key: cord-1047172-8awssp1d authors: López, Leonardo; Rodó, Xavier title: A modified SEIR model to predict the COVID-19 outbreak in Spain and Italy: simulating control scenarios and multi-scale epidemics date: 2020-12-25 journal: Results Phys DOI: 10.1016/j.rinp.2020.103746 sha: 4b76caff586cae6d2c67113fcd5e24242f90858a doc_id: 1047172 cord_uid: 8awssp1d After the spread of the SARS-CoV-2 epidemic out of China, evolution in the pandemic worldwide shows dramatic differences among countries. In Europe, the situation of Italy first and later Spain has generated great concen, and despite other countries show better prospects, large uncertainties yet remain on the future evolution and the e, or attack strategies. Here we applied a modified SEIR compartmental model accounting for the spread of infection during the latent period, in which we also incorporate effects of varying proportions of containment. We fit data to reported infected populations at the beginning of the first peak of the pandemic to account for the uncertainties in case reporting and study the scenario projections for the individual regions (CCAA). The aim of this model it’s to evaluate the confinement rate at the first stages of the epidemic outbreak in order to evaluate the scenarios that minimize the incidence but also the mortality. Results indicate that with data for March 23, the epidemics follow an evolution similar to the isolation of 1 , 5 percent of the population, and if there were no effects of intervention actions it might reach a maximum of over 1.4 M infected around April 27. The effect on the epidemics of the ongoing partial confinement measures is yet unknown (an update of results with data until March 31st is included), but increasing the isolation around ten times more could drastically reduce the peak to over 100 k cases by early April, while each day of delay in taking this hard containment scenario represents a 90 percent increase of the infected population at the peak. Dynamics at the sub aggregated levels of CCAA show epidemics at the different levels of progression with the most worrying situation in Madrid and Catalonia. Increasing alpha values up to 10 times, in addition to a drastic reduction in clinical cases, would also more than half the number of deaths. Updates for March 31st simulations indicate a substantial reduction in burden is underway. A similar approach conducted for Italy pre-and post-intervention also begins to suggest a substantial reduction in both infected and deaths has been achieved, showing the efficacy of drastic social distancing interventions. At last we show the real evolution of the pandemic up to the end of May and the beginning of July in order to calculate the real confinement rate from data to compare with the scenarios formulated at March. Epidemiological reports for the current situation of COVID-19 out of China indicate Europe is one of the centers of the pandemic, and that nearly 200 countries/regions worldwide have reported cases [1] . Particularly in Italy and Spain, a large outbreak of SARS-CoV-2 infection is underway. Given the lack of previous exposure to this virus, future predictions of both the levels of virus spread in the global naive population and the acquired herd immunity cannot be properly anticipated. Data on the last similar pandemic one century ago remains limited for obvious reasons, despite aftermath reports exist [2] . Policymakers in the most affected countries face a difficult situation when trying to balance between draconian public health actions and keeping the economy alive, as the impact on the health of severe economic measures is also well-known [3] . The fact that the 1918 Spanish flu pandemic had a second deadly epidemic wave, presumably caused by mutations in the H1N 1 viral strain [4] , has also stimulated a vivid debate on whether actions to take now should take into account this uncertain future. Under this scenario, optimal action on the COVID-19 pandemic is hard to fathom. And more so, the extent of pre-symptomatic and asymptomatic infections is not narrowly constrained (e.g. up to 86 percent [5] ). In China, regions as well as local governments, including Hubei, tightened preventive measures to curb the spreading of COVID-19 since Jan. 2020 [6] . Many cities in Hubei province were locked down and many measures were implemented, such as tracing close contacts, quarantining infected cases, promoting social consensus on self-protection (e.g. wearing a face mask in the public area, minimum social distances). However, in other areas, the extent and efficacy of the so-called confinement or self-isolation are doubtful, and Facebook data on mobile location and movement showed yet massive people displacements under semi-confinement measures. At a time when the success of large-scale social distancing interventions is critical, access to accurate information to ascertain mobility is lacking [7] . Similarly, credible serology tests that could show whether someone has had the infection and recovered are not yet