COVID-19: Estimating Spread in Spain Solving an Inverse Problem with a Probabilistic Model

COVID-19: Estimating spread in Spain solving an inverse problem with a probabilistic model

Marcos Matabuena1,4, Carlos Meijide-Garc´ıa3, Pablo Rodr´ıguez-Mier2, and V´ıctorLebor´an1

1CiTIUS (Centro Singular de Investigaci´onen Tecnolox´ıas Intelixentes), Universidade de Santiago of Compostela, Santiago de Compostela, Spain 2Toxalim (Research Centre in Food Toxicology), Universit´ede Toulouse, INRAE, ENVT, INP-Purpan, UPS, Toulouse, France 3Universidade de Santiago of Compostela, Santiago de Compostela, Spain [email protected]

May 5, 2020

Abstract

We introduce a new probabilistic model to estimate the real spread of the novel SARS-CoV-2 virus along regions or countries. Our model simulates the behavior of each individual in a population according to a probabilistic model through an inverse problem; we estimate the real number of recovered and infected people using mortality records. In addition, the model is dynamic in the sense that it takes into account the policy measures introduced when we solve the inverse problem. The results obtained in Spain have particular practical relevance: the number of infected individuals can be 17 times higher than the data provided by the Spanish government on April 26 th in the worst case scenario. Assuming that the number of fatalities reflected in the statistics is correct, 9.8 percent of the population may be contaminated or have already been recovered from the virus in Madrid, one of the most affected regions in Spain. However, if we assume that the number of fatalities is twice as high as the official numbers, the number of infections could have reached 19.5%. In Galicia, one of the regions arXiv:2004.13695v2 [q-bio.PE] 3 May 2020 where the effect has been the least, the number of infections does not reach 2.5 %. Based on our findings, we can: i) estimate the risk of a new outbreak before Autumn if we lift the quarantine; ii) may know the degree of immunization of the population in each region; and iii) forecast or simulate the effect of the policies to be introduced in the future based on the number of infected or recovered individuals in the population.

1 1 Introduction

The spread of the Coronavirus is generating an unprecedented crisis worldwide. In Europe, for example, it is considered to be the most significant challenge that the continent has faced since World War II. In the light of this situation of emergency, the governments must work quickly to avoid the collapse of the healthcare system, reduce the mortality associated with the virus, and avoid the possible effects of an economic recession [1, 2]. Given the strong capacity of the virus to spread and the lack of preventive measures, many countries have been systematically forced to lockdown the population temporarily. Although these policies may be useful in controlling the spread of the virus in the immediate future, they are unsustainable over time from an economic point of view. In this regard, forecasting the evolution and consequences of the pandemic based on the exposure of the population becomes a critical factor in decision-making [3, 4]. However, to rigorously predict these effects, it is necessary to assess the current spread of the epidemic, which is often unknown. At the beginning of the XX century, the first mathematical models to study the dynamics of an epidemic were introduced. Probably the most well-known is the susceptible-infected-recovered model (SIR). SIR models and its variations [5, 6] divide the population into compartments, and using differential (deterministic) equations, the number of individuals in each of the compartments over time is estimated. Since then, many new variations of these models have been introduced in the literature (see for review [7, 8, 9, 10] or more contemporary examples [11, 12]). However, those approaches suffer from a critical shortcoming. In essence, they focus on explaining the dynamics of the epidemic at the population level and exclude the complex interactions that occur at the microscopic level between individuals [13, 14, 15]. Furthermore, these models tend to be adjusted with fixed parameters that are not updated over time, according to how the epidemic evolves and the different policies introduced. In the current era of precision medicine, where more efficient solutions are being sought that optimize the health of the individual [16], it is surprising not to find in the current state of the art models of epidemic spread that follow more individual approaches [17, 18], as is rightly pointed out in [19]. Precisely in [19], the authors propose to adopt new models that exploit the enormous individual information recorded by biosensors [20] and other devices. A crucial aspect to consider in the success of mathematical modeling is the need to adjust the parameters of the models with accurate data. However, this is difficult to achieve in practice, because governments are often overloaded, and information is only recorded on patients with symptoms. The observational nature of this data [21], together with a delay or errors in the recording of information [22], requires specific techniques to correct the biases related to the data [23].

1.1 Main contributions and results Motivated by the need to developed new models that overcome these drawbacks and that estimate the expansion of the SARS-CoV-2 in a realistic way in Spain,

2 we introduce a new probabilistic model. The contributions and most signiﬁcant results of this paper are introduced below.

• To the best of our knowledge, we propose one of the few probabilistic models that estimate the evolution of an epidemic from each individual of the population. This model combines the ﬁnite Markov model, along with continuous probability distributions and a dynamic Poisson model. • The model is designed in such a way that it does not need to use statistics on the number of infections that have a signiﬁcant measurement error, and uses mortality records instead.

• The model estimates for each day the number of susceptible, contaminated, recovered, and fatalities. For this, we solve an inverse problem, and we introduce biological expert knowledge related to the SARS-CoV-2 disease into the probabilistic model about the time each individual spends in the diﬀerent states and the transition of probabilities. • Using the data from several regions of Spain, we estimate the real number of infections as of April 26st. The results show a higher proportion of infections than that provided by the Spanish government. • The eﬀect of the Coronavirus has been uneven among regions of Spain. In Madrid, more than 10 percent of the population was infected, or it recovers from the virus, while in Galicia, this percentage is less than 2.5.

• These results highlight differences between regions; i) the degree of immunity of the population can be different; ii) the number of recovered people who can return to normality is also different; iii) based on this analysis, a reasonable strategy to be considered could be to lift the quarantine on a case-by-case basis based on the degree of exposure of each region to the virus. • Finally, we repeat the estimations assuming that the number of fatalities is twice than reported by the official records. In this case, the number of infections can increases between 40 y 110 percent by region. However, we must interpret these results with caution.

1.2 Outline The mathematical details of the model are relegated to the end of the article, to facilitate the readability for a general target audience. The structure of the article is as follows: First, we describe the evolution of Coronavirus in Spain, together with some demographic and economic characteristics of the Spanish population. Subsequently, we ﬁt our model in diﬀerent scenarios, and we show the evolution of susceptible patients, infected and recovered ones over time. Next, we discuss the results and limitations of the model. Finally, we introduce the mathematical content of our model and the computational details of the implementation and parameter optimization.

3 2 COVID-19 in Spain

Spain was one of the ﬁrst countries in the world to experience the eﬀects of COVID-19 after China and Italy. However, despite the delay in the start of the outbreak, with respect to these countries, the consequences are now more dramatic. To give a better context to the evolution of Coronavirus in Spain and to compare it with other countries, we introduce some historical background:

• January 31st. The first positive result was confirmed on Spanish territory in La Gomera. At that time, there were around 10, 000 confirmed cases worldwide. • February 12th. The Mobile World Congress, one of the most remarkable technological congresses in the world to be held in Barcelona, was canceled.

