The Covid-19 disease caused by the novel coronavirus SARS-CoV-2 has rapidly spread out around the world since the first case reported from Wuhan (China) on 31 December 2019.
On 30 January 2020 the World Health Organization (WHO, 2020) declared the outbreak as a Public Health Emergency of International Concern, and on 11 March the WHO declared COVID-19 a pandemic. As of the revision of this manuscript the disease has spread to over 200 countries and territories, and it has affected more than 20 million people with more than 700,000 deaths worldwide (Dong et al., 2020; Johns Hopkins, 2020; Worldometers, 2020) .
In this global context of pandemic, Spain experienced one of the worst situations. With a population of 47 million people, it has one of the highest rate of affected per million inhabitants, and one of the highest number of deaths per capita. Spain registered the first positive on 31 January 2020, from a German tourist in Canary Islands. Throughout February, Spain confirmed multiple cases related to the Italian cluster. On late February, the transmission was classified as local. In mid-March, cases had been registered in all 50 provinces, and Spain together with Italy were the most affected countries in the EU. On 14 March, a state of alarm and nationwide lockdown and quarantine were imposed to control the spread of the virus.
Although Spain was seven days behind Italy in the outbreak, it did not enforced measures until it had more than 7000 cases, the same than Italy the week before. On 3 April Spain surpassed Italy in total cases. After two months of strict containment measures, the epidemic was apparently under control, and from 14 May Spain begin easing measures (CNE, 2020) .
In this article, we study the Covid-19 outbreak in Spain, with a double objective: first, to explain the dynamics of the epidemic with the simplest possible model, and, second, to draw some lessons from a mathematical perspective that may be helpful to better understand the growth of an epidemic and to implement control measures. We use a deterministic SIR model, as well as a SEIR equivalent model, to describe the spread of the disease during the three and a half months from the start of the outbreak to its stabilization in Spain. We advocate the simplicity of the model as an advantage to effectively account for the coronavirus epidemic.
From determination of a single parameter, the growth rate, we successfully simulate the epidemic curve and make predictions. As the method is directly applicable to other countries, we also study the case of Italy and Germany for comparison. After estimating the reproductive number, we discuss the feasibility of mitigation strategies and then evaluate the effectiveness of the suppression measures adopted in Spain. Likewise, we simulate scenarios to explore the potential impact of variations in the effectiveness. Finally, in this work we present simple analytical expressions that allow, by a simple calculation, to estimate the final size of the outbreak and the benefit of early intervention. This approach could help governments nowcasting the behavior of the outbreaks and designing adequate and prompt containment measures.
J o u r n a l P r e -p r o o f 2. Methods
We used a susceptible-infectious-recovered SIR model (Kermack & McKendrick, 1927) to simulate the epidemic: Since the outbreak progression remains under the early growth dynamics, the approximation ( ) S t N ≈ holds and the growth is purely exponential:
where 0 I is the initial infected population, and λ is the Malthusian coefficient for the exponential growth rate, defined as the solution to the Euler-Lotka equation (Britton & Tomba, 2019) :
From Eqs.
(1-2), the removed population is obtained by integration (Appendix A):
We also used a susceptible-exposed-infectious-recovered SEIR model, with governing
where S , E , I , and R were the number of susceptible, exposed or latent (infected but not yet infectious), infectious, and removed individuals at time t, τ i was the infection period, and τ l the latent period.
The solution of the differential equations yields (Appendix B) J o u r n a l P r e -p r o o f 4 ( )
and the subindex 1 corresponds to the positive sign in the solution. The growth rate is 1 λ λ = .
Therefore, the early SEIR dynamics is equivalent to the exponential growth obtained with the SIR model choosing properly the parameters:
Definitions of epidemiological concepts in connection with mathematical well-defined parameters are often not obvious (Svensson, 2007) . Fig. (1) shows the definitions of some epidemic metrics as well as a graphical interpretation. Although the incubation period (time until symptoms) is usually assumed to be similar to the latent period, the novel coronavirus is believed to be infectious during incubation before onset of symptoms Nishiura et al., 2020) , so the serial interval is close to the incubation period. Recent studies (see review in table I) showed that the serial interval of COVID-19 ranges between 4 and 8 days, with average of 6 days, and the incubation period is between 2 and 14 days, with average of 5.5 days. 
In SEIR models the latent period (τ l ) and the infectious period (τ i ) appears as fitting parameters, or as constants related with real values that are difficult to measure. In SIR models the infection period may be interpreted as the generation time ( G T ) (Britton & Tomba, 2019; Svensson, 2007) .
Typically, what we observe is not the infection transmission but the clinical onsets (symptoms), i.e. the serial interval ( S T ). Although the relationship between G T and S T is model-dependent, both have the same mean (Britton & Tomba, 2019) . For this reason, we used the serial time as a proxy for the generation time (Fine, 2003) :
J o u r n a l P r e -p r o o f and we took the average value of 6 days for the serial interval found in the literature review.
