) that all effects on their own serve to decrease Rt. This gives us confidence that our choice of prior distribution is not driving the effects we see in the
main analysis. Lockdown has a very large effect, most likely due to the fact that it occurs after other
interventions in our dataset. The relatively large effect sizes for the other interventions are most likely
due to the coincidence of the interventions in time, such that one intervention is a proxy for a few
others. Figure 15: Effects of different interventions when used as the only covariate in the model. 8.4.4

To assess prior assumptions on our piecewise constant functional form for Rt we test using a
nonparametric function with a Gaussian process prior distribution. We fit a model with a Gaussian
process prior distribution to data from Italy where there is the largest signal in death data. We find
that the Gaussian process has a very similartrend to the piecewise constant model and reverts to the
mean in regions of no data. The correspondence of a completely nonparametric function and our
piecewise constant function suggests a suitable parametric specification of Rt. Nonparametric fitting of Rf using a Gaussian process:

8.4.5 Leave country out analysis

Due to the different lengths of each European countries’ epidemic, some countries, such as Italy have
much more data than others (such as the UK). To ensure that we are not leveraging too much
information from any one country we perform a ”leave one country out” sensitivity analysis, where
we rerun the model without a different country each time. Figure 16 and Figure 17 are examples for
results for the UK, leaving out Italy and Spain. In general, for all countries, we observed no significant
dependence on any one country. Figure 16: Model results for the UK, when not using data from Italy for fitting the model. See the


Figure 17: Model results for the UK, when not using data from Spain for fitting the model. See caption
of Figure 2 for an explanation of the plots. 8.4.6 Starting reproduction numbers vs theoretical predictions

To validate our starting reproduction numbers, we compare our fitted values to those theoretically
expected from a simpler model assuming exponential growth rate, and a serial interval distribution
mean. We fit a linear model with a Poisson likelihood and log link function and extracting the daily
growth rate r. For well-known theoretical results from the renewal equation, given a serial interval
distribution g(r) with mean m and standard deviation 5, given a = mZ/S2 and b = m/SZ, and

a
subsequently R0 = (1 + %) .Figure 18 shows theoretically derived R0 along with our fitted

estimates of Rt=0 from our Bayesian hierarchical model. As shown in Figure 18 there is large
correspondence between our estimated starting reproduction number and the basic reproduction
number implied by the growth rate r.

R0 (red) vs R(FO) (black)

Figure 18: Our estimated R0 (black) versus theoretically derived Ru(red) from a log-linear
regression fit. 8.5 Counterfactual analysis — interventions vs no interventions


Figure 19: Daily number of confirmed deaths, predictions (up to 28 March) and forecasts (after) for
all countries except Italy and Spain from our model with interventions (blue) and from the no
interventions counterfactual model (pink); credible intervals are shown one week into the future. DOI: https://doi.org/10.25561/77731

Page 28 of 35

30 March 2020 Imperial College COVID-19 Response Team

8.6 Data sources and Timeline of Interventions

Figure 1 and Table 3 display the interventions by the 11 countries in our study and the dates these
interventions became effective. Table 3: Timeline of Interventions. Country Type Event Date effective
School closure
ordered Nationwide school closures.20 14/3/2020
Public events
banned Banning of gatherings of more than 5 people.21 10/3/2020
Banning all access to public spaces and gatherings
Lockdown of more than 5 people. Advice to maintain 1m
ordered distance.22 16/3/2020
Social distancing
encouraged Recommendation to maintain a distance of 1m.22 16/3/2020
Case-based
Austria measures Implemented at lockdown.22 16/3/2020
School closure
ordered Nationwide school closures.23 14/3/2020
Public events All recreational activities cancelled regardless of
banned size.23 12/3/2020
Citizens are required to stay at home except for
Lockdown work and essential journeys. Going outdoors only
ordered with household members or 1 friend.24 18/3/2020
Public transport recommended only for essential
Social distancing journeys, work from home encouraged, all public
encouraged places e.g.

Question: What are the descriptive statistics for the onset-to-death for coronavirus?
Gamma distributed with a mean of 18.8 days and a
coefficient of va riation 0.45