massively deployed, thereby assessment of SARS-CoV-2 prevalence in the population is not established [8] . We followed the approach of Peng et al. [6] and implemented a modified SEIR model that accounts for the spread of infection during the latent period. This novel capacity to significantly spread during the latent period is a distinctive feature of the current SARS-CoV-2 epidemic if compared to SARS in 2003. Therefore, many classical models such as SIR [9, 10, 11] , SEIR [11] and SEIJR [12] are not appropriate to describe the outbreak of SARS-CoV-2 in China and elsewhere. Thanks to new data on the average latent period and average time of treatment, this time delay process can be successfully incorporated in a novel dynamical system framework to describe these unique dynamics. It has been discussed that at times of ongoing epidemics, and due to errors and under-reporting, accumulated numbers of diagnosed cases and even the number of deaths might reflect more reliably the extent of epidemic progression than the daily reported new cases. Alternately, and as in Chen 2020 [9] and Peng et al. [6] we also employ the accumulated numbers in time as variables, as it was done also for modeling MERS-CoV in the recent past [13] . According with some studies [14] many of the first cases in China (before January 1, 2020), were linked to the Huanan Seafood Wholesale Market, as compared with 8.6% of the subsequent cases. The mean incubation period was 5.2 days, with the 95th percentile of the distribution at 12.5 days. In its early stages, the epidemic doubled in size every 7.4 days. With a mean serial interval of 7.5 days, the basic reproductive number was estimated to be 2.2. On the basis of this information, there is evidence that human-to-human transmission has occurred among close contacts since the middle of December 2019. Isolating key segments of populations (e.g. vulnerable populations, workers providing essential services, or territory with too fast-growing incidences) to protect them from an uncontrolled epidemic progression, is under vivid discussion as the real extent of viral spread in the population is not well known. Both the limited diagnostic capacity and the lack of clear strategies and mechanisms to quarantine infectious individuals, stand as two of the main limitations to control disease spread. Countries such as South Korea [15, 16] and Taiwan [17, 18] rapidly deployed aggressive contact tracing and quarantine systems [14] , understanding that early deployment of resources to try to control initial seeding foci, often not only balances public health criteria but also economics. Given the differences in public health strategies and the varying capacity of the national health systems in each country to tackle the extent of the infection in the population, the growing proportions of undiagnosed infected that eventually show up, are seen to exert elevated stress on the already saturated health's system capacities. It is therefore relevant to accurately evaluate the effects of the social distancing actions imposed by governments, such as the individual or territory isolation and interruption of labor activity and/or intra and inter-urban transportation. Such strong provisions of isolation for suspected cases and infection due to contact with undiagnosed individuals are also taken into account in our modeling approach. This same SEIR model has been utilized to compare the effects of the lock-down of Hubei province on the transmission dynamics in Wuhan and Beijing [6] . Many works have been done to evaluate different aspects of COVID-19 dynamics. In Mohammad et al. [19] authors use a dynamical model based on the fractional derivative of nonlinear equations that describe the outbreak of COVID-19 according to the available infection data in the press. They simulate the available total cases reported and based on the data they use to show graphical illustrations of the numerical solutions of all parameters of the model being handled under different situations. The results of the model contribute to the ongoing research to reduce the spreading of the virus and infection cases but the complexity of the model can be very high. The model formulated by Khan and Atangana [20] the authors describes a mathematical model for CVID-19 putting the eye in the details of interaction among the bats and unknown hosts, then among the peoples and the reservoir of the infection (seafood market). In this work, the seafood market is considered the main source of infection. They assume the purchasing of items from the seafood market can infect either asymptomatically or symptomatically. They consider the available infection cases for January 21, 2020, till January 28, 2020. This model is centered on the very early stages of the pandemic and try to solve the source of this worldwide infectious disease. This model is based on a previous work by Singh et al. [21] where the authors analyze the moderate epidemiological model to describe computer viruses with an arbitrary order derivative