• March 8th. Multitudinous marches were celebrated in Spain. Also, sports competitions and other events were held as usual. • March 13th. Madrid reported 500 new cases of Cov-19 in one day (64 deaths total). Wuhan had gone into lockdown with 400 new cases per day (17 deaths total).

• March 14th. With the increase in the outbreak of infections, the government declared a quarantine throughout the country. • March 21st. Due to an overloaded health system, the ﬁrst patients started to arrive at new makeshift hospitals.

• April 3rd. Spain accounts for a total of 117, 710 conﬁrmed cases, surpass- ing Italy for the ﬁrst time. • April 6th. Spain becomes the country in the world with more deaths per million inhabitants.

• April 9th. The FMI forecasts that 170 countries are going to be into recession this year in the worst crisis since the Great Depression. • April 18th. The Spanish government changes protocols for the daily statistics of COVID-19. • April 21th. The president of the United States, Donald Trump, declared: ”It’s incredible what happened to Spain, it’s been shattered.”.

Figure 1, shows accumulated number of cases and deceases respectively in the previous periods in Spain, Italy, China, United Kingdom and the United States according to the data supplied by the diﬀerent governments. Following the statistics of the Population Reference Bureau, Spain is the 20th country with the world’s oldest population [26]. The country demographic structure, poverty rates, and epidemiological proﬁles is essential to compare mortality between countries. In the Coronavirus disease, relative and absolute case-fatality risk (CFR) [27] increases dramatically with age and with comor- bidity, as evidenced by the current literature. Relative risk can increase by more than 900% in patients over 60 [28].

4 Cumulative number of cases worldwide time−series Cumulative number of deaths worldwide time−series ● ● ● ● ● ● ● 50000 ● ● ● ● ● ● ● 750000 ● ● ● ● 40000 ● ● ● ● ● ● ● ● ● 30000 500000 ● ● ● ● ● ●● ●● ● ●● ● ●● ● ● ● ●● ● ●● ●● ● ● ●● ● ● ● ●●● ● 20000 ●● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● 250000 ● ● ●●● ● ● ● ● ● ●●● ● ● ● ●● ●●● ● ● ● ●●● ●●●● ● ● ● ● ●●●●●●● ● ● ● ● ●●●●●● 10000 ● ● ● ●●●● ●●● ● ● ● ● ●●● ●● ● ● ● ● ●● ●●● ● ● ●●● ● ● ● ● ●●● ●●● ● ● ● ● ●●●● ●● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●●●●●●●●●● ●●●●●●●● ●● ●● ●● ● ● ● ● ●● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●● ●●● ●● ●●●●●●●●●●●● ●● ● ●● ● ●●●● ●● ●●● ●●● ●●●●●●● ● ● ● ● ●●● ●●● ●● ●●● ●●●● ●●● ●● ● ●●● ●●● ●●●● ●●●● ●●●● ●●●●●●● ●●● ●●● ●●●●● 0 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 0 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● feb. mar. abr. feb. mar. abr. Date Date

● China ● Italy ● Spain ● United Kingdom ● US ● China ● Italy ● Spain ● United Kingdom ● US

Figure 1: Spread and number of deaths of Coronavirus in Spain, Italy, China, and the United States. Number of accumulated infected patients (left) and the number of accumulated deaths (right) [24].

Galicia Pa´ısVasco Castilla y León Cataluña Madrid España Population 2, 698, 763 2, 181, 916 2, 553, 301 7, 609, 497 6, 685, 470 47, 100, 396 At-risk-of-poverty rate 18.8 8.6 16.1 13.6 16.1 21.5 Population density 91.28 305.19 25.47 239.01 830.02 93.08 Percentage of population by age group 0 − 9 7.50 8.93 7.51 9.78 9.92 9.28 9 − 18 7.56 8.78 7.79 9.74 9.44 9.37 18 − 30 10.39 10.66 10.67 12.72 12.80 12.42 30 − 45 21.18 20.28 19.47 22.24 23.24 22.07 45 − 60 22.77 23.28 23.60 21.77 22.23 22.61 60 − 80 22.60 21.41 22.30 18.24 17.33 18.70 from 81 on 8.01 6.67 8.65 5.50 5.03 5.56

Table 1: Demographic and socioeconomics characteristics of the Spanish population throughout some regions: Galicia, Pa´ısVasco, Castilla y Le´on,Catalu˜na, Madrid [25].

Subsequently, we perform a descriptive analysis in the regions of Spain that we analyze in this paper: Galicia, Pa´ısVasco, Castilla y León,Madrid, Cataluña. Table 1 contains the essential demographic and socioeconomic characteristics of these regions. We can see that Castilla y Leónis the region with the highest proportion of elderly people. At the same time, Castilla y Leónhas the most delocalized population centers, and the Pa´ısVasco is the region with the low- est poverty rate. Spain is a multicultural country where there are significant economic, geographical, social, and demographic differences throughout the regions. All these peculiarities make Spain an interesting country to extrapolate the effects of the spread of the Coronavirus to other regions and countries. Finally, in Figure 2, we show the evolution of infections and fatalities among the regions under consideration. As we can see, Madrid is the most affected region, while Galicia is the least affected, despite its older population. However, it is important to note that the outbreak began later, and the containment was

5 Cumulative number of deaths by day in Spain time−series Cumulative number of cases by day in Spain time−series ● ● ●● ●● ●● ● ● ● ● 60000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● 6000 ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 40000 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 4000 ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 20000 ● ● ● ● ●● ●● ● ● ● ● ● ●●● 2000 ● ●● ● ● ● ● ● ●●● ● ● ● ●● ●●●● ● ●● ●●●● ● ●●● ● ● ●●● ● ●●● ● ●● ●●● ●●● ● ● ●● ● ● ●● ●●● ● ● ●●● ●● ●●● ● ●●● ●●●● ● ● ●●● ●●● ●●●● ● ●● ●●● ● ● ●●●●●●● ● ●● ●● ●● ●● ●●●● ● ● ●●● ● ●● ●●●● ● ● ●● ● ●●● ●● ● ●● ●● ● ● ●● ●● ● ● ● ●●● ● ● ●● ●● ● ●● ●●● ●●●●●● ●● ● ●● ● ● ● ● ●● ●●●●●●●●●● ● ●● ●● ●● ●● ●●●● ●●●●●● ● ● ●●●●●●● ●●●●● ●●●●● ●●● ●●●●●●●●● 0 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 0 ●●●●●●●●●●●●●●●●●●●●●●● mar 01 mar 15 abr 01 abr 15 mar 01 mar 15 abr 01 abr 15 Date Date

● Castilla y León ● Cataluña ● Galicia ● Madrid ● País Vasco ● Castilla y León ● Cataluña ● Galicia ● Madrid ● País Vasco

Figure 2: Evolution of accumulated infected (left) and death patients (right) in Galicia, Pa´ıs Vasco, Castilla y Le´on,Madrid, Catalu˜na carried out earlier than in Madrid.

