Unraveling the COVID-19 hospitalization dynamics in Spain using Bayesian inference

The model estimates daily admissions and bed occupancy based on the daily incidence data reported by the authorities. To do so, we estimate the probability of being admitted into a hospital upon being confirmed as a positive case, as well as the delay between detection and admission, and the length of stay (LoS). We employ a Bayesian inference model trained on data from July 2020 to late November 2020, inspired by [4,47]. This technique allows us to integrate prior information with data and is flexible enough as to be easily applied to other regions. With this information, we can reproduce the hospital dynamics in the autonomous region of Aragon -taken here as a case study- from December 2020 to June 2021, and produce forecasts on hospitalization based on any technique for predicting the daily incidence. In addition, we show the generality of the methodology by applying it to all the autonomous regions of Spain. This allows us to study their evolution, and analyze their differences, strategies for case detection and the effects of the first stages of the vaccination campaigns.

We use the open data provided by the Spanish Health Ministry, which reports the number of confirmed cases by symptom onset, or notification date minus 3 days if it is not available, and the number of daily hospital admissions [39]. We further complement this dataset with the one provided by the regions of Aragon and Catalonia for some specific analysis [40, 41]. In particular, the main dataset provided by the Spanish Health Ministry does not contain information on discharges or bed occupancy, limiting regional comparisons to new admissions.

There is an alternative repository focused on hospitalization but has several gaps since it is not updated on Saturdays, Sundays and special holidays [42]. Furthermore, the data on discharges “might not be collected exhaustively.” Thus, for the analysis involving bed occupancy, we resort to the regional repository. Note also that the quality, accessibility and depth of each regional repository varies greatly, and not always matches the data provided by the Ministry, as we show later for the case of Catalonia. Another example is the incidence data provided by the government of Aragon, which is based on the notification date, while the incidence data provided by the Spanish Health Ministry is based on symptoms’ onset. Therefore, we perform the comparisons among different regions using the standardized data provided by the Spanish Health Ministry.

For each region r, given the number of daily cases on day i, \({C}_i^r\), we estimate the daily number of hospital admissions at time t aswhere \({p}_H^r\) is the regional probability of hospitalization and ϕ^r(t − i) is a delay function that yields the probability of being admitted to the hospital on day t given that the case was notified on day i. Data on the characteristics of all hospitalized individuals between early August and late November provided by the Health Department of the region of Aragon indicates that ϕ can be approximated by a Half-Cauchy distribution:

where β is the scale parameter. Similarly, to determine the daily number of discharges, we can apply a convolution between the number of daily admissions and the distribution of the length of stay, ψ,In this case, the data showed that ψ can be well approximated by a log-normal distribution:where μ is the location parameter and σ the standard deviation. So that the bed occupancy in region r, on day t, given the daily incidence up to that day will be given byIn the Supplementary Information (Figs. S8-S10), we also explore the results when the admission delay is approximated by an Exponential distribution and the length of stay by a Gamma distribution.

To estimate the parameters, we implement a Bayesian statistical model using PyMC3 3.11.5 with Python 3.8 [43]. Note that all parameters cannot be estimated simply from bed occupancy data (Eq. (5)) since a larger probability of hospitalization with small LoS can yield the same occupancy as a small probability of hospitalization with large LoS. Thus, we first estimate the set of model parameters for the daily admissions, \({\uptheta}^r=\left\{{p}_H^r,{\upbeta}^r,{\alpha}^r\right\}\), using Bayesian inference with Markov-chain-Monte-Carlo (MCMC). We assume that the likelihood of observing the real-world data point, \({\hat{A}}^r(t)\), is given by NegBinomial(μ = A^r (t| θ), α) to allow for over-dispersion. We run 4 independent chains using NUTS with 2000 burn-in steps and 5000 steps to approximate the posterior distribution of the parameters with uninformed priors.