having a non-singular kernel. Gao et al. [22] investigate to find the optimal values of a mathematical model that reproduce the transfer of 2019−nCoV from the reservoir to people. This model, named Bats-Hosts-Reservoir-People coronavirus (BHRPC) model, is based on bats as essential animal beings. They simulate the spreading under the optimal parameters. Whereas the obtained results show the effectiveness of the theoretical method considered for the governing system, the results also present much light on the dynamic behavior of the Bats-Hosts-Reservoir-People transmission network coronavirus model. As in the previous case, in this paper authors concentrate on the virus spillover from the animal reservoir more than on the human to human transmission of the virus. Since the beginning of the pandemic, many efforts have been made by the scientific community to assess the possible impact of the pandemic to analyze strategies to mitigate its effect on the population [23] . In this sense, several approaches have been proposed. From machine learning techniques to understand the spreading and improve the forecasting of the infection [24] , to classic modeling approaches [25, 26] based on differential equations as previously mentioned. All of these efforts are not in vain since all the efforts of the community answer different aspects of the same problem. In this work, we present a simple and easy-to-implement mathematical model that allows us to easily evaluate the best population confinement strategy during the initial stages of an outbreak. This allows us to know the impact that this strategy has on the final size of the outbreak and deaths. To do this, we focus on the first COVID-19 wave that spanned from early February to mid-July in Europe. We only use the data from the first wave of the pandemic due to there is a high discrepancy in the way this data was reported after July. For Spain, this problem is notable, mainly for the recovered cases but also for the reported cases. The manuscript is organized as follows: in Section 2, we describe the dynamic model and datasets used to simulate the evolution of the outbreak of SARS-CoV-2 in Spain and its administrative regions (CCAA), as well as in Italy, we also describe the methods used for the model optimization and the main parameters. The effect on the epidemic curves of the efficacy of the different intervention measures aggregated for Spain is presented in Section 3.1. Section 3.2 provides estimates of the future number of diagnosed people, fatalities, and recovered individuals for each CCAA prediction as of March 23 and the main active foci in Section 3.3. Section 3.4 addresses model uncertainties and limitations of our study, and we add results for Italy pre-and post-interventions in Section 3.5. We also discuss the efficacy of possible actions in each case and territory in terms of health burden and report and provide an update of the situation up to March 31 and May 30 for Spain and for the case of Italy we provide a first actualization of the model for March 31 and a second until July 4. We used a generalized SEIR modeling framework similar to Peng et al. [6] , which enables the testing of control interventions. Compared with statistical methods mathematical modeling based on dynamical equations can often provide essential information on the epidemic dynamics. This is particularly true when basic epidemiological parameters are unknown or largely uncertain and more mechanistic understanding is needed, such as for the current COVID-19 disease. The population is assumed constant due to the rapid disease spread, i.e. the births and natural death have the same value. The recovery rate and death rate are time-dependant. The model also assumes that susceptible individuals (S) are contagious upon coming into contact with infected (I) individuals not detected. It is assumed, however, that the infected are all quarantined (Q) and that they do not have contact with susceptible individuals. In turn, the susceptible population can be protected by confinement by moving to the protected population compartment (C). This assumption on Q means that hospital infections are not considered under this framework, therefore resulting in a potential underestimation of the real contagion extent. However, we chose this option in an attempt to be conservative and our results should be interpreted as the current best-case scenario. It is assumed also that the protected population does not have contact with the infected individuals and therefore cannot be infected. The COVID-19 dynamics is modeled by the following equations system: where S(t) is the susceptible population, C(t) is the confined susceptible population, E(t) is the exposed population, I(t) is the infectious population, Q(t) is the population under Quarantine (infected reported cases), R(t) is the recovered population and D(t) is the dead population by the virus. The main parameters of the model are the protection rate (α), the infection rate (β), the incubation rate (γ), the