3 Results

Next, we show the estimation performed with the model until April 26st in Galicia, Pa´ısVasco, Castilla y Le´on,Madrid, Catalu˜na,assuming that these two scenarios hold:

1. We assume that the number of real deaths due to Coronavirus is the one reflected by the official records. 2. We assume that many people have died of Coronavirus, but they have not been included in the records because a diagnostic test was not performed. In particular, we shall suppose that the number of deaths is twice as high as those indicated in the official records each day.

We graphically represent along time the result of estimating how many individuals belong to each of the following states:

• I1(t): Number of infected individuals who are incubating the virus on day t.

• I2(t): Number of infected people who have passed the incubation period and do not show symptoms on day t.

• I3(t): Number of infected people who have passed the incubation period and do show symptoms on day t.

• R1(t): Number of recovered cases which are still able to infect on day t.

• R2(t). Number of recovered cases which are not able to infect anymore on day t.

6 • M(t): Number of deaths on day t.

In addition, to understand the meaning of the results, we show: i) the number of people who may be contaminated or who have already transmitted the virus as a percentage of the population size; and ii) the rate of new infections each day. Finally, we introduce conﬁdence bands of our estimations 1.

1Our conﬁdence bands shall not be confused with classic frequentist statistics conﬁdence bands

7 8 3.1 Scenario 1 Galicia

I1 (Infected, incubation) I2 (Infected, asymptomatic)

25000 25000

20000 20000

15000 15000

10000 10000

5000 5000

0 0 02 09 16 23 30 06 13 20 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 I3 (Infected, symptomatic) 2020 R1 (Recovered, still infectious)

25000 25000

20000 20000

15000 15000

10000 10000

5000 5000

0 0 02 09 16 23 30 06 13 20 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

R2 (Recovered) Model fit vs Observed fatalities

Observed fatalities 400 35000

30000

300 25000

20000 200

15000

10000 100

5000

0 0

02 09 16 23 30 06 13 20 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020 % of the population infected Daily rate of infections ( t)

1.4 3000

1.2 2500

1.0 2000

0.8 1500 0.6

1000 0.4

500 0.2

0.0 0

02 09 16 23 30 06 13 20 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

Figure 3: Results in Galicia: we assume9 that there are as many fatalities as in the oﬃcial records. Castilla y Le´on

I1 (Infected, incubation) I2 (Infected, asymptomatic) 100000 100000

80000 80000

60000 60000

40000 40000

20000 20000

0 0 24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 I3 (Infected, symptomatic) 2020 R1 (Recovered, still infectious) 100000 100000

80000 80000

60000 60000

40000 40000

20000 20000

0 0 24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

R2 (Recovered) Model fit vs Observed fatalities

1750 Observed fatalities

140000 1500 120000 1250 100000

1000 80000

750 60000

500 40000

20000 250

0 0

24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020 % of the population infected Daily rate of infections ( t)

6 10000

5 8000

6000

4000 2

2000 1

0 0

24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

Figure 4: Results in Castilla y Le´on:we assume that there are as many fatalities as in the oﬃcial records. 10 Pa´ısVasco

I1 (Infected, incubation) I2 (Infected, asymptomatic)

50000 50000

40000 40000

30000 30000

20000 20000

10000 10000

0 0 17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 I3 (Infected, symptomatic) 2020 R1 (Recovered, still infectious)

50000 50000

40000 40000

30000 30000

20000 20000

10000 10000

0 0 17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

R2 (Recovered) Model fit vs Observed fatalities

Observed fatalities 1200 100000

1000 80000

800

60000

600

40000 400

20000 200

0 0

17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020 % of the population infected Daily rate of infections ( t) 5000 5

4000 4

3000 3

2000 2

1 1000

0 0

17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

Figure 5: Results in Pa´ısVasco: we assume that there are as many fatalities as in the oﬃcial records. 11 Madrid

I1 (Infected, incubation) I2 (Infected, asymptomatic)

500000 500000

400000 400000

300000 300000

200000 200000

100000 100000

0 0 17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 I3 (Infected, symptomatic) 2020 R1 (Recovered, still infectious)

500000 500000

400000 400000

300000 300000

200000 200000

100000 100000

0 0 17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

R2 (Recovered) Model fit vs Observed fatalities

700000 Observed fatalities 8000

600000

500000 6000

400000

4000 300000

200000 2000

100000

0 0

17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020 % of the population infected Daily rate of infections ( t)

80000 10 70000

8 60000

50000 6 40000

4 30000

20000 2 10000

0 0

17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

Figure 6: Results in Madrid: we assume that there are as many fatalities as in the oﬃcial records. 12 Catalu˜na

I1 (Infected, incubation) I2 (Infected, asymptomatic) 250000 250000

200000 200000

150000 150000

100000 100000

50000 50000

0 0 24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 I3 (Infected, symptomatic) 2020 R1 (Recovered, still infectious) 250000 250000

200000 200000

150000 150000

100000 100000

50000 50000

0 0 24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

R2 (Recovered) Model fit vs Observed fatalities

400000 Observed fatalities

350000 4000

300000

250000 3000

200000

2000 150000

100000 1000

50000

0 0

24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020 % of the population infected Daily rate of infections ( t)

5 25000

4 20000

3 15000

2 10000

1 5000

0 0

24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

Figure 7: Results in Catalu˜na:we assume that there are as many fatalities as in the oﬃcial records. 13 14 3.2 Scenario 2 Galicia

I1 (Infected, incubation) I2 (Infected, asymptomatic) 50000 50000

40000 40000

30000 30000

20000 20000

10000 10000

0 0 02 09 16 23 30 06 13 20 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 I3 (Infected, symptomatic) 2020 R1 (Recovered, still infectious) 50000 50000

40000 40000

30000 30000

20000 20000

10000 10000

0 0 02 09 16 23 30 06 13 20 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

R2 (Recovered) Model fit vs Observed fatalities

Observed fatalities 70000 800

60000

600 50000

40000 400 30000

20000 200

10000

0 0

02 09 16 23 30 06 13 20 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020 % of the population infected Daily rate of infections ( t)

5000 2.5

4000 2.0

3000 1.5

2000 1.0

0.5 1000

0.0 0

02 09 16 23 30 06 13 20 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

15 Figure 8: Results in Galicia: we assume there are twice as many deaths as the oﬃcial records. Castilla y Leon

I1 (Infected, incubation) I2 (Infected, asymptomatic) 200000 200000

175000 175000

150000 150000

125000 125000

100000 100000

75000 75000

50000 50000

25000 25000

0 0 24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 I3 (Infected, symptomatic) 2020 R1 (Recovered, still infectious) 200000 200000

