Moriña et al. BMC Medical Research BMC Medical Research Methodology (2023) 23:75 https://doi.org/10.1186/s12874-023-01894-9 Methodology RESEARCH Open Access Estimated Covid‑19 burden in Spain: ARCH underreported non‑stationary time series David Moriña1*, Amanda Fernández‑Fontelo2, Alejandra Cabaña2, Argimiro Arratia3 and Pedro Puig2,4 Abstract Background The problem of dealing with misreported data is very common in a wide range of contexts for differ‑ ent reasons. The current situation caused by the Covid-19 worldwide pandemic is a clear example, where the data provided by official sources were not always reliable due to data collection issues and to the high proportion of asymptomatic cases. In this work, a flexible framework is proposed, with the objective of quantifying the severity of misreporting in a time series and reconstructing the most likely evolution of the process. Methods The performance of Bayesian Synthetic Likelihood to estimate the parameters of a model based on AutoRegressive Conditional Heteroskedastic time series capable of dealing with misreported information and to reconstruct the most likely evolution of the phenomenon is assessed through a comprehensive simulation study and illustrated by reconstructing the weekly Covid-19 incidence in each Spanish Autonomous Community. Results Only around 51% of the Covid-19 cases in the period 2020/02/23–2022/02/27 were reported in Spain, show‑ ing relevant differences in the severity of underreporting across the regions. Conclusions The proposed methodology provides public health decision-makers with a valuable tool in order to improve the assessment of a disease evolution under different scenarios. Keywords Continuous time series, Mixture distributions, Under-reported data, ARCH models, Infectious diseases, Covid-19, Bayesian synthetic likelihood Background [1]. Many methodological efforts have been made to deal The Covid-19 pandemic that is hitting the world since late with misreported Covid-19 data, following ideas intro- 2019 has made evident that having quality data is essential duced in the literature since the late nineties [2–7]. These in the decision-making chain, especially in epidemiology proposals range from the usage of multiplication factors [8] but also in many other fields. There is an enormous global to Markov-based models [9, 10] or spatio-temporal mod- concern around this disease, leading the World Health els [11]. Additionally, a new R [12] package able to fitting Organization (WHO) to declare public health emergency endemic-epidemic models based on approximative maxi- mum likelihood to underreported count data has been *Correspondence: recently published [13]. However, as a large proportion of David Moriña the cases run asymptomatically [14] and mild symptoms dmorina@ub.edu could have been easily confused with those of similar dis- 1 Department of Econometrics, Statistics and Applied Economics, Riskcenter-IREA, Universitat de Barcelona (UB), Barcelona, Spain eases at the beginning of the pandemic, it is reasonable to 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona expect that Covid-19 incidence has been notably underre- (UAB), Cerdanyola del Vallès, Spain ported. Very recently several approaches based on discrete 3 Department of Computer Science, Universitat Politècnica de Catalunya (UPC), Barcelona, Spain time series have been proposed [15–17] although there is 4 Centre de Recerca Matemàtica (CRM), Barcelona, Spain a lack of continuous time series models capable of deal- ing with misreporting, a characteristic of the Covid-19 © The Author(s) 2023. 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The Creative Commons Public Domain Dedication waiver (http://​creat​iveco​ mmons.​org/​publi​cdoma​in/​zero/1.