quarantine rate (δ), the natural death and birth rate(µ) (1/(80 * 365)), the recovery rate (λ(t)), the mortality rate by the virus (k(t)) and finally, τ is the length of the protection by confinement (1/30) The α parameters represent the rate of people being protected from infected populations at time t and it is used to model the different actions of control of the epidemic by isolation of a healthy population. The parameters λ(t) and k(t) are time-dependent following the approach of [27] and for simplicity are modeled as shown next: where k 0 , k 1 , λ 0 and λ 1 are the fitted coefficients. The model parameters and their meaning are summarized in Table 1 . The data for fitting the model to Spain was obtained from public data sources [28, 29] and correspond to the reported cases, recovery cases, and deaths. Data is available for the entire country as well as for individual CCAA. Data for Italy is retrieved from the World Health Organization [30] . The model was fitted in a non-linear approach by calculating the normalized least-squares error of the model approximation and the infected reported cases. The optimization algorithm used was trust-region-reflective with a maximum number of iterations of 500 and a step size of 6.4e − 7. The error for the model fitted for all Spain was 3e − 3 + − 0.6e − 4 (ci = 95%).The fitted parameters are summarized in Table 2 Length of the protection The ordinary differential equation (ODE) system was solved by using a nonlinear data-fitting approach that minimizes a least-squares error function, by using the lsqcurvefit function of the Matlab optimization toolbox. The problem to minimize is shown in where given an input data, x data (Q,R and D), provided by the integration of the ODE system 1,y data is the observed output (Active reported cases, Recovered and Deaths) sourced from public repositories. Confirmed cases may include presumptive positive cases and probable cases. The ODE system 1 was integrated by using a 4th order Runge-Kutta method. These kind of methods are easy to implement and very stable. For straightforward dynamical systems, this method can converge fast if we use an integration step in the order 1e3. Although from the computational point of view this method is expensive compared to simpler methods such as Euler's, the error of approximation is considerably less. On the other hand, this type of method has greater stability, especially when applied to non-linear systems [31] . The fitted parameters are the confinement rate , the infection rate , the incubation rate , the quarantine rate , the coefficients and that determine the recovery rate and the coefficients and that determine the mortality rate. Thus, the optimization problem described in equation 3 is subject to the constraints specified by the vectors lb = [0; 0; 0; 0; 0; 0; 0; 0] and ub = [1; 2; 2; 2; 1; 1; 1; 1] (lower limit and upper limit, respectively). The algorithm chosen to minimize the problem was a trust-region57 because this type of algorithm is especially useful for solving non-linear optimization constrained problems. To obtain a representative measure of the optimal set of parameters, 500 iterations of the method were performed taking initial values for the parameters, randomly within a uniform distribution. he same algorithm was run using incremental data set sizes (from 10 points to 50 points, the equivalent of 80% of the data set). This procedure was carried out point by point 100 times with stochastic initial conditions in order not to always fall into the same local minimum. Figure 1 shows a measure of the uncertainties of the fit using both a data set of 80% of the total values and how the remaining values (not used in the training of the model), were correctly predicted. The mean error is of the order of 3e − 3 + − 6e − 4 (CI95%) The model fitting to data aggregated for Spain on reported infected population up to March 23 yields a good approximation to the exponential curve, as well as to the reported evolution in deaths and recovered 2. The model was fitted from day 1 of the epidemic at the end of February because the initial protection rate is assumed to be 0. Progression of infected, deaths, and recovered track very approximately observations. With data up to March 22, the model predicts the peak in cases around the end of April or early days in May with an error of the approximation of less than 5%. The current scenario may already include to some extent the effects of the partial 'confinement' measures imposed on March 13, therefore it is likely that total projected infections would be much higher should these restrictions not exist. More than 1.6million people would have been recovered from SARS-CoV-2 infection by mid-May and total deaths for the entire country at that time would rise over 100 thousand people 3. To approach the degree of colonization of the virus at smaller levels of aggregation than the entire country, we applied the same model framework to each of the 17 individual CCAA. The former was an attempt to model