175000 175000

150000 150000

125000 125000

100000 100000

75000 75000

50000 50000

25000 25000

0 0 24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

R2 (Recovered) Model fit vs Observed fatalities

300000 3500 Observed fatalities

3000 250000

2500 200000

2000 150000 1500

100000 1000

50000 500

0 0

24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020 % of the population infected Daily rate of infections ( t) 25000

20000 10

8 15000

6 10000

5000 2

0 0

24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

Figure 9: Results in Castilla y Le´on:we assume there are twice as many deaths as the oﬃcial records. 16 Pa´ısVasco

I1 (Infected, incubation) I2 (Infected, asymptomatic)

100000 100000

80000 80000

60000 60000

40000 40000

20000 20000

0 0 17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 I3 (Infected, symptomatic) 2020 R1 (Recovered, still infectious)

100000 100000

80000 80000

60000 60000

40000 40000

20000 20000

0 0 17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

R2 (Recovered) Model fit vs Observed fatalities

Observed fatalities 2500 200000

2000

150000

1500

100000 1000

50000 500

0 0

17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020 % of the population infected Daily rate of infections ( t) 10000

8000 8

6000 6

4000 4

2 2000

0 0

17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

Figure 10: Results in Pa´ısVasco: we assume there are twice as many deaths as the oﬃcial records. 17 Madrid

I1 (Infected, incubation) I2 (Infected, asymptomatic)

800000 800000

600000 600000

400000 400000

200000 200000

0 0 17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 I3 (Infected, symptomatic) 2020 R1 (Recovered, still infectious)

800000 800000

600000 600000

400000 400000

200000 200000

0 0 17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

R2 (Recovered) Model fit vs Observed fatalities 17500 1400000 Observed fatalities

15000 1200000

12500 1000000

10000 800000

7500 600000

400000 5000

200000 2500

0 0

17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020 % of the population infected Daily rate of infections ( t)

20 100000

80000 15

60000

40000

5 20000

0 0

17 24 02 09 16 23 30 06 13 20 17 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

Figure 11: Results in Madrid: we assume there are twice as many deaths as the oﬃcial statistics. 18 Catalu˜na

I1 (Infected, incubation) I2 (Infected, asymptomatic) 500000 500000

400000 400000

300000 300000

200000 200000

100000 100000

0 0 24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 I3 (Infected, symptomatic) 2020 R1 (Recovered, still infectious) 500000 500000

400000 400000

300000 300000

200000 200000

100000 100000

0 0 24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

R2 (Recovered) Model fit vs Observed fatalities

800000 Observed fatalities

700000 8000

600000

500000 6000

400000

4000 300000

200000 2000

100000

0 0

24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020 % of the population infected Daily rate of infections ( t) 60000 10

50000

40000

6 30000

4 20000

2 10000

0 0

24 02 09 16 23 30 06 13 20 24 02 09 16 23 30 06 13 20

Mar Apr Mar Apr 2020 2020

Figure 12: Results in Catalu˜na:we assume there are twice as many deaths as the oﬃcial statistics. 19 3.3 Analysis of results The most relevant results are described below:

• Madrid is the most affected the region by COVID-19. 19.5 % of the population could have been infected or recovered from the virus if we assume that the number of deaths is double that reported by the Government. At present, there may be 120, 000 patients recovered. • Galicia is the region which suffered the mildest effects. The percentage of infected people is less than 2.5 %.

• Castilla y León,Pais Vasco, Cataluñacould have suffered the effects of COVID-19 with a proportional magnitude. The percentage of infections could be between 6 − 10 % of the population. • The peak of new infections has probably occurred at the start of quarantine, while the peak of people who can contaminate took place between March 17-24. • The number of new infections have been dramatically reduced after the introduction of containment measures. Now, it seems that the situation is back under control.

20 4 Discussion

The Coronavirus pandemic is causing an unprecedented crisis in health, economic, and social terms throughout the world. To help designing better politi- cies in the future, estimating the number of people who were infected by the virus is a crucial priority. To address this issue, we have proposed a new probabilistic model to estimate the spread of an epidemic by simulating the behaviour of each individual in a population.

COVID-19 in Spain The predictions obtained over several regions of Spain have practical relevance. The results reveal that the number of infections is much higher than what is reported by the Spanish government. Two factors may explain these discrepancies: i) The tests were carried out in a limited way among the general population; and ii) no random sampling was done among the different regions throughout the territory shading the real magnitude of the pandemic. It is also worth not- ing that its impact was uneven across regions. Madrid was the most affected region, while Galicia was the least affected. However, according to our model, in Cataluña,Castilla y León,and the Pa´ısVasco, the effects were surprisingly similar, even though the outbreak of infections theoretically started at different time points. In addition, the geographical dispersion of Castilla y Leónis more significant (see Table 1). Perhaps the older population of Castilla y León(see Table 1) has a higher case fatality rate than the other communities, and our model is overestimating the number of infected people in that region. As previ- ously mentioned, taking into account the demographic structure of the regions, together with the epidemiological profiles and other socioeconomic characteristics, is fundamental in the analysis of results.

More deaths than those reported in the official records Even though many patients die due to the virus, this is not the official cause of death. This problem occurs due to a lack of diagnostic tests. Spanish and International press echoed this problem in the current Coronavirus crisis, and more than twice as many deaths than those officially announced have been reported in certain regions. A recent study reported that the actual number of deaths might be even three times higher in Italy [29]. Due to this fact, it is vital to take into account this uncertainty present in the models to perform an accurate estimate of the number of actual infections. However, this uncertainty is not probabilistic. To alleviate this problem, we have assumed that there could be up to twice the number of official deaths. If we consider this correction, the number of infected increases from 40 to 110 percent by region.

Practical considerations of this results These ﬁndings allow us have a better sense of the degree of exposure to the virus of the population and to design a restoration to normality based on the following factors.

• The degree of immunity of each population.

21 • The number of people who might contaminate today. • The number of recovered patients.

In this sense, it is vital to gradually de-escalate the conﬁnement and always proceed according to the individual characteristics of each region. For the moment, there is no available vaccine for use with humans, and there is uncertainty about treatments. Therefore it is advisable to take measures such as social distancing [30] to prevent the infection rate from skyrocketing and to avoid another outbreak.

The notorious peak Wondering about when the peak of infection would have been reached was a hot topic in mass media. However, such peak can be i) the maximum peak of new infections. ii) the maximum peak of people who can contaminate. In the ﬁrst case, our models prove the existence of this peak between 9 − 16 March, while in the second case, it was estimated to be in March 17 − 24. From a perspective of virus containment, the most important is the second one from as it indicates the number of potential transmitters.