​0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data. Moriña et al. BMC Medical Research Methodology (2023) 23:75 Page 2 of 8 data and typically present in infectious diseases modeling. shown in [17], the number of affected individuals at time In this sense, a new approach for longitudinal data not t , A(t) = I(t) + R(t) can be approximated by accounting for temporal correlations is introduced in [18] and a model capable of dealing with temporal structures M ∗ (β0 , β1 , β2 , t)A0 ekt A(t) =  , (3) using a different approach is presented in [19]. A typical M (β0 , β1 , β2 , t) + A0 ekt − 1 ∗ limitation of these kinds of models is the computational effort needed in order to properly estimate the parameters. where M ∗ (β0 , β1 , β2 , t) = β0 + β1 · C1 (t) + β2 · C2 (t), Synthetic likelihood is a recent and very powerful alter- being C1 (t) and C2 (t) dummy variables indicating if time native for parameter estimation in a simulation based t corresponds to a period where a mandatory confinment schema when the likelihood is intractable and, conversely, was implemented by the government and if the number the generation of new observations given the values of the of people with at least one dose of a Covid-19 vaccine in parameters is feasible. The method was introduced in [20] Spain was over 50% respectively. At any time t the condi- and placed into a Bayesian framework in [21], showing tion S(t) + I(t) + R(t) = N is fulfilled. The expression (3) that it could be scaled to high dimensional problems and allow us to incorporate the behavior of the epidemics in a can be adapted in an easier way than other alternatives like realistic way, defining µǫ (t) = A(t) − A(t − 1), the new approximate Bayesian computation (ABC). The method affected cases produced at time t . takes a vector summary statistic informative about the The Bayesian Synthetic Likelihood (BSL) simulations parameters and assumes it is multivariate normal, estimat- are based on the described model and the chosen sum- ing the unknown mean and covariance matrix by simula- mary statistics are the mean, standard deviation and the tion to obtain an approximate likelihood function of the three first coefficients of autocorrelation of the observed multivariate normal. process. Parameter estimation was carried out by means of the BSL [22, 23] package for R [12]. Taking into account Methods the posterior distribution of the estimated parameters, the Auto Regressive Conditional Heteroskedasticity (ARCH) most likely unobserved process is reconstructed, resulting models are a well-known approach to fitting time series in a probability distribution at each time point. The prior of data where the variance error is believed to be serially cor- each parameter is set to be uniform on the corresponding related. Consider an unobservable process Xt following an feasible region of the parameter space and zero elsewhere. AutoRegressive ( AR(1)) model with ARCH(1) errors struc- ture, defined by Results This section presents the results of the analyses using the Xt = φ0 + φ1 · Xt−1 + Zt , (1) proposed methodology over a real data set and they are compared to the most common alternatives. The perfor- 2 + ǫ , being ǫ ∼ N µ (t), σ 2 . where Zt2 = α0 + α1 · Zt−1 t t ǫ ǫ mance of the method is also studied by means of a compre- The process Xt represents the actual Covid-19 inci- hensive simulation study, with and without covariates. dence. In our setting, this process Xt cannot be directly The performance and an application of the proposed observed, and all we can see is a part of it, expressed as methodology are studied through a comprehensive simu-  Xt with probability 1 − ω lation study and a real dataset on Covid-19 incidence in Yt = q · Xt with probability ω, (2) Spain on this Section. where q is the overall intensity of misreporting (if Simulation study 0 < q < 1 the observed process Yt would be underre- Although the estimation method is already known and ported while if q > 1 the observed process Yt would be has been tested before, to the best of our knowledge it has overreported) and ω can be interpreted as the overall never been used in the context of ARCH time series, and frequency of misreporting (proportion of misreported therefore a thorough simulation study has been conducted observations). To model consistently the spread of the to ensure that the model behaves as expected, including disease, the expectation of the innovations ǫt is linked ARCH (1), AR(1), MA(1) and ARMA(1, 1) structures for to a simplified version of the well-known compartimen- the hidden process Xt defined as tal Susceptible-Infected-Recovered (SIR) model. At any Xt = 𝜙0 + 𝜙1 ⋅ Xt−1 + Zt , Zt2 = 𝛼0 + 𝛼1 ⋅ Zt−1 2 + 𝜖t (ARCH(1)) time t ∈ R there are three kinds of individuals: Healthy Xt = 𝜙0 + 𝛼 ⋅ Xt−1 + 𝜖t (AR(1)) individuals susceptible to be infected ( S(t)), infected Xt = 𝜙0 + 𝜃 ⋅ 𝜖t−1 + 𝜖t (MA(1)) (4) individuals who are transmitting the disease at a certain Xt = 𝜙0 + 𝛼 ⋅ Xt−1 + 𝜃 ⋅ 𝜖t−1 + 𝜖t (ARMA(1, 1)) speed ( I(t)) and individuals who have suffered the dis-   ease, recovered and cannot be infected again ( R(t)). As where ǫt ∼ N µǫ (t), σǫ2 . Moriña et al. BMC Medical Research Methodology (2023) 23:75 Page 3 of 8 The values for the parameters φ1, α0, α1, α, θ , q and ω performance were observed related to the value of these ranged from 0.1 to 0.9 for each parameter. Average abso- parameters or sample size, besides a poorer coverage for lute bias, average interval length (AIL) and average 95% lower sample sizes, as could be expected. credible interval coverage are shown in Table 1. To sum- marize model robustness, these values are averaged over Real incidence of Covid‑19 in Spain all combinations of parameters, considering their prior The betacoronavirus SARS-CoV-2 has been identified as distribution is a Dirac’s delta with all probability concen- the causative agent of an unprecedented world-wide out- trated in the corresponding parameter value. break of pneumonia starting in December 2019 in the city For each autocorrelation structure and parameters of Wuhan (China) [1], named as Covid-19. Considering combination, a random sample of size n = 1000 has that many cases run without developing symptoms or just been generated using the R function arima.sim, and the with very mild symptoms, it is reasonable to assume that parameters m = log(M ∗ ) and β have been fixed to 0.2 and the incidence of this disease has been underregistered. 0.4 respectively. Several values for these parameters were This work focuses on the weekly Covid-19 incidence reg- considered but no substantial differences in the model istered in Spain in the period (2020/02/23–2022/02/27). Table 1 Model performance measures (average absolute bias, average interval length and average coverage) summary based on a simulation study Structure Parameter Bias AIL Coverage (%) ARCH(1) 0 φ -0.377 3.586 68.77% 1 φ 0.122 0.525 66.08% 0 α -0.296 1.351 74.72% 1 α -0.085 0.920 77.34%  ω -0.020 0.234 83.71%  q -0.022 0.167 85.06%  m -0.226 0.783 79.17% β -0.734 3.581 76.83% σǫ -1.540 3.323 63.65% AR(1) 0 φ -0.983 5.189 70.10%  α 0.043 0.814 92.46%  ω -0.003 0.111 94.10%  q -0.001 0.014 89.03%  m 0.001 0.190 75.17% β 0.007 0.192 74.49% σǫ -1.689 4.718 81.07% MA(1) 0 φ -1.241 5.171 68.31% θ 0.051 0.818 90.40%  ω -0.005 0.108 95.06%  q -0.001 0.014 87.24%  m -0.002 0.187 76.95% β 0.004 0.190 80.38% σǫ -1.619 4.679 83.95% ARMA(1,1) 0 φ -1.834 5.107 61.01%  α 0.062 0.799 89.39% θ 0.011 0.873 96.86%  ω -0.001 0.014 88.32%  q -0.005 0.109 94.97%  m 0.002 0.184 78.49% β 0.004 0.183 78.01% σǫ -1.828 4.631 74.74% Moriña et al. BMC Medical Research Methodology (2023) 23:75 Page 4 of 8 Fig. 1 Registered and predicted weekly new Covid-19 cases in each Spanish region It can be seen in Fig. 1 that the registered data (turquoise) Figure 2 shows the evolution of the registered (tur- reflect only a fraction of the actual incidence (red). The quoise) and predicted (red) weekly number of Covid-19 grey area corresponds to 95% probability of the poste- cases in Spain in the period 2020/02/23–2022/02/27. rior distribution of the weekly number of new cases (the As can be seen in Figs. 1 and 2, there are 4 weeks lower and upper limits of this area represent the percen- (2021–12-26, 2022–01-02, 2022–01-09 and 2022–01-16) tile 2.5% and 97.5% respectively), and the dotted red line for which the predicted values coincide with those corresponds to its median. registered in all simulations, so no underreporting is In the considered period, the official sources reported detected these weeks. This behavior might be due to 11,056,797 Covid-19 cases in Spain, while the model predicts the breakout of a new variant with different character- a total of 21,639,627 cases (only 51.10% of actual cases were istics (for instance producing more symptomatic cases reported). This work also revealed that while the frequency and therefore reducing the underreporting) around of underreporting is extremely high for all regions (values of these dates.  