the epidemics spatially and approach scale-derived differences with country aggregated data (e.g. municipality-level data is not yet available). Results are shown in Figure 5 and display varying developmental stages of the COVID-19 epidemic outbreak, thereby recovering the different demographic topologies in the variety of CCAA and Figure 6 shows the extended fitting to have a better understanding of different development stages at the community level. The largest increases of infected -in absolute terms-are expected according to the CCAA curves, in the coming weeks for Madrid, Catalunya, Castilla La Mancha, Paìs Valencià, Castilla León and Andalucía. All the former display -under the current scenario evolution-peak cases occurring from late April to early May. To compare Figure 2above with the effect of community dis-aggregation on the evolution of reported cases for Spain, simulations were performed individually and results aggregated together as shown in Figure 4 . Results are comparable despite they show relatively larger values for total infected than those arising from the simulations on the country's total cases. The situation in Spain is at this time yet very severe, based on public data until March 25. To this end, we tested the potential effects of intervention strategies to control and reduce social contacts. Effects of the adopted policies on March, 14 are not yet known or they can barely begin to show up as substantial changes in the epidemic curve at this date (see Appendix for an update to March 31. To help in the evaluation, we generated different scenarios with varying values of α, to simulate the effects of social protection. Increases in α were structured in scenarios representing up to 10 times larger values than in that obtained from observations up to March 23. The results of these scenarios are summarised in Figure 3 . A drastic decrease in the resulting number of cases can be seen, if we increase the actual daily protection rate around ten times. Results with the current epidemic evolution for Spain, show that over 1, 400, 000 quarantined infections would be reported by the end of April, with an estimated burden of over 100, 000 deaths. A four times increase in the levels of the Results are shown in Figure 7 for Madrid and in Figure 8 for Catalunya, respectively. Similar conclusions on the large effect of imposing stricter epidemic containment measures can be seen in both regions, even if imposed at the current stage of the epidemic development. Because the strength of this type of model, both to adjust and to predict future dynamics, depends on the accuracy in case reporting, it is important to analyze the uncertainties in the adjustment and evaluate the degree of variation of the parameter estimates. To get a better idea of its performance, the model was fitted with a successively longer data series, adding a new day to the data window with which the fitting was computed. The shortest time series used was of 5 days from February 29 to March 3 and the longest covered up to March 31. The temporal span of the time series used can seriously affect the value of the fitted parameters. This is especially true at the beginning of the epidemic when the relationship between infected, dead, and recovered is not entirely clear. For example, if the number of confirmed cases is small, it is difficult to ascertain whether the quarantine has been rigorously applied or not. Also, such under-representation would suggest that the number of infectious people is much larger than the number of confirmed cases. Results of the different dynamics and the parameter space can be seen in Figure 1 . Those parameters most affected are the ones directly related to the infection rate and the recovery rate, as well as the protection rate α. To see the performance of applying the same model framework in the other European country with more quarantine cases to date, we fitted the model to the COVID-19 epidemic in Italy. This way, as the situation in Italy precedes that of Spain by around one week, evaluation of control scenarios can be done at a more advanced stage of confinement measures. As the first cases in Italy were reported at the end of February, two different sets of fittings were performed to compare the confinement restrictions imposed by the Italian authorities on March 10th. The NMSE in this case was 2.7e − 3 + − 3.6e − 4 (ci = 95%) and the parameters for each case are summarized in Table 3 . Current estimates of future trends in new infections in the weeks after March 23 largely compromise the capacity of the Spanish health system, as it happened for Italy. This is especially critical in particular for intensive care units from the end of March [32] . Our projections for the forthcoming evolution in the COVID-19 epidemic curve for Spain show a pessimistic scenario unless current social distancing measures are effective (see Appendix for an update as of March 31. With the estimates at hand, over 1.4 million people would be clinically diagnosed at the