Methodological aspects of epidemic modeling From a methodological standpoint, indeed, there are already simulators that work at the individual level [31, 32, 33, 34, 35]. However, from our point of view, the models analyzed have some of the following potential limitations: i) some models use fixed parameters without prior optimization according to the target population; ii) the parameters do not usually evolve dynamically according to the policies introduced; and iii), many of these models were designed without adding biological expert knowledge. More generally we can find two philosophies when addressing epidemic forecasting in the literature (see [18]): i) Mechanistic models: Using an epidemiological approach, these models explain the evolution of an epidemic from the causal mechanisms of transmissionc [13]; ii) Phenomenological models: From entirely data-driven approaches such as regression models or time series, the development of epidemics is predicted using historical data [18]. Phenomenological models can be an appropriate solution in the case of epidemics such as influenza [36], for which there exists long-term data collected on hospitalization and mortality rates. Additionally, the etiology, together with the previous incidence rates, are better characterized. With all this, extrapolations of the real impact of the epidemic in the whole population can be done from data-driven approaches. In addition, current evidence indicates that influenza incubation times are shorter than those of COVID-19, which reduces an individual’s potential exposure to the virus. For all these reasons, we believe that in the case of the new COVID-19, it is necessary to use Mechanistic models as our proposal. Often, defining a clear boundary between Mechanistic models and Phe- nomenological models may not be simple, and even both methods can be seen as complementary approaches. For example, SIR models and their variants from a theoretical point of view try to explain the effects of an epidemic from the mechanisms of transmission, however they are often used as time series [37].

22 At this point, we would like to point out that the underpinning assumptions of these models are often unrealistic [5]. Moreover, from the perspective of Epistemology and Epidemiology, the epidemic spread is based on the complex interactions between individuals, which these models cannot capture due to their population-based nature [18]. The use of individual models should be an aspiration in the development of new methods. In the current era of big data, in developed countries, a substan- tial part of the population is geolocalized through mobile devices. Moreover, it is increasingly common to use biosensors that monitor patient health [20]. Knowledge about interactions between individuals and patient health can provide a more realistic picture of the spread and evolution of the epidemic [38]. However, in this context, it is necessary to systematically apply techniques that correct the biases that appear in the observational nature of these data. At the same time, it is also important to preserve the privacy of the citizens. The absence of reliable information is one of the main problems with the COVID-19 pandemic. To alleviate this problem, when ﬁtting model parameters, it is important to understand the social, economic, and demographic characteristics along with the epidemiological proﬁles of the region to introduce expert knowledge into the model and reduce uncertainty in statistical learning. Also, with this strategy, we reduce the computational demands and we increase model interpretability.

Collaborative attitude Information is one of the most valuable weapons in combating a pandemic of this magnitude. Understanding the mechanisms of virus transmission [39, 40], the prevalence of asymptomatic patients in the population [41], risk and prognostic factors [22] are topics that can reduce the number of deaths. Due to the absence of prior knowledge of all these factors, a collaborative approach between countries and institutions to share their knowledge should be a top priority. At an international level, we could highlight the initiative promoted by Eran Segal from Israel, to which several countries have joined [42]. Segal and his team designed a survey in which anonymous volunteers participated to estimate the real evolution of the Coronavirus in Israel [43]. This questionnaire was adapted and applied in other countries. There is now an international consortium in action to share the results of these surveys, together with other clinical information from patients. However, it is important to note that the data obtained from the questionnaires do not represent a random sample of the population, which limits the quality of the data collected.

Expert knowledge in parameter fitting From a general perspective of inverse problems and statistical learning, the estimation of the model parameters is an ill-posed problem [44, 45, 46, 47, 48, 49]. In the case of epidemic spread models, these difficulties are even more significant (see, for example, [50, 51]). In practice, the evolution of several compartments is predicted simultaneously, and usually, there is only information about the actual growth of one of the compartments. In this regard, it is vital to follow Ockham’s Razor rule: the simplest solution is most likely the right. Reducing the number of variables to be fitted in the

23 models and introducing expert knowledge into these models guided by current biological evidence is for ill-posed estimation problems. In the case of COVID-19, the current epidemiological evidence is inconsis- tent. It is especially noteworthy about the case fatality rate (CFR) and the proportion of asymptomatic patients [52]. In the ﬁrst case, variations between 0.4 and 15 percent have been reported in the literature. Meanwhile, with asymptomatic patients, these range from 20 percent to 80 percent. The primary ratio- nale for these discrepancies is the application of non-observational data analysis methods for a sample composed largely of elderly patients. In this sense, after carrying out our estimates taking into account the demographic structure of Spain and using the published articles that we believe reﬂect the reality in a more precise way, we estimate that the rate of asymptomatic patients ranges between 75 and 85 percent. Likewise, CFR may vary between 0.4 and 0.8 percent. A recent article has estimated that asymptomatic patients to be 78 percent [53].

Combining a probabilistic model with a deterministic inverse problem Our probabilistic model is non-stationary and has a complex dependency structure. In addition, the time series of each day’s death records can be seen as a realization of the stochastic process. To fit the model, we estimate a deterministic inverse problem that makes the observed deaths is the average of the random process. From there, we define a window, and we try with a non-probabilistic approach to capture the trajectories of fatalities close to the real one. In this way, we can obtain trajectories of the random process that allow us to explain the real situation of the infected in each region. In other words: From day one, with the chosen configuration of parameters, we can have many different scenarios in our probabilistic model; however, at the present moment, we already observed a concrete trajectory. Then, using this trajectory as a reference, we look for a set of close trajectories that measure other alternative scenarios that approximate the real number of deaths.

Model limitations The main limitation of our proposal is that some of the parameters on which the model depends on are set and may need to be tuned more carefully to the study population. However, the current biological evidence does not allow further reﬁnement. Also, optimizing more parameters with a data-driven approach is dangerous due to the quality of data about infections reported by governments. Therefore, it is better to restrict oneself to current biological expertise information. The models are an approximation of reality, and as Cox would say: all models are wrong, some are useful. We are predicting the unknown in this epidemic without reliable data. However, with statistical simulation models [54, 55], we can see the eﬀect of a deviation of the input parameters, the extreme cases, and the model predictions in other scenarios such as the end of containment policies. Another limitation of our model is that despite simulating data for each individual in the population, we were not using information about the demographic

24 structure and epidemiological proﬁle along regions. We also not introduce ge- olocation information as in the interaction between each individual in the population [38]. However, the inclusion of all these aspects, may not lead to greater accuracy as no reliable model output data are available, and, furthermore, we would be entering into privacy issues that are beyond the scope of this study.