over 0.80 in all cases), the intensity of this underreporting ω The registered values  predicted by  the model can also is not uniform across the considered regions: Aragón and be obtained as Yt = 1 − ω +ω ·q ·X t , and compared Ceuta are the CCAAs with highest underreporting inten- to the actual registered values Yt . That allows comput- sity (  q = 0.28) while Región de Murcia and País Valencià ing standard forecasting error measures as Root Mean are the regions where the predicted values are closest to Squared Error (RMSE) or Mean Absolute Percent- the number of reported cases (  q = 0.46). Detailed under- age Error (MAPE). Globally, the RMSE was 113,145.4 reporting parameters estimates for each region can be found and MAPE was around 8%, ranging between 4 to 13% in Table 2. Although the main impact of the vaccination across regions. The specific RMSE and MAPE for each programmes can be seen in mortality data, the results of region are described in Table S1 in the Supplementary this work also showed a significant decrease in the weekly Material. number of cases as well in all CCAA except Euskadi, as can The global differences in underreporting magnitudes also be seen in Table 2 through the estimates corresponding across regions can be represented by the percentage of to parameter β2. Figure 2 represents the predicted and reported cases in each CCAA (compared to model esti- registered Covid-19 weekly incidence globally for Spain. mates), as shown in Fig. 3. Moriña et al. BMC Medical Research Methodology (2023) 23:75 Page 5 of 8 Table 2 Estimated underreporting parameters and impact Table 2 (continued) of the considered covariates (β1 and β2 are the coefficients CCAA​ Parameter Estimate (P25-P75) for confinement and vaccination respectively) for each Spanish CCAA. Reported values correspond to the median Illes Balears ω 0.98 (0.96—0.99) and percentiles 25% and 75% of the corresponding posterior q 0.36 (0.33—0.39) distribution β1 0.74 (0.43, 1.01) CCAA​ Parameter Estimate (P25-P75) β2 -0.72 (-1.16, -0.34) Andalucía ω 0.97 (0.95—0.99) Región de Murcia ω 0.93 (0.45—0.98) q 0.44 (0.41—0.48) q 0.46 (0.34—0.80) β1 -1.67 (-2.31, -0.39) β1 0.62 (-0.02, 1.36) β2 -1.71 (-2.66, -0.68) β2 -1.97 (-3.07, -0.59) Aragón ω 0.98 (0.97—0.99) Madrid ω 0.98 (0.96—0.99) q 0.28 (0.27—0.32) q 0.37 (0.34—0.40) β1 0.76 (0.17, 1.43) β1 0.35 (-0.39, 0.59) β2 -1.06 (-1.36, -0.69) β2 -0.35 (-0.77, -0.07) Asturies ω 0.97 (0.90—0.99) Nafarroa ω 0.99 (0.97—1.00) q 0.40 (0.37—0.53) q 0.30 (0.26—0.32) β1 0.44 (0.11, 0.69) β1 -1.71 (-1.92, -0.53) β2 -0.90 (-1.77, -0.63) β2 -2.05 (-3.20, -1.33) Cantabria ω 0.97 (0.95—0.99) Euskadi ω 0.99 (0.97—0.99) q 0.30 (0.28—0.35) q 0.27 (0.25—0.31) β1 -0.44 (-0.71, 0.002) β1 -0.42 (-0.69, -0.21) β2 -0.53 (-1.29, -0.25) β2 -0.10 (-0.24, 0.00) Castilla y León ω 0.98 (0.95—0.99) La rioja ω 0.98 (0.96—0.99) q 0.36 (0.32—0.41) q 0.31 (0.28—0.35) β1 -0.84 (-1.33, -0.23) β1 -0.83 (-1.08, -0.35) β2 -1.22 (-1.88, -0.60) β2 -0.43 (-0.71, -0.22) Castilla – La Mancha ω 0.98 (0.96—0.99) Melilla ω 0.97 (0.95—0.99) q 0.33 (0.31—0.36) q 0.34 (0.31—0.37) β1 0.06 (-0.18, 0.44) β1 -0.48 (-0.82, -0.11) β2 -0.80 (-1.11, -0.40) β2 -1.59 (-2.05, -0.93) Canarias ω 0.98 (0.96—0.99) País Valencià ω 0.95 (0.40—0.98) q 0.35 (0.32—0.38) q 0.46 (0.40—0.67) β1 -0.92 (-2.06, -0.29) β1 1.45 (1.24, 1.83) β2 -1.34 (-1.78, -1.06) β2 -1.70 (-2.64, -0.52) Catalunya ω 0.98 (0.96—0.99) q 0.30 (0.27—0.34) β1 -0.25 (-0.52, 0.21) β2 -1.51 (-1.97, -0.94) Discussion Ceuta ω 0.98 (0.95—0.99) Although it is very common in biomedical and epide- q 0.28 (0.25—0.31) miological research to get data from disease registries, β1 0.007 (-0.52, 0.34) there is a concern about their reliability, and there have β2 -1.38 (-1.93, -0.84) been some recent efforts to standardize the protocols Extremadura ω 0.98 (0.95—1.00) in order to improve the accuracy of health information q 0.40 (0.36—0.44) registries (see for instance [24, 25]). However, as the β1 1.45 (1.24, 1.83) Covid-19 pandemic situation has made evident, it is not β2 -0.72 (-1.30, -0.37) always possible to implement these recommendations Galiza ω 0.84 (0.33—0.98) in a proper way. q 0.41 (0.35—0.56) Another work analyzing the cumulated burden of β1 -0.20 (-0.53, 0.18) Covid-19 in Spain [26] estimated that only around 21% β2 -2.03 (-3.07, -1.34) of