Growth parameters were inferred from case-incidence reports (Lipsitch et al., 2003; Wallinga & Lipsitch, 2007; Dimitrov & Meyers, 2020) . We fixed the removal rate
interpreting τ as the mean generation time, or serial interval, for COVID-19 (τ = 6 days, see Section 2.3). From Eqs.
(1-2) the new daily cases reported are
We linearized this expression to get
Then, the exponential growth rate λ is obtained from Eq. In the case of Italy and Germany the source of information was the Italian Ministero della Salute (MSI, 2020), the Robert Koch Institut in Germany (RKI, 2020), and the WHO (2020).
We simulated the outbreak trend in Spain after the government-issued containment measures, assuming that the transmissibility (rate a) was reduced by 60%, 65%, 70%, 75%, and 80% after lockdown. Since no intensive testing nor contact tracing were implemented in Spain, we kept the removal rate fixed (b = 1/6 days -1 ). These simulated scenarios correspond to reproductive numbers from 1 to 0.5. Also long-term mitigation scenarios were simulated with r above 1.
The reproductive number after the intervention is
where E is the effectiveness of the measures. Only when r falls below 1 (negative growth rate) the epidemic would fade out. Thus, the minimum value of the effectiveness to control the epidemic is J o u r n a l P r e -p r o o f min 0
It can be shown (Appendix C) that the total affected population at the end of the epidemic is
where a R is the number of affected people when the containment measures take effect, and a r the reproductive number at that time (Fig. 2) . We also demonstrate in the Appendix C that the total affected population reduces by the factor ( )
if the intervention is performed t ∆ days before, where a λ is the growth rate at the moment of the intervention.
Assuming daily time increments, the instantaneous growth rate was calculated as
Time-varying estimates of the growth rate and the reproductive number were made by computing the weighted moving average (WMA) with a 6-day sliding window (Abbott et al., 2020) . Rates obtained from this nowcasting were used to forecast the epidemic curve with the SIR model. 
Results were expressed with the estimate of the parameter plus/minus the standard error.
A confidence interval (CI) of 95% was also reported. The coefficient of determination, 2 R , was calculated in lineal regressions.
The goodness-of-fit of the model was evaluated by means of two different statistics:
The root-mean-squared-error, as a measure of the residuals, defined as
are the real total cases and the total cases predicted with the model, respectively.
The coefficient of determination, defined as
where R is the mean value of the total cases. This metric measures the proportion of the variance in the observations that is predictable from the model.
Solving of differential equations, data fitting and parameters calculations were performed in MATLAB software, v 6.5 (The MathWorks, Inc., Natick, MA, USA), by using the statistical toolbox and custom-written routines.
The baseline scenario lasted from the outbreak onset (23 February For the equivalent SEIR model two different sets of parameters were inferred from Eq.
(7) for the latent and the infectious periods: In order to choose between models, we used the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) to score the SIR and the SEIR models when applied to the fitting of the total case series (see Appendix E for more details). The SEIR model fits the observations only slightly better than the SIR model, and it is more complex since has one parameter more. Both criteria resulted higher in the SEIR model than in the SIR: 
J o u r n a l P r e -p r o o f 4%, respectively. These numbers are still very large because of the high case-fatality of the disease (above 10% in Spain). Fig. 7 . Simulation of four mitigation scenarios. The curves show the percentage of the affected population for decreasing reproductive numbers starting from 1.8 (value at the stage of the intervention) to 1. In these scenarios, spread slows and population immunity builds up through the epidemic.
Simulations of suppression scenarios with reproductive numbers below 1 were made on 
When containment measures started to show effect, the number of confirmed cases was 57 000 a R , and the reproductive number 1.8 a r = . By using Eq. (15), we can predict the final number of cases at the end of the epidemic. Table II shows the results for the different scenarios after the intervention. The expected number in Spain is approximately 300000. Results are the same as those obtained with the SIR simulation (Fig. 4) . With this simple calculation, the projection in the long term is estimated. Even if the trend of the epidemic is not known a priori, these calculations anticipate what the variation range of the epidemic curve would be.
Because the dynamics is nonlinear, a small variation in effectiveness may produce a large difference in growth (Fig. 8) depending on the value of the reproductive number. Above 80% effectiveness (r below 0.5), the decay in total cases is very small; however, for r between 0.75 and 1, the final number of cases changes significantly. Due to the early exponential growth, a R increases as the intervention is delayed. Therefore, anticipation of control measures suppresses to a much greater degree. Fig. (9) shows simulations considering that measures in Spain had been enforced four or seven days before.
If the intervention is advanced t ∆ days, the final cases reduce by the factor ( ) t ∆ = 4 and 7 days, the reduction of cases is 57% and 38%, respectively (as seen in Fig. 9 ). ) if the intervention were 7.7 days before.