disease maximum and over 100.000 resulting in deaths. Imposing stricter measures under the current uncertainties seems a logical preventive option given the considerable gains in terms of infection and casualties, even at moderate increases in alpha. Assessment of total country changes through aggregated and CCAA disaggregated data estimates, provides similar dynamical evolution despite different absolute values. Discrepancies to larger values might partly come from the heterogeneous population landscape in the different CCAA (e.g. cities with high population density), as opposed to the homogeneous contact assumption in the aggregated model estimates. Gaining extra time for exerting alternative interventions, such as temporary increases in public health responses, or increasing the capacity of massively screening the population to gain an accurate value of the real prevalence of SARS-CoV-2, or to develop disease treatment strategies seem a logical conclusion. While two of the most populated CCAA (namely Catalunya with roughly 7.7M people and Madrid Community, with 6.7M people) are already in the lead in terms of epidemic progression, other similarly populated regions lag only a few days entering a similar slope in their respective epidemic growths. The largest population density in Madrid compared to other CCAA together with its more advanced epidemic progression cast urgent attention on the evolution of this highly active epidemic focus. Those regions where the outbreak was initiated earlier (e.g. Madrid, Catalunya, Euskadi, and Navarra) or appear to have more connectivity to the initial foci and largest population density tend to show earlier peaks. Those regions yet in the early stages of the exponential increase are those where strong initial social isolation measures (therefore protection of the population), would have the largest effects in resulting infected ( Figure 5 ). The projected rise in the number of infections shown in Figure 2 is very sensitive to the latest data used, as these data are the ones governing the changes in trends, therefore errors in reporting or other large uncertainties associated with these values may dramatically alter the outcome in terms of the number of infected and associated mortality. For the curve to happen as shown, the simulation assumes the initial conditions hold, therefore if that were not the case and measures were lifted before the epidemic reaches R 0 ≈ 1, infections might rise again. Improvements can be achieved with more complete integration of time-delay coordinates in classical SEIR models [6] . This way, both the incubation period and the period before recovery, as well as the precise role of the asymptomatic population, can be better represented at times when this information is highly uncertain. A substantial gain in the epidemic projection in the form of reduced infections in the population would already be achieved with a four-times rise in the efficacy of the control situation we had around March 9 − 16. This was the time when social distancing/confinement was imposed (March 13). Time will show to which extent current measures manage to increase the value of alpha. Application of strict containment measures clearly shows a drastic effect on the epidemic progression and a substantial reduction in both infected and to a lesser extent, deaths ( Figure 3 ). The control scenarios presented contemplate a significant increase in the percentage of the population under confinement, ranging from 1.3 percent daily to 13 percent daily. These percentages imply that after a week of starting this control, the percentage of protected susceptible population is 9.1% and 91%, respectively, from the start of the measurement. These numbers explain the drastic differences in the two dynamics observed and the huge gain in lowering the toll of infected. Update of the epidemic situation in Spain partially showing the effect of partial lockdown and the current values for alpha are shown in the Appendix as for March 31. Later we update the model for May 30 in the case of Spain and its communities and until July 4 for Italy. This actualization suggest that the model first predictions and scenarios formulated are not to far from the reality. The Covid-19 pandemic is exerting unprecedented stress on the public health systems of many countries. Those at major risk now are Italy and Spain, and for them, the efficacy of the partial confinement or total lockdown effects are yet unknown. Under this situation, we implemented a modified SEIR compartmental model accounting for infection from undiagnosed individuals and for different levels of population isolation, to evaluate the effects of contacts reduction in the epidemic temporal dynamics. In this manuscript, we present a simple model that allows us to evaluate confinement strategies for the initial stages of the pandemic. The scenarios are counterfactual since, in reality, the confinement strategies carried out by each community and by each country were very heterogeneous, but the results