Future work and model extensions Our immediate work is to extend our estimations to other countries. Also, we want to simulate the effect of an immediate end coronavirus lockdown [56] in each of these location. We also want to carry out simulations to find out how to minimize the effects of a possible outbreak in Autumn [4], for example, with social distance [57] or weeks of partitioned work: combining remote working with face-to-face activity. In a complementary way, it would be interesting to predict the effects of these measures on the economy. Some economists claim that the economic consequences of this crisis may be more damaging to our health in the long term than the effects caused directly by the virus [58]. From a methodological point of view, another exciting extension is to de- velop a mixed model or an empirical Bayes framework (see an example of this methodology [59]) to make a simultaneous estimation across many regions and to be able to make a statistical inference about the parameters of the models. In this case, it may be interesting to optimize the parameters with Bayesian optimization [60], to alleviate the computational issues of this approach. An- other variation of the model may be to employ other probability distributions such as the Conway-Maxwell-Poisson distribution [61] to handle underdisper- sion situations in the number of newly infected. However, our current election of Poisson distribution is well motivated given the asymptotic properties of the Erdos-Rényi model [62]. Finally, another possible improvement of the model could be to introduce an ensemble forecasting as they do in meteorology and manage better the possible uncertainty in the initial conditions [63].

Our future prediction in the world Given the current emergency, and the need to provide predictions on the spread of the epidemic worldwide, we have decided to set up a website (https:// github.com/covid19-modeling/forecasts) where we regularly update our estimates across many regions of the world in a fully accessible manner. In addition to this, we will be incorporating our simulations of what the eﬀect of current end coronavirus lockdown would be in each of these regions.

25 5 Mathematical model 5.1 Model elements We suppose that D = {0, 1, ··· , n} is the set of days under study. Consider the following random processes whose domain is deﬁned on D.

• S(t): Number of people susceptible to become infected on day t.

• I1(t): Number of infected individuals who are incubating the virus on day t.

• I2(t): Number of infected people who have passed the incubation period and do not show symptoms on day t.

• I3(t): Number of infected people who have passed the incubation period and do show symptoms on day t.

• R1(t): Number of recovered cases which are still able to infect on day t.

• R2(t). Number of recovered cases which are not able to infect anymore on day t. • M(t): Number of deaths on day t.

Henceforth, we will denote by I(t) = I1(t) + I2(t) + I3(t) the number of infected people at time t ∈ D and R(t) = R1(t)+R2(t) the number of recovered people. The above random processes describe the evolution of population in diﬀer- ent compartments. However, unlike the classics models [6, 10], we divided the infected and recovered individuals in a more broader and realistic taxonomy for the particular case of the COVID-19. There are two main reasons for this: First, the patients tested by healthcare are usually those found in I3. In this case, there exists a corpus of prior knowledge about how they evolve over time and in case of death, which is their survival time. Second, there is evidence that there are recovered patients who can still infect others.

5.2 Model deﬁnition The causal mechanism of new infected individuals (see Figure 13) is introduced below. For each day t ∈ D, we assume that the new infected population I1(t) − I1(t−1), is generated by the individual interaction of the susceptible people with infected patients and the recovered cases who can still contaminate. Formally, we assume that if a person can contaminate, it does so according to a random variable X ∼ Poisson(Ri(t)), being Ri(t) the average number of new infections that can cause each person in the day t. It is important to remark, that it is natural to assume that the function Ri(t) follows a decreasing trend over time in our setting, basically for two reasons: i) quarantine policies have been systematically introduced along with diﬀerent countries and regions. ii) the number of susceptible people decreases over time, while the number of infected people can increase. This all makes it more complicated to interact with a non-infected people.

26 M R2

I3 1 − β R1

S I1 1 − α I2

Figure 13: Diagram of state changes in our model.

Once a new infected person arrives to the model (see Figure 13), we assume that the transitions between the diﬀerent graph states are modeled by a probability law that veriﬁes the following conditions: i) the transition probabilities are independent of the absolute instant when such transition takes place ii) the probabilities depend only on the current state of the patient regardless of the previous path in the graph. In particular, given the States = {I1,I2,I3,R1,R2,M}, and α, β ∈ [0, 1], we have: P(I2|I1) = α, P(I3|I1) = 1 − α, P(M|I3) = β, P(R1|I3) = 1 − β, P(I3|I2) = 1, P(R2|R1) = 1; all other transitions take a value equal to zero in probability. More schematically, the transition matrix between events is shown in the Equation 1.

 I1 I2 I3 R1 M R2   I1 0 1 − α α 0 0 0     I2 0 0 0 1 0 0    P =  I3 0 0 0 1 − β β 0  . (1)   R1 0 0 0 0 0 1     M 0 0 0 0 1 0  R2 0 0 0 0 0 1 Additionally, the Table 2 shows the random variables that model the time between transitions together with the references that have been used. In particular cases, as survival time, we have made our estimates based on data from some of these researchers and incorporating some expert knowledge of others.

Transition Random variable Used references I1 → I2 Gamma(5.807, 0.948) [64, 65] I1 → I3 Gamma(5.807, 0.948) [64, 65] I2 → R1 Uniform(5, 10) I3 → R1 Uniform(9, 14) [65] I3 → M Gamma(6.67, 2.55) [66, 67, 65, 68] R1 → R2 Uniform(7, 14) [69, 70]

Table 2: Random variables of the time of each transition

Finally, in a similar way, in Table 3, we show the values used to model the

27 transition probabilities.

Coeﬃcient Value Used references α 0.8 [53, 71, 72, 73] β 0.06 [74, 52, 75, 76, 77, 78]

Table 3: Probability of each transition

5.3 Model implementation The proposed model is not known to have an closed-form solution. In a real- world setting, it is necessary to use statistical simulation methods to approximate the mean trajectory or quantile functions over time. Also, we must ﬁt some parameters of the model to characterize the behavior of the study population. For this purpose, we use a sample of the deceased patients M1, M2,··· , Ms of each day in the timeframe O = {1, ··· , s}. Next, we suppose that our model (M) is dependent on a vector of pa- p1 p2 rameters θ = (θ1, θ2) ∈ R × R (with p1 + p2 = p), where θ1 is a vector of dimension p1, deﬁned in beforehand, and θ2 must be estimated from the sample. Furthermore, let us assume that the initial state of the system is 7 characterized by S = (S(0),I1(0),I2(0),I3(0),R1(0),R2(0),M(0)) ∈ N and m m T = (T1(0),T2(0),T3(0),T4(0),T5(0),T6(0)) ∈ N × · · · × N . S has the number of elements for each compartment of the model on day 0. T also contains the amount of remaining days to complete the transition they are in for each individual in the initial state, being m a integer number that represents the maximun numbers of registred days. To simplify the notation, from now on we denote the average dead trajectory by the function Mean(θ1, θ2, S, T )(t) for each day t ∈ D ˆ The next step is to estimate θ2. To do this, we solve the following optimization problem:

s X θˆ = arg min ωi(M − Mean(θ , θ , S, T )(i))2, (2) 2 p i 1 2 θ2∈S⊂ 2 R i=1 where ω = (ω1, ··· , ωs) is a weighted vector that can help to improve model estimation. Examples of this weights may be: i Ps i Ps ω = Mi/ i=1 Mi or ω = (1/Mi)/( i=1 1/Mi)(i = 1, . . . , s). At this point, it is relevant to note that the above optimization problem (2), as formulated, includes the possibility of introducing constraints in the space of parameters. It is important because we have prior knowledge of what the range of the parameters is. We may know constraints involving them as well. In Equation (2), we have used the real mean trajectory. However, in practice, this is unknown, and we must approximate it using simulation. Next, we run B 0 1 PB i diﬀerents simulations and we denote by M(θ1, θ2, S, T ) = B i=1 M (θ1, θ2, S, T ), i the estimated mean trajectory. M (θ, θ2, S, T )(i = 1, ··· ,B) denote the result of each simulation. So the optimization problem to be solved is:

