Model 1

The approach estimates the current reproduction number (the average number of secondary cases generated by a typical infected individual, $R_t$) and to use that to forecast future incidence of death. The current reproduction number is estimated assuming constant transmissibility during a chosen time-window (here, one week).

Estimating current transmissibility

Here we relied on a well-established and simple method (3) that assumed the daily incidence, I_t (here representing deaths), could be approximated with a Poisson process following the renewal equation (4):

$$I_t \sim Pois\left( R_t \sum_{s=0}^tI_{t-s}w_s\right)$$

where $R_t$ is the instantaneous reproduction number and $w$ is the serial interval distribution. From this a likelihood of the data given a set of model parameters can be calculated, as well the posterior distribution of $R_t$ given previous observations of incidence and knowledge of the serial interval (5).

We used this approach to estimate $R_t$ over three alternative time-windows defined by assuming a constant $R_t$ for 10 days prior to the most recent data-point. We made no assumptions regarding the epidemiological situation and transmissibility prior to each time-window. Therefore, no data prior to the time-window were used to estimate $R_t$, and instead we jointly estimated $R_t$ as well as back-calculated the incidence before the time-window. Specifically, we jointly estimated the $R_t$ and the incidence level 100 days before the time-widow. Past incidence was then calculated using the known relationship between the serial interval, growth rate and reproduction number. The joint posterior distribution of $R_t$ and the early epidemic curve (from which forecasts will be generated) were inferred using Markov Chain Monte Carlo (MCMC) sampling.

The model has the advantage of being robust to changes in reporting before the time-window used for inference.

Forward projections

We used the renewal equation (4) to project the incidence forward, given a back-calculated early incidence curve, an estimated reproduction number, and the observed incidence over the calibration period. We sampled sets of back-calculated early incidence curves and reproduction numbers from the posterior distribution obtained in the estimation process. For each of these sets, we simulated stochastic realisations of the renewal equation from the end of the calibration period leading to projected incidence trajectories.

Projections were made on a 7-day horizon. The transmissibility is assumed to remain constant over this time period. If transmissibility were to decrease as a result of control interventions and/or changes in behaviour over this time period, we would predict fewer deaths; similarly, if transmissibility were to increase over this time period, we would predict more deaths We limited our projection to 7 days only as assuming constant transmissibility over longer time horizons seemed unrealistic in light of the different interventions implemented by different countries and potential voluntary behaviour changes.

Model 2

Estimating current transmissibility

The standard approach to inferring the effective reproduction number at $t$, $R_t$, from an incidence curve (with cases at t denoted I_t) is provided by (5). This method assumes that $R_t$ is constant over a window back in time of size k units (e.g. days or weeks) and uses the part of the incidence curve contained in this window to estimate $R_t$. However, estimates of $R_t$ can depend strongly on the width of the time-window used for estimation. Thus mis-specified time-windows can bias our inference. In (6) we use information theory to extend the approach of Cori et al. to optimise the choice of the time-window and refine estimates of $R_t$. Specifically:

• We integrate over the entire posterior distribution of $R_t$, to obtain the posterior predictive distribution of incidence at time t+1 as P(I_t+1 | I₁^t) with I₁^t as the incidence curve up to t. For a gamma posterior distribution over $R_t$ this is analytic and negative binomial ((6) for exact formulae).

• We compute this distribution sequentially and causally across the existing incidence curve and then evaluate every observed case-count according to this posterior predictive distribution. For example at t = 5, we pick the true incidence value I₅* and evaluate the probability of seeing this value under the predictive distribution i.e. P(I₅ = I₅* | I₁⁴).

This allows us to construct the accumulated predictive error (APE) under some window length k and under a given generation time distribution as:

$$\text{APE}_{k} = \sum_{t = 0}^{T - 1}{- \log{P\left( I_{t + 1} = I_{t + 1}^{*}\ \right|\ I_{t - k + 1}^{t})\ \ }}$$

The optimal window length k* is then $k^{*} = \arg{\min_{k}{\text{APE}_{k}}}$. Here T is the last time point in the existing incidence curve.

Forward Projections

Forward projections are made assuming that the transmissibility remains unchanged over the projection horizon and same as the transmissibility in the last time-window. The projections are made using the standard branching process model using a Poisson offspring distribution.

Model 3

Objectives

• Estimate trends in case ascertainment and the ratio of deaths to reported cases.
• Use these to forecast the number of deaths in the coming week.

Assumptions

We assume

• that deaths due to COVID-19 are perfectly reported;
• a known distribution for delay from report to death (gamma distribution with mean 10 days and standard deviation 2 days).

Let $D_{i, t}$ be the number of deaths in location $i$ at time $t$. Let $I_{i, t}^r$ be the reported number of cases in location $i$ at time $t$ and $I_{i, t}^{true}$ be the true number of cases. We assume that the reporting to death delay $\delta$ is distributed according to a gamma distribution with mean $\mu$ and standard deviation $\sigma$. That is,

$$\delta \sim \Gamma(\mu, \sigma).$$

Let $r_{i, t}$ be the ratio of deaths to reported cases in location $i$ at time $t$. We assume that deaths are distributed according to a Binomial distribution thus: $$D_{i, t} \sim Binom\left( \int\limits_0^{\infty}{\Gamma(x \mid \mu, \sigma)I_{i, t - x}^{r}dx} , r_{i, \mu}\right).$$

This allows us to obtain a posterior distribution for $r_{i, t}$.

To obtain forecast of deaths, we rely on reported cases to obtain $\int\limits_0^{\infty}{\Gamma(x \mid \mu,\sigma)I_{i, t - x}^{r}dx}$. As cases reported in the coming week may die within the same week ( i.e. for $x \in \{0,7\}$, $\Gamma(x \mid \mu,\sigma) > 0$), we estimate new reporting cases in the coming week by sampling from a Gamma distribution with mean and standard deviation estimated from the number of observed cases in the last week.

While this assumes no growth or decline in the coming week, this baseline assumption is justifiable as a null-hypothesis scenario as it does not influence our results dramatically given the contribution to deaths due to those being very small (i.e. less than 2%).

We therefore obtain the forecasted number of deaths as: $$D_{i, t} \sim Binom\left( \int\limits_0^{\infty}{\Gamma(x \mid \mu, \sigma)I_{i, t - x}^{r}dx} , r_{i, \mu}\right).$$

where $r_{i, \mu}$ is the estimated ratio of deaths to reported cases for the last week of data, and $\int\limits_0^{\infty}{\Gamma(x \mid \mu,\sigma)I_{i, t - x}^{r}dx}$ relies on observed reported cases up to the last day with available and estimated reported cases as described above.

Ensemble Model

For all countries included in the analysis, the ensemble model is built from Models 1, 2 and 3.

Individual Model Outputs

Code

All code used for this analysis can be found at: https://github.com/mrc-ide/covid19-forecasts-orderly and https://github.com/mrc-ide/us-covid19-forecasts-orderly

References

The forecasts produced use the reported daily counts of deaths per country available on the WHO website: https://covid19.who.int

Notes

• Some countries have been excluded from the analysis despite meeting the threshold because the number of deaths per day did not allow reliable inference.

• We have excluded US states from the analysis because a majority of the states are not reporting data on a daily basis.