obtained suggest that the model accurately adjusts to reality and that the projections are reliable. Among the advantages of the implemented model, it should be noted that despite the simplicity of the hypotheses, the adjustments obtained were accurate and the projections made do not differ much from those other more complex models. Also, the instantaneous increment of cumulative diagnosed people depends on the history of cumulative infected people, by which the latent period can be taken into consideration. The results of the implemented control scenarios for March 23 show that drastic isolation of the susceptible population should be implemented as soon as possible. Even not so drastic increases in alpha (two or three times the current rate) imply also significant reductions in the incidence of cases. The adjustments made in different CCAA also serve to verify the efficacy of the isolation hypothesis for the most affected communities (Madrid and Catalunya). They also serve as a basis for timely action in those communities that do not yet have a significant number of cases that jeopardize their health systems. Policymakers should weigh in the values and ethical considerations of employing now maximum strength in actions to help reduce the slope of the epidemic curve against the enormous associated economic cost. However, our study indicates that a three-week interruption of labor activities, thereby a drastic reduction in contacts, could end the current epidemic in around two months and drastically reduce both the burden of this disease and much lower the toll of lives. Our results could also provide useful suggestions for the prevention and control of the COVID-19 outbreaks in different countries and locations such as Argentina and the USA lagging behind the current epidemic wave in Spain and Italy. These represent two very different scenarios. In the case of the USA, no radical control measure was initially implemented by the federal government and the epidemic seems to be in a phase of uncontrolled growth. On the Argentina side, the health authorities seem to have taken note of what is happening in Europe and more strict movement restrictions have been implemented, although it is difficult to determine the degree of commitment by the population. However, our model does not consider space explicitly as we approached aggregated data. This lack of spatial granularity may affect the accuracy of simulations when the spread of infections in a territory reaches and takes off in a populated city. The former might alter results-producing more new cases than expected, exactly what the aggregation for CCAA data clearly shows. Spatial modeling should also be incorporated explicitly and an extension of this modeling approach to incorporate the movement of individuals should follow. Results should be interpreted with care as projections at these initial stages of the epidemic are very dependent on the quality of data, with small changes in observed values producing large variations in trends. However, even with this limitation in mind, the magnitude of positive increments in cases suggests these results are strong. Variable isolation strength measures can be tested with this model and inform governments of the most probable effects of their actions on the initial course of the COVID-19 disease. In general terms, the model is quite stable, as shown in the uncertainty analysis carried out. On the other hand, it can be seen in the successive updating of the parameters that the initial projections of the proposed model were not too far from the real data reported. This gives a special value to the work since it somehow justifies the previous discussion. Although the scenario is counterfactual and it is not possible to determine whether a more severe increase in the confinement rate would have reduced the number of cases if it can be said that the model did not depart too far from the initial projections. In this appendix, we show an update of the fitted model with the data reported up to March 31 first, and then we fit the model with data reported until May 30. By doing this we can compare the model first predictions we can compare the firsts model's predictions with the actual data reported at the end of the first epidemic wave. Figure 10 shows the fitting and projections of the model with the new data. On the left panel, the results for March 31 are showed while in the right panel the model fitting until May 30 is showed. If we compare these results with those in the previous sections, the model now predicts an epidemic peak much lower than before. Also watching the update to May 30 can be seen that the peak it's a little lower than the first predictions but the result it's acceptable if we have into account the few data points used at the beginning of the fitting (less than 20 data-points). This may be mainly because In Figure 11 and to better understand the scenario of Match 31 is showed in dashed lines in the context of the scenarios previously outlined in Figure 3 , the dynamics of the update until May 30 it's not showed in this figure due to the similarity of the previous dynamic in terms of the peak. The corresponding populations to the parameters indicated above are added as dashed lines for each panel (Reported, Recovered, and Deaths). In the same way than we did for Spain, in this section we show results of the model fitting with the data updated until March 31st and until the end of the first epidemic wave in May 30. In Figure 12 the new fittings are showed and in Table 4 the fitted parameters are listed. Notice the closeness to the epidemic peak in some of the CCAA, a fact that can be real or be due to the yet low prevalence of the disease in parts of Spain. Figure 13 the fittings until May 30, the end of the first epidemic wave, are showed and in Table 5 the fitted parameters for this update are listed. As can be seen, there is a great discrepancy in the reported data for some communities that are certainly not reflected not only in the adjustments at the CCAA level but also at the level of the whole of Spain. This uncertainty in the data makes it difficult to adjust the model, but even taking this aspect into account, the adjustments are quite good and the updated results are not too far from the initial results, especially at the national level. C Updated Results for Italy: Increasing the initial protection rate With the purpose of making an hypothetical evaluation of the impact of a more severe protection rate for Italy we perform several control scenarios since March when the control measures were completely implemented were run where the alpha value was gradually increasing from 3.5% daily up to the post-control scenario assumption of 7.2% daily. Results can be seen in Figure 14 . Also and to compare these early projections with the real data reported at the end of the first wave, we fit the model with data reported until July 4 since on that date the end of the first wave is clear. In figure 15 the results of the adjustment obtained for the first wave of the pandemic in Italy can be observed. As can be seen, the first approximation of the model was quite successful. This can be seen especially in the final size of the peak of reported cases. On the other hand, the adjusted parameters do not vary significantly from those reported in Table3). For July 4 the mean fitted parameters are: α = 0.035, β = 1.10, γ = 0.712, δ = 0.25, λ 0 = 0.023, λ 1 = 0.013, k 0 = 0.020 and k 1 = 0.017. The lessons of the pandemic The global macroeconomic impacts of covid-19: Seven scenarios The spanish influenza pandemic: a lesson from history 100 years after 1918 Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (sars-cov2) Epidemic analysis of covid-19 in china by dynamical modeling Aggregated mobility data could help fight covid-19 A time delay dynamical model for outbreak of 2019-ncov and the parameter identification Estimation of parameters in a structured sir model Mathematical understanding of infectious disease dynamics Studies on mathematical models for sars outbreak prediction and warning A realistic two-strain model for mers-cov infection uncovers the high risk for epidemic propagation Early transmission dynamics in wuhan, china, of novel coronavirus-infected pneumonia Transmission dynamics of the covid-19 outbreak and effectiveness of government interventions: A data-driven analysis Estimations of the coronavirus epidemic dynamics in south korea with the use of sir model Characteristics of and important lessons from the coronavirus disease 2019 (covid-19) outbreak in china: summary of a report of 72 314 cases from the chinese center for disease control and prevention The epidemiology and pathogenesis of coronavirus disease (covid-19) outbreak The dynamics of covid-19 in the uae based on fractional derivative modeling using riesz wavelets simulation Modeling the dynamics of novel coronavirus (2019-ncov) with fractional derivative A fractional epidemiological model for computer viruses pertaining to a new fractional derivative New investigation of bats-hosts-reservoir-people coronavirus model and application to 2019-ncov system Modeling and forecasting of epidemic spreading: The case of covid-19 and beyond Applications of machine learning and artificial intelligence for covid-19 (sars-cov-2) pandemic: A review A sir model assumption for the spread of covid-19 in different communities Mathematical modeling of covid-19 transmission dynamics with a case study of wuhan Generalized seir epidemic model Reportes coronavirus ministerio de sanidad WHO daly covid-19 report Comparison of various robust and efficient load-flow techniques based on runge-kutta formulas A mathematical model for the spatiotemporal epidemic spreading of covid19 We acknowledge support from the Spanish Ministry of Science and Innovation through the "Centro de Excelencia Severo Ochoa 2019 − 2023" Program (CEX2018 − 000806 − S), and support from the Generalitat de Catalunya through the CERCA Program. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.