28 s X θˆ = arg min ωi(M − M(θ , θ , S, T )(i))2. (3) 2 p i 1 2 θ2∈S⊂ 2 R i=1 Schematically, the overall optimization process is described below.

0 1. Deﬁne an initial θ2 and run B times M(θ1, θ2, S, T ). We denote by 1 0 B 0 M (θ1, θ2, S, T ), ··· , M (θ1, θ2, S, T ) to each of the obtained results.

0 1 PB i 0 2. Estimate the mean trajectory M(θ1, θ2, S, T ) = B i=1 M (θ1, θ2, S, T )

ˆ 0 Ps i 0 3. Estimate the mean square error RSS = i=1 ω (M(θ1, θ2, S, T )(i) − 2 Mi) .

j M+1 ˆ 0 ˆ 1 4. To construct a succession of vectors {θ2}j=1 so that RSS > RSS > 2 M+1 RSSˆ > ··· > RSSˆ . For example with a stochastic optimization algorithm.

M+1 5. Stop after M + 1 iterations and return θ2 as the optimal parameter of the problem.

In our particular setting, θ2 contain the parameter of a Ri(t) function deﬁnes −(bt+ct2+dt3+et4+ft5) in Section 5.2. From now on, we asume that Ri(t) = min{C, ae } where a ∈ [0, 3], b ∈ [−1, 1], c ∈ [0, 1], d ∈ [0, 1], e ∈ [0, 1], f ∈ [0, 1] with θ2 = (a, b, c, d, e, f) ∈ [0, 3] × [−1, 1] × [0, 1] × [0, 1] × [0, 1] × [0, 1] and C is a positive constant ﬁxed 0.005.

5.4 Model optimization In our setting, we have found optimal parameters taking into account random- ness in the approximation of the mean. To do this, we must resort to stochastic optimization algorithms. Many algorithms of this kind can be found in literature, but based on the good results obtained in a preeliminary analysis we have decided to use a state-of-the-art evolutionary algorithm: the CMA-ES [79]. CMA-ES is an evolutionary-based derivative-free optimization technique that can optimize a wide variety of functions, including noisy functions as the one we use in our method. One survey of Black-Box optimizations found it outranked 31 other optimization algorithms, performing especially keen on “diﬃcult functions” or larger dimensional search spaces [80]. From a theoretical point of view, CMA-ES can be seen as a particular case of Expectation-Maximization algorithm (EM) [81].

5.5 Model Inference Our model has been ﬁtted independently for each region. Therefore, we cannot make statistical inference in the usual sense since we only observe a trajectory of a stochastic process with a complex dependence structure. Hence, to identify the model, we assume that the observed trajectory of accumulated deaths is the mean of our random process.

29 0 0 Given a day t ∈ O, Mt0 accumulated number of observed deaths in t and the multivariate random process {X(t) = (I1(t),I2(t),I3(t),R1(t),R2(t),M(t))}t∈D, the death trajectories of the probabilistic model close to the observed path on t0, are for us a set of possible explanations of the spread of the coronavirus in a study region. We now formal- ize this idea. Let Iα = [Mt0 − αMt0 ,Mt0 + αMt0 ], where α ∈ (0, 1). Considerer, 0 {ω ∈ Ω: M(t , ω) ∈ Iα}, being Ω sample space of process {X(t)}t∈D. We build 0 our conﬁdence bands of level α as paths that belong {ω ∈ Ω: M(t , ω) ∈ Iα}. Our conﬁdence bands contain the trajectories of infected and recovered patients that generate many deaths similar to that which occurs in reality. There- fore, in practice, α = 0.1 is usually a fair value, and t0 is selected as one of the latest datums in the historical records.

5.6 Software details and resources Our proposal has been implemented in several programming languages: C++, Python, and R, although the results shown in this article have been obtained with Python. We optimized the parameters using library pycma [82], and Numpy has been used for mathematical operations. In the different statistical analyses performed, we have used R. Plots have been made both in R with ggplot2 library and in Python with Matplotlib. Finally, the training data used to fit the models can be downloaded from at [83] and [84]. In the immediate future we are going to release the code used in this paper for the benefit of the scientific community at (https://github.com/covid19-modeling).

Acknowledgements

The authors thank Oscar Hernan Madrid Padilla for their helpful suggestions. We are also grateful to Felipe MuñozLópez for his help in the computer imple- mentations at the early stages of this research. This work has received financial support from the Conseller´ıade Cultura, Ed- ucacióne OrdenaciónUniversitaria (accreditation 2019-2022 ED431G-2019/04) and the European Regional Development Fund (ERDF), which acknowledges the CiTIUS-Research Center in Intelligent Technologies of the University of Santiago de Compostela as a Research Center of the Galician University Sys- tem.

Competing Interests

The authors declare no competing interests.

References

[1] Ilona Kickbusch and Gabriel Leung. Response to the emerging novel coronavirus outbreak. BMJ, 368, 2020.

30 [2] Scott P. Layne, James M. Hyman, David M. Morens, and Jeﬀery K. Taubenberger. New coronavirus outbreak: Framing questions for pandemic prevention. Science Translational Medicine, 12(534), 2020.

[3] Martin Enserink and Kai Kupferschmidt. Mathematics of life and death: How disease models shape national shutdowns and other pandemic policies. Science Magazine, 2020. [4] Stephen M Kissler, Christine Tedijanto, Edward Goldstein, Yonatan H Grad, and Marc Lipsitch. Projecting the transmission dynamics of sars- cov-2 through the postpandemic period. Science, 2020. [5] A Huppert and G. Katriel. Mathematical modelling and prediction in infectious disease epidemiology. Clinical Microbiology and Infection, 19(11):999– 1005, Nov 2013.

[6] Matt J Keeling and Pejman Rohani. Modeling infectious diseases in humans and animals. Princeton University Press, 2011. [7] Sandip Mandal, Ram Rup Sarkar, and Somdatta Sinha. Mathematical models of malaria-a review. Malaria journal, 10(1):202, 2011.

