To inform the EpiBeds model structure, we first analysed the detailed COVID-19 Hospitalisation in England Surveillance System (CHESS) and Severe Acute Respiratory Infection (SARI) datasets (see Section SM.1.1 in S1 Supplementary Material) to identify the most relevant hospital pathways and to estimate the distributions of the time individuals spent along each step of these hospital pathways. We classed patients using five states: Hospitalised (not been to ICU), in Critical care (ICU), Monitored (discharged from ICU but still in hospital), Recovered, and Deceased. After hospital admission, patients are either discharged, admitted to ICU, or die (without entering ICU), and from ICU individuals may go on to be discharged from ICU (but remain in hospital in the monitored state) or die. We then estimated the distributions of the time individuals take for each transition (hereafter referred to as “length of stay” or “delay distribution”, with the former preferred for in-hospital events and the latter preferred for out-of-hospital events), in particular: hospital admission to ICU admission, ICU admission to ICU discharge, ICU admission to death, ICU discharge to hospital discharge, hospital admission to death and hospital admission to hospital discharge. For hospital admission to death and hospital admission to discharge, we only considered patients who are not admitted to ICU, to prevent overlap with the ICU-related pathways.

Our aim was to produce a set of ordinary differential equations (ODEs) that best describe hospital progression. We therefore assumed length of stay distributions were gamma distributed, so that they could be approximated by Erlang distributions (see Section SM.1.2 in S1 Supplementary Material). Since treatment policies and practices, and patient demographics, are likely to have changed over time, we estimated the waiting time distributions separately for the first (1^st March 2020 to 15^th September 2020) and second (1^st August 2020 to 31^st December 2020) waves in the UK (Table 1), with monthly cumulative estimates given in Table D in S1 Supplementary Material. Our estimates are consistent with previous results for length-of-stay distributions (Fig A in S1 Supplementary Material), particularly findings for the UK, Europe and Japan ([7–16]). Note that the first and second wave periods have some overlap, as some historic data was needed to fit the second wave.

Comparing the first wave to the second wave, we observe substantial changes in the lengths of stay on ICU. The length of stay from entering ICU to dying slightly increased between the two waves, whilst the length of stay from entering ICU to leaving ICU decreased by a factor of two. Similarly, the length of stay from leaving ICU to discharge decreased by a factor of two. There are various potential drivers for this. First, treatment changes could have reduced the length of time patients require critical care treatment, and prolonged the time until death. Second, younger patients, who were more common in the second wave, take less time to recover and longer to die. The lengths of stay without ICU does not show the same drop in the time to recovery as seen in ICU, but has a similar increase in the time to death, possibly because of improved quality of treatment.

Informed by the estimated length of stay distributions (Table 1), we constructed a compartmental model describing the progression of individuals through the hospital pathways (Fig 1). To account for considerable differences in the duration of different hospital transitions even from the same state, we divided individuals into compartments both in terms of their current status (e.g., Hospitalised or Critical care) and in terms of their future outcome (e.g., will recover, will die). This approach requires more parameters than the more common approach based on competing hazards but is more flexible (resulting in more general phase-type sojourn times in each state) and can be directly parameterised with the available data. Since the mean and standard deviation of the estimated lengths of stay in each compartment are similar, this indicates the gamma distributions are approximately exponential (shape parameter 1). Therefore, flows between hospital compartments are suitably described by constant transition rates (equal to the inverse of the mean of the exponentially distributed sojourn time in the compartment, see Section 4.1 [17]). The resultant hospital flow is shown by the red and orange compartments in Fig 1.

Since the infectious burden in the population determines the rate at which cases will be admitted to hospital, we also used compartments to describe the process of infection in the general population, based on an SEIR (Susceptible Exposed Infectious Recovered) model structure. Symptomatic individuals therefore go through three states of infection, Exposed (but not yet infectious), Infectious (but not yet symptomatic), and Late infection (infectious and symptomatic), with a proportion of symptomatic individuals requiring hospitalisation (L_H) and the other proportion recovering naturally (L_R). The latter distinction is motivated by the fact that the processes of biological recovery and hospital seeking behaviour are conceptually different, hence involving different progression rates: for an infected individual, the time to recovery reflects the natural course of a non-severe infection, while the time to hospital admission is driven by hospital seeking behaviour, current policy, and health-care logistic availability. A proportion of individuals are assumed to remain asymptomatic throughout infection; these individuals follow an infection pathway that is distinct from, but mimics, that of symptomatic individuals.