the cases were reported in the period 2020/01/01– 2020/06/01, but it should be taken into account that it Moriña et al. BMC Medical Research Methodology (2023) 23:75 Page 6 of 8 Fig. 2 Registered and predicted weekly new Covid-19 cases globally in Spain Fig. 3 Percentage of reported cases in each CCAA​ seems reasonable to assume that the underreporting in the longer period considered in this work. Addition- intensity was higher at the early stages of the pandemic, ally, the presented methodology allows for a real time and therefore a lower overall underreporting is expected monitoring and not only cumulated over a time period. Moriña et al. BMC Medical Research Methodology (2023) 23:75 Page 7 of 8 Having accurate data is key in order to provide public Supplementary Information health decision-makers with reliable information, which The online version contains supplementary material available at https://​doi.​ can also be used to improve the accuracy of dynamic org/​10.​1186/​s12874-​023-​01894-9. models aimed to estimate the spread of the disease [27] Additional file 1: Table S1. Root Mean Squared Error (RMSE) and Mean and to predict its behavior. Absolute Percentage Error (MAPE) for the predicted number of cases in each Spanish CCAA. Conclusions The proposed methodology can deal with misreported Acknowledgements (over- or under-reported) data in a very natural and Not applicable. straightforward way and is able to reconstruct the most Authors’ contributions likely hidden process, providing public health decision- DM, AF-F, AC, AA and PP participated in the development of the statistical makers with a valuable tool in order to predict the evolu- model. DM and PP derived the described properties, and DM implemented tion of the disease under different scenarios. the model in R software and conducted the analyses. All authors have read and approved the manuscript. Using a flexible approach for the underlying hidden process, such as ARCH time series, are a natural exten- Funding sion to recent developments (see for instance [19]) pro- Research funded by Fundación MAPFRE. This work was partially supported by grant RTI2018-096072-B-I00 from the Spanish Ministry of Science and Innova‑ posed for fitting underreported time series but restricted tion and by the Spanish State Research Agency, through the Severo Ochoa to the case when the underlying process has an ARMA and María de Maeztu Program for Centers and Units of Excellence in R&D structure and allow us to model phenomena presenting (CEX2020–001084-M). A.F-F acknowledges Agencia Estatal de Investigación for the financial support IJC2020-045188I/AEI/10.13039/501100011033. AC more complex behavior like Covid-19 in the long time was partially financed by PID2021-123733NB-I00 (Ministerio de Ciencia e Inno‑ period considered in the present work. vación, Spain). AC and AA were partially supported by Project “EcoDep” CY- The analysis of the Spanish Covid-19 data shows that AAP2020-0000000013 (“Investissements d’Avenir” ANR-16-IDEX-0008, France). in average only around 51% of the cases in the period Availability of data and materials 2020/02/23–2022/02/27 were reported, and that there The datasets generated and/or analyzed during the current study are available are important differences in the severity of underre- in the GitHub repository, https://​github.​com/​dmori​nya/​Mapfre/​blob/​main/​ Papers/​Paper%​202%​20(BSL)/​BMC%​20MRM/​GitHub/​Data/​cases.​xls. porting across the Spanish regions. The impact of the vaccination program can also be assessed, achieving a Declarations significant decrease in the Covid-19 incidence in almost all regions after 50% of the population had one dose at Ethics approval and consent to participate least (although these results would probably be notably Not applicable (no human or animal experimentation and data and source codes fully available). different if including SARS-CoV-2 immunity-escape vari- ants like BA.4 or BA.5, which are currently predominant Consent for publication in many countries), while the impact of the mandatory Not applicable. lockdown could only be detected by the model in 7 out Competing interests of 19 regions. The authors declare no competing interests. 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