Or, in the actual situation of Spain, the expected number of cases (297000) would be the same having acted 7 days before with much less restrictive measures that drop r only to 0.94. 
The same methodology was applied to the outbreak in Italy and Germany, and also in different regions in Spain. The confirmed cases and the modelled curves are plotted in Figs. (11) and (12).
Italy implemented the government measures on March 8. The incidence peak occurred 13 days later. The average reproductive number before intervention was 0 2. 
We have proposed a simple and effective method to study the dynamics of the SARS-CoV-2 outbreak, based on the exponential growth and a SIR model. There is a plethora of epidemic models with different advantages and limitations: deterministic compartmental models, logistic and Gompertz models, likelihood-based methods, and stochastic simulations.
Despite giving an overly simplified representation, deterministic models can describe the mechanisms in more detail (Ma, 2020) . Among them the SEIR model is the most widely adopted for characterizing the COVID-19 epidemic Tang et al., 2020; Kucharski et al., 2020; Fang et al, 2020) . We have shown that the SIR and the SEIR models are equivalent by choosing the adequate parameters. A drawback to some models is they involve too many parameters (Ma, 2020) . This can lead to an overfitting problem, when the model closely fits a data set but have poor predictive performance in other situations. Our SIR model is in good agreement with the epidemic in Spain as well as in other countries (Italy and Germany), and it has the advantage of being simple and including a single parameter.
We found two distinct phases in the one-month baseline scenario. The epidemic grew faster over the first two weeks ( Mitigation strategies may reduce the final size of the epidemic to between 70% and 4%, with the best case corresponding to reproductive number of 1 (60% transmissibility reduction), as also noted by Wu et al. (2020) . In view of the high mortality rate of the disease, 6.8%
worldwide and above 10% in Spain, it is unrealistic to wait for a major immunisation of the population and the "herd immunity" becomes at least controversial. A prevalence study has been conducted in Spain to determine how many people have developed antibodies after virus exposure. The results show that only 5% of Spaniards have been infected (ISCIII, 2020) .
The proposed SIR model allows predictions to be made on the curve evolution and on the epidemic size under different scenarios according to the effectiveness of the suppression measures. These simulations are very useful, since they inform a priori about the benefit that the measures would provide depending on whether they were more or less strict, and also indicate to what extent they can relax without leaving the suppression scenarios. It is important to note that, due to the nonlinearity of the dynamics, a small variation in effectiveness will produce large changes in epidemic growth.
Throughout the suppression period we followed-up the instantaneous decrease rate and the reproductive number. We noticed that there is a delay of between 11 and 14 days from the implementation of the containment measures to the incidence peak, which is in agreement with the known reporting delay of about 14 days between infectiousness onset and confirmation This study applied to Spain, and in less detail to Italy and Germany, is useful to predict the trend of Sars-Cov-2 epidemic and provide a quantitative guide for other countries at high risk of outbreak. In addition to the epidemic curve modelling and predictions with the SIR or SEIR models, we have proposed two analytical formulas (Eqs. 15-16) that allows us to estimate with a simple calculation (without computing the dynamic equations) the final size of the epidemic in the predicted or other scenarios, and also the potential benefit that can be reached depending on when the control measures start. This provides a theoretical framework for the decision-making of epidemic interventions and prevention.
This research is supported by a grant (project COV20/00736) from the Carlos III Health
Institute of the Government of Spain, in the framework of the Covid-19 Fund.
In the early growth dynamics S N ≈ and, thus, the equation for infectious Both populations grow exponentially but with lag δ :
Appendix B. Exponential growth in the SEIR model 
In the early growth dynamics S N ≈ , and the equation for exposed is
The solution of Eqs. The infectious group in the SIR model is equivalent to the sum of the infectious and 
The differences between the epidemic curves of the SEIR and the SIR models were measured with the root-mean-squared-error:
( ) Fig. (13) plots the RMSE as a function of the period of time taken. In the early exponential growth phase, corresponding to the scenarios studied, the difference between models is practically zero (RMSE < 0.01%. Up to the incidence peak (with no-intervention), the RMSE is 0.2%. In the phase of decreasing incidence, the RMSE presents a maximum value of 5%, and afterwards the two epidemic curves converge again (see Fig. 6 ).
In order to compare the relative quality of the models, the Akaike Information Criterion and the Bayesian Information Criterion were used in this paper. These estimators pose the trade-off between the goodness of the fit to the observations and the simplicity of the model. Both criteria include a penalty that is an increasing function of the number of estimated parameters, and that deals with complexity and the risk of overfitting.
The Akaike Information Criterion is defined as (Akaike, 1974) 2 2ln AIC k L = − (E.1) and the Bayesian Information Criterion is defined as (Schwarz, 1978) ln 2 ln
where k is the number of parameters estimated by the model, n the number of observations, and L the maximum likelihood.
Given a set of models, the preferred model is the one with the minimum AIC or BIC value.