With these parameters and Eq. (1) we can obtain the daily number of admissions as a function of the number of new detected cases. This information, together with Eq. (5) and the observed bed occupancy, \({\hat{B}}^r(t)\), can be used to estimate the parameters of the log-normal distribution governing discharges. In this case we assume that the likelihood of observing the real data point is given by Normal(μ = B^r(t| θ), σ), but otherwise the training procedure is the same as for the admissions.

The previous model can translate new infections into new admissions, use the distribution of LoS and, finally, estimate bed occupancy for the short-term (2-3 days). Thus, to increase the prediction window, it is necessary to forecast future incidence. There are many techniques available to do this, ranging from classical compartmental models to Bayesian inference akin to the one used here, detailed agent-based models, metapopulation models, branching processes, machine learning and deep learning techniques or other time-series forecasting tools [44‐50]. Since this is not the main purpose of this work, we resort to a simple heuristic that yields satisfactory results and is easier to communicate to non-technical stakeholders.

First, we computed the evolution of the effective reproduction number in the region of Aragon from early July to mid-November. Then, we explored how this quantity changed from day to day by dividing the value at time t, R(t), over its value on the previous day, R(t − 1). This analysis revealed that during the growing phase of an outbreak, the daily increase on R(t) is below 5%. Similarly, during the shrinking phase induced by the non-pharmaceutical interventions imposed by the authorities, the daily decrease is close to 5%, see Fig. S3. Given this observation, we can forecast the value of the reproduction number aswith the sign chosen depending on whether the outbreak is growing or decreasing. Then, we can estimate the number of new cases by multiplying this quantity iteratively by the number of cases observed or estimated in the previous days and introduce it in Eq. (5) to forecast bed occupancy.

To estimate the model parameters, we choose the period between July 1, right before the second wave of infections started in Spain, and December 1, right before the beginning of the Christmas wave. In Fig. 1, we show the number of daily admissions obtained using Eq. (1), while in Fig. S1 we depict the corresponding prior and posterior distributions. The period before December 1 can be regarded as the training period for the model. Afterwards, the prediction is labeled “informed forecast” since it requires knowledge of the number of new cases.

We estimate an admission probability of \({p}_H^{\textrm{AR}}=\) 0.090 [0.086-0.094] (95% C.I.), while the scale parameter of the Half-Cauchy distribution governing hospital admission is β^AR= 3.557[2.579-4.564] (95% C.I.). More specifically, this distribution yields a median interval of 3.5 days and an IQR of 7 days for the delay between case detection and hospital admission.

As described in the Methods section, we use the information on bed occupancy to estimate the model parameters for daily discharges. In Fig. 2, we show the observed daily occupancy of hospital beds in Aragon together with the estimated one. We observe that the agreement is exceptionally good, even after vaccination started, until the beginning of April 2021. From that point on, the estimated occupancy is lower than the observed one. Given that the estimated admissions still agree with the observed values of this variable, this implies that the length of stay for the age groups admitted to the hospital during those dates is larger than the average. Whether this is the effect of protecting the oldest individuals from severe infection, or due to the circulation of new variants, is something that cannot be addressed unless more disaggregated data is released (e.g., discharges by age-group).

We estimate that the Log-Normal distribution used to approximate the distribution on the LoS has μ^AR= 2.476 [2.441-2.512] (95% C.I.) and σ^AR= 0.620 [0.554-0.688] (see Fig. S2 and Table S1 for a summary of the estimated parameters). This distribution yields a median of 12 days with an IQR of 10 days.

In Fig. 3, we show the forecasts performed during the Christmas wave. Each Monday, the algorithm ran on the data collected up to that date and forecasted bed occupancy as a function of the expected change of R(t). Three scenarios were considered: pessimistic (+ 5%), neutral (+ 0%) and optimistic (− 5%), which could be chosen depending on domain expertise and the unfolding of events, such as the imposition of new containment measures (see the section Surveillance data in Aragon in the Supplementary Material for the rationale behind this choice). A summary of all the estimated parameters used in the forecast is presented in Table S1.