The structure for the generalised epidemic was constructed to reflect delay distributions from the literature, using constant rates to represent exponentially distributed sojourn times, and sequences of compartments to represent gamma-distributed sojourn times (more details in Section 4.1). This is known as “linear chain trickery” and is a way of representing gamma-distributed sojourn times by using the Erlang distribution. Hence, to describe a gamma-distributed incubation period (i.e., the time from infection to symptom onset) with mean 4.85 days and shape parameter 3 [18], we used three subsequent compartments (E₁,E₂, I) with identical constant rates between them, with mean sojourn time 1.6 days in each compartment [17]. This assumes pre-symptomatic transmission of 1.6 days, which is roughly consistent with literature estimates that show most pre-symptomatic transmission occurs in the two days prior to symptom onset [19]. The delay between symptom onset and hospitalisation is gamma distributed with shape parameter approximately equal to two [18], and we therefore used two compartments for late-infection symptomatic individuals who will be hospitalised (L_H). For cases that recover without hospitalisation, in the absence of better data on the duration of infectivity since symptom onset, we made the parsimonious choice of a single late infection compartment with an exponentially distributed length of stay with mean 3.5 days, such that the overall period during which an individual is actively infectious (I plus the L compartments) is consistent with the 5-day mean generation time estimated in [20]. The resultant compartmental model is illustrated in Fig 1, with the state variables and parameters described in Tables 2 and 3. The equations are reported in Section 4.1.

We assumed only non-hospitalised infectious individuals contribute to new infections, with asymptomatic individuals less infectious than individuals who are pre-symptomatic or symptomatic. Due to behavioural changes, changes in test specificity, and the possibility that asymptomatic cases may correspond to individuals who simply have a long incubation period, identification of the relative infectivity of an asymptomatic case is challenging. We assume relative infectivity of 25%, based on [30,31]. We assume that asymptomatic cases make up 55% of infections, which we determined by adjusting age specific estimates of the asymptomatic rate to the age distribution in England [24]. Although infections from hospitalised patients could have an effect on the overall epidemic, most notably with health care workers as transmission links, detailed genetic data are required to characterise this process [32]. We also assume that nosocomial cases do not substantially alter hospital flow, i.e., upon testing positive nosocomial patients follow similar pathways to community-acquired cases. In the hospital admissions data, we either count patients from admission (if they were tested in the community) or from the date of their first positive swab result (if they were tested in hospital). This second cohort will include all nosocomial cases, who we treat as being admitted from the community.

To evaluate the performance of EpiBeds as a tool for real-time monitoring of the evolving epidemic in England, we performed two-week projections made on days 1 and 15 of each month, from March to December 2020, based on the data available at that time. We illustrate these projections in Fig 4, superimposed to the complete data for both waves. The posterior parameter estimates vary at every projection due to the additional data at each successive time point. We do not report the specific parameter estimates from each model fitting, but only the projections for the data streams. See Section SM.1.3.4 in S1 Supplementary Material for details on the setup when generating these results.

For the first forecasts (start date 1^st April 2020), a transmission change was added on 24^th March 2020 to allow EpiBeds to adjust transmission based on lockdown. Such a short fitting window resulted in large uncertainty, with both growing and declining epidemics falling within the 90% prediction interval. By the 15^th April 2020 forecast, a peak had been observed in the admissions data, but EpiBeds was unable to reconcile the four data streams, which resulted in the forecasts underestimating the reduction in the transmission rate and overshooting the data. This poor performance could be driven by multiple factors, such as challenges with estimating length of stay early in the pandemic (Sections SM.1.2 and SM.3 in S1 Supplementary Material), changing demographics after entering the first lockdown, and data quality issues in some of the data streams (Section SM.1.1 in S1 Supplementary Material). After this point, forecasts remained reliable into the summer.

As transmission started to rise again, EpiBeds was able to accurately forecast the rise in all four data streams. However, throughout September and October, there was a demographic change, from younger to older age groups. This led to the ICU probability gradually declining and the mortality rate increasing, and the forecasts overestimated and underestimated, respectively, these two data streams. In November, the demographic distribution of cases stabilised, and EpiBeds was able to reconcile all four data streams. Noticeably, the 1^st December 2020 forecast completely missed the trend in the data. This was partly to be expected since 2^nd December 2020 marked the end of the second England-wide lockdown, and prior to this transmission rates were also likely to have been increasing due to behaviour changes and the emergence of the more transmissible Alpha variant [34].

Overall, 77% of data points, across all 4 data streams, fell within the 90% prediction intervals admissions 76%; hospital beds 80%; ICU beds 73%; deaths 80%). In many cases when data points fall out of the 90% prediction interval occur where an intervention was introduced during the forecasting window. Others potentially arise from data quality issues between the data streams, particularly during the first wave. Overall, this shows reasonably good model performance, and in practice throughout the pandemic EpiBeds has provided reliable forecasts in all regions where it was used. Our results highlight the context dependence of model performance, with lower predictive ability when transmission rates change frequently, and conversely greatly predictive ability when transmission rates are relatively stable.