The model can be used to compare the characteristics and temporal evolution of the outbreaks in each autonomous region of Spain. This comparison is especially interesting since each region manages its own healthcare service and response to the pandemic, which might lead to different values for the parameters of the model. Thus, we have implemented the model to estimate new admissions for each of the 17 autonomous regions in Spain (the 2 autonomous cities, Ceuta and Melilla, have been discarded due to their small size). In Fig. S5, we show the number of new daily admissions up to May 1, 2021, in each region. As in the previous case, we compute their corresponding parameters \(\left({p}_H^r,{\upbeta}^r\right)\) using data from July 1, 2020, to December 1, 2020. The model can estimate very well the number of new admissions in all regions, although there are some territories in which the estimation is slightly worse (see Table S2).

In Fig. 4, we present a summary of the \({p}_H^r\) and β^r values obtained in the estimation of new admissions for all the autonomous regions of Spain. We observe that most regions cluster together around the same values, with a value for the probability of an infected individual needing hospital care upon detection around 9% and a delay of 3 to 4 days between detection and admission.

Focusing on the estimation of new hospital admissions in Aragon, a few remarks are in order. First, the region of Aragon was severely hit during the training period, with two infection waves, while other Spanish regions only had one. Second, the number of daily admissions can vary substantially, yielding a large prediction credible interval. Third, we observe that the prediction is reliable until late May since the probability of hospitalization estimated in summer 2020 still gives satisfactory results until that date. Lastly, the rising wave of infections in early summer 2021 was associated specifically with individuals between 15 and 25 y.o., and several outbreaks were linked to summer parties celebrated after the end of the academic year, leading to the closure of nightlife in several regions [51]. A similar pattern was also observed in the Netherlands [27]. This significant difference in the age profile explains why the incidence is not translated into hospital admissions.

There are several hypotheses that may explain why the probability of hospitalization barely changes even after the introduction of vaccination during the first half of 2021. First, early vaccination was focused on the most vulnerable, a relatively small group of the population, which can be attended in medicalized nursing homes rather than in hospitals. Thus, reducing infections among that small group had a minor impact in both cases and hospitalizations. Second, another possible explanation is that vaccination reduced the severity of cases, hindering case detection. Indeed, since vaccination reduces the chance of suffering severe disease, it is possible that those infections might have been undetected, not contributing to lower the value of p_H. Third, the vaccines’ effects might have been hidden by the presence of a more virulent variant of SARS-CoV-2, yielding a scenario similar to the one in late 2020. Lastly, it has been proposed that the profile of hospitalized cases shifted towards un-vaccinated groups, compensating the effect of vaccination and leaving p_H unaltered. Surveillance data shows that, indeed, the fraction of hospitalizations corresponding to the oldest age groups (older than 80 y.o.) steadily declined since vaccination started. Conversely, the hospitalization rate within the 40 to 49 and 50 to 59 y.o. groups increased (see Fig. S4), especially in May. Note that the vaccination of these groups started precisely in May, which explains the observed pattern. Recently, daily incidence data disaggregated by age has been made publicly available by the Ministry. Further studies should use this data to better estimate the hospitalization dynamics by age-group and its relationship with vaccination.

We observe that, albeit the prediction of new cases is relatively simplistic, the forecast of occupancy is remarkably good, especially during the first days of the prediction. In particular, the procedure can correctly estimate the maximum bed occupancy and the date when the tendency shifts downwards. Admittedly, the ±5% scenarios may not be suitable for other regions or other circumstances, as they were specifically tailored for the situation in Aragon (see the section Surveillance data in Aragon in the Supplementary Material). Expert opinion and local knowledge have been commonly used in the unfolding of the pandemic in the absence of data [18, 27, 29, 48]. However, their power is limited, and models should be updated as soon as new data is available [52, 53]. Fortunately, our model can be applied to the output of any incidence prediction algorithm, and thus it can be easily adapted to other regions, or more complex scenarios.

As seen in Fig. S5 and Table S2, the estimation of new hospital admissions is slightly worse in a handful of regions. A lower quality of the fit usually signals that the region changed its testing capacities, rather than other external factors such as emergence of new variants. For instance, in the Principality of Asturias, we observe that after mid-November the model tends to overestimate new admissions. The reason for this discrepancy can be that either some interventions reduced the value of p_H, or that the surveillance system was improved, and more cases were detected. It turns out that Asturias was one of the three regions, together with the Canary Islands and the Balearic Islands, which did not use antigen tests in their surveillance system. This situation changed on November 10, 2020, which would explain why our predictions overestimate occupancy after mid-November 2020. An analogous situation can be observed from mid-January 2021 in the Balearic Islands, which can be related to a massive testing campaign using antigen tests that started on mid-January 2021 and extended up to February 2021. Lastly, we also observe a slight overestimation of occupation in the region of Madrid. In this region, the health department of the autonomous community redefined contact tracing on September 28, 2020, excluding social encounters from the surveillance system. This change was reverted on November 20, 2020. The first change would force p_H to increase, while the latter would produce a larger number of detected cases. The combination of both would result in a larger estimation of bed occupancy from late November, therefore nicely explaining the divergence obtained in the model.

Focusing on the estimated values for the admission delay and admission probability (Fig. 4), the regions with the most different values are those for which the prediction is worst, as previously describe. It is interesting to mention that Catalonia (CT) has the lowest admission probability, which could indicate that the region is detecting more cases than the rest. However, as shown in Fig. S6, this is not indeed the case as the observed difference is a consequence of the data alone. Certainly, there are significant differences between the values reported by the Spanish Health Ministry and the regional government of Catalonia. Using the values provided by the latter, the hospitalization probability is 8.5%, in line with the rest of the regions.

Finally, it would also be possible to hypothesize that the observed differences among the regions in which the prediction was successful to the characteristics of their populations. For instance, one of the drivers of these differences could be that some regions have a larger fraction of the population within the eldest age-groups, which are more prone to need hospital care. However, in Fig. S7, we show that the correlation between these variables is very weak and cannot explain the difference.

The methodology presented here, as we have shown, can easily be used evaluate the effect of interventions on the hospitalization dynamics during a pandemic. Many models found in the literature usually fit the incidence to agnostic growth models, with parameters that do not have a clear interpretation in terms of epidemic spreading, and even assume that there are no interventions during a wave. The over-parametrization of these growth models may even lead to convergence problems [3]. At the same time, the model we proposed to translate cases into occupancy is independent from the one used to estimate incidence. Hence, it is possible to substitute the latter by any other model, including complex compartmental models with advanced features such as contact tracing schemes, seasonality or multi-scale interactions.

Another important strength of the model is the use of Bayesian inference, as it allows us to integrate prior information with data in a flexible way. Besides, it just requires aggregated statistics which are relatively easy to obtain (new cases, new hospitalizations and current occupancy) even at the regional level. This way, and thanks to the interpretability of its parameters, it provides a unifying framework to study the hospitalization dynamics of multiple areas, enabling the evaluation of the effect of different policies. As we have discussed, the model can be used to detect changes in the dynamics, and to study several waves at the same time, something that models based on growth equations fitted to a single wave cannot do.

Our model also has several limitations. The use of aggregated data facilitates its applicability, but also limits its potential if individual data is available. For instance, models based on queues could use individual information to predict precisely when a specific person will be discharged. Our choice to model the incidence was heavily based on the observations in the area of interest, and might not be directly applicable in others. Note, however, that it is completely decoupled from the model of the hospital dynamics, and thus it can be improved without changing the main model. Another limitation is that to train the model it is necessary to have information of a period of a few weeks. Even though the predictions can then be used for periods much larger than the one used for training, as shown in the paper, it is not suited for the very first days of an emerging pandemic. Lastly, in Bayesian frameworks the choice of priors can be problematic, but we showed that in this case with uninformative priors convergence is correctly achieved.