The structure of the ODE model was informed by the delay distributions (Section 2.1 and Section SM.1.2 in S1 Supplementary Material). In an ODE compartmental model with constant progression rates, the permanence waiting times in each compartment are exponentially distributed with mean equal to the inverse of the rate. Since all the hospital length of stay distributions were approximately exponential (because the mean is similar to the standard deviation), we modelled the hospital compartments using constant rates. For the background epidemic, we represented the Gamma-distributed incubation period (shape parameter 3, [4]) and the Gamma-distributed time between symptom onset to hospitalisation (shape parameter 2, [5]), by using three and two compartments in sequence. This is a way of representing Gamma-distributed permanence times by using the Erlang distribution. Specifically, an Erlang distribution with shape parameter n corresponds to the sum of n independent and identically distributed exponential distributions [6]. If all rates are equal to r, the mean permanence time in a sequence of n compartments will be Erlang distributed with mean n/r and shape parameter n [7]. This results in EpiBeds described by the flowchart in Fig 1 and by the following system of ODE’s (where the time index in all parameters and variables has been dropped for brevity) dSdt=−λ dES1dt=(1−pA)λ−3rEES1 dES2dt=3rEES1−3rEES2 dEA1dt=pAλ−3rEEA1 dEA2dt=3rEEA1−3rEEA2 dIsdt=3rEES2=3rEIS dIAdt=3rEEA2−3rEIA dLRdt=3rEIS(1−pH)−rLHLR dARdt=3rEIA−rARAR dLH1dt=3rEISpH−2rLHLH1 dLH2dt=2rLHLH1−2rLHLH2 dHDdt=2rLHLH2pT−rHDHD dMRdt=rCMCM−rMRMR dDdt=rHDHD+rCDCD dNdt=−rHDHD−rCDCD dRdt=rHRHR+rMRMR+rLRLR+rARAR where λ=Sβ(f(IA+AR)+IS+LR+LH1+LH2).

Here, f is the reduction in transmission for asymptomatic cases, which is taken to be f = 0.25.

From the solution to the ordinary differential equations, the control and effective reproduction numbers can be calculated. The control reproduction number is given by Rc(t)=β(t)k, where k=(1−pA)(13rE+pHrLH+1−pHrLR)+pAf(13rE+1rAR).

From the control reproduction number, the effective reproduction number can be calculated as Re(t)=Rc(t)S(t)N(t).

To fit the ODE model to data, we generated a likelihood function which we then optimised using MCMC. Specifically, adding Negative Binomial noise to the ODEs describing the model enabled us to calculate a likelihood function for observing the data given our model parameters. This is based on the probability that the deviation between our model and the data can be explained by noise. For each of the four data streams we constructed a likelihood function, which were then multiplied together to build the overall likelihood function. In addition, we included an informative prior for the probability of dying in ICU, p_D, giving an overall likelihood function: L=∑ln(f(dA,yAσA−1,1σA)) +∑ln(f(dB,yBσB−1,1σB)) +∑ln(f(dC,yCσC−1,1σC)) +∑ln(f(dD,yDσD−1,1σD)) −(12ln(2πσprior2))−1(2σprior2)(pD−μprior)2

Where A, B, C and D refer to the four different data streams fitted and σ is the overdispersion parameter of the Negative Binomial observation noise, d is the data, y is the solution to the ODEs, μ_prior is the mean prior estimate of p_D and σ_prior is the standard deviation of the prior p_D estimate. The continuous variables y are defined as yA=rLHLH2 yB=HD+HC+HR+CD+CM+MR yC=CD+CM yD=rHDHD+rCDCM and were evaluated at each day for which a data point d was available. The sums are over all days for which data is available. Adding an informative prior for p_D was required to constrain the values for p_C and p_T.

To fit the model, we manually tuned a random walk MCMC algorithm implemented in Julia, with the input data depending on whether the first wave or second wave was being fitted. We start the epidemic on 20^th January 2020, with I₀ initial cases in the E_A and E_S states. This allowed sufficient time for the other compartments to reach roughly stable proportions before the first data point on 1^st March 2020. Prior values for EpiBeds parameters are specified as described in Sections SM.1.3.1 –SM.1.3.2 in S1 Supplementary Material, coupled with initial conditions for the free parameters with uninformative priors. The ODE was then solved for the input parameters, generating the time-series output that are added to the likelihood functions. Based on these likelihoods, the parameter values are scored and resampled, allowing EpiBeds to explore the parameter space. Code for simulating EpiBeds, and generating the scenarios shown in the paper, are available at [11], along with trace plots for all MCMC results included in this paper. Unfortunately, input data cannot be shared, since this was provided through a data sharing agreement, but similar publicly available data are available at [2].

When fitting the data, we considered the first and second waves separately. Due to changes in length of stay and patient outcomes over time, we cannot fit a single set of parameters over the whole pandemic. To fit the first wave of the epidemic, we used all four data streams, using data starting on 1^st March 2020. When fitting the second wave, we removed beds, ICU, and deaths data prior to 1^st August 2020. Prior to this date, EpiBeds is only constrained by the hospital admissions data, and only the first term of the likelihood (which does not depend on the outcome probabilities p_C, p_T and p_D) is used. After 1^st August 2020, we introduce the other three data streams and compute the other likelihood terms. This then constrains the probabilities to fit the relationship between these data streams in the second wave.