[8] Linda JS Allen. Some discrete-time si, sir, and sis epidemic models. Math- ematical biosciences, 124(1):83–105, 1994. [9] Junkichi Satsuma, R Willox, A Ramani, B Grammaticos, and AS Carstea. Extending the sir epidemic model. Physica A: Statistical Mechanics and its Applications, 336(3-4):369–375, 2004.

[10] Frank Ball, Catherine Lar´edo,David Sirl, and Viet Chi Tran. Stochastic Epidemic Models with Inference, volume 2255. Springer Nature, 2019. [11] Boseung Choi and Grzegorz A Rempala. Inference for discretely observed stochastic kinetic networks with applications to epidemic modeling. Bio- statistics, 13(1):153–165, 2012.

[12] Dave Osthus, Kyle S Hickmann, Petrut¸a C Caragea, Dave Higdon, and Sara Y Del Valle. Forecasting seasonal inﬂuenza with a state-space sir model. The annals of applied statistics, 11(1):202, 2017. [13] J. S. Koopman and J. W. Lynch. Individual causal models and population system models in epidemiology. American journal of public health, 89(8):1170–1174, Aug 1999. 10432901[pmid]. [14] J. Gomez-Gardees, L. Lotero, S. N. Taraskin, and F. J. Perez-Reche. Ex- plosive contagion in networks. Scientiﬁc Reports, 6(1):19767, 2016. [15] Matt J. Keeling and Ken T. D. Eames. Networks and epidemic models. Journal of the Royal Society, Interface, 2(4):295–307, Sep 2005. 16849187[pmid]. [16] Michael R Kosorok and Eric B Laber. Precision medicine. Annual review of statistics and its application, 6:263–286, 2019.

31 [17] Nicholas G. Reich, Logan C. Brooks, Spencer J. Fox, Sasikiran Kandula, Craig J. McGowan, Evan Moore, Dave Osthus, Evan L. Ray, Abhinav Tushar, Teresa K. Yamana, Matthew Biggerstaff, Michael A. Johansson, Roni Rosenfeld, and Jeffrey Shaman. A collaborative multiyear, multi- model assessment of seasonal influenza forecasting in the united states. Proceedings of the National Academy of Sciences, 116(8):3146–3154, 2019. [18] Logan C. Brooks, David C. Farrow, Sangwon Hyun, Ryan J. Tibshi- rani, and Roni Rosenfeld. Nonmechanistic forecasts of seasonal influenza with iterative one-week-ahead distributions. PLOS Computational Biology, 14(6):1–29, 06 2018. [19] CécileViboud and Alessandro Vespignani. The future of influenza forecasts. Proceedings of the National Academy of Sciences, 116(8):2802–2804, 2019.

[20] Xiao Li, Jessilyn Dunn, Denis Salins, Gao Zhou, Wenyu Zhou, Sophia Miryam Schussler-Fiorenza Rose, Dalia Perelman, Elizabeth Col- bert, Ryan Runge, Shannon Rego, Ria Sonecha, Somalee Datta, Tracey McLaughlin, and Michael P. Snyder. Digital health: Tracking physiomes and activity using wearable biosensors reveals useful health-related information. PLoS biology, 15(1):e2001402–e2001402, Jan 2017. 28081144[pmid].

[21] Sander Greenland. Multiple-bias modelling for analysis of observational data. Journal of the Royal Statistical Society: Series A (Statistics in Soci- ety), 168(2):267–306, 2005. [22] Marc Lipsitch, Christl Donnelly, Christophe Fraser, Isobel Blake, Anne Cori, Ilaria Dorigatti, Neil Ferguson, Tini Garske, Harriet Mills, Steven Riley, Maria Van Kerkhove, and Miguel Hernan. Potential biases in estimating absolute and relative case-fatality risks during outbreaks. PLoS neglected tropical diseases, 9:e0003846, 07 2015. [23] Heejung Bang and James Robins. Doubly robust estimation in missing data and causal inference models. Biometrics, 61:962–73, 01 2006.

[24] John Hopkins Coronavirus Resource Center. https://coronavirus.jhu. edu/map.html. [25] Instituto Nacional de Estad´ıstica(INE). https://www.ine.es/. [26] Population Reference Bureau. https://www.prb.org/ countries-with-the-oldest-populations/. [27] AC Ghani, CA Donnelly, DR Cox, JT Griﬃn, C Fraser, TH Lam, LM Ho, WS Chan, RM Anderson, AJ Hedley, et al. Methods for estimating the case fatality ratio for a novel, emerging infectious disease. American journal of epidemiology, 162(5):479–486, 2005.

[28] Xianxian Zhao, Bili Zhang, Pan Li, Chaoqun Ma, Jiawei Gu, Pan Hou, Zhifu Guo, Hong Wu, and Yuan Bai. Incidence, clinical characteristics and prognostic factor of patients with covid-19: a systematic review and meta-analysis. medRxiv, 2020.

32 [29] Paolo Buonanno, Sergio Galletta, and Marcello Puca. News from the covid- 19 epicenter. Available at SSRN 3567093, 2020. [30] Spencer Woody, Mauricio Garcia Tec, Maytal Dahan, Kelly Gaither, Michael Lachmann, Spencer Fox, Lauren Ancel Meyers, and James G Scott. Projections for first-wave covid-19 deaths across the us using social- distancing measures derived from mobile phones. medRxiv, 2020. [31] Keith R. Bisset, Ashwin M. Aji, Madhav V. Marathe, and Wu-chun Feng. High-performance biocomputing for simulating the spread of contagion over large contact networks. BMC genomics, 13 Suppl 2(Suppl 2):S3–S3, Apr 2012. 22537298[pmid]. [32] Dennis L. Chao, M. Elizabeth Halloran, Valerie J. Obenchain, and Ira M Longini, Jr. Flute, a publicly available stochastic influenza epidemic simulation model. PLOS Computational Biology, 6(1):1–8, 01 2010. [33] Keith Bisset, Jiangzhuo Chen, Xizhou Feng, Sritesh Kumar, and Madhav Marathe. Epifast: A fast algorithm for large scale realistic epidemic simulations on distributed memory systems. pages 430–439, 01 2009. [34] Y Ma, K. R. Bisset, J. Chen, S. Deodhar, and M. V. Marathe. Formal spec- ification and experimental analysis of an interactive epidemic simulation framework. In 2011 IEEE International Conference on High Performance Computing and Communications, pages 790–795, Sep. 2011. [35] Liang Mao and Ling Bian. Spatial-temporal transmission of influenza and its health risks in an urbanized area. Computers, Environment and Urban Systems, 34(3):204–215, May 2010. PMC7124201[pmcid]. [36] Centers for Disease Control and Prevention. https://www.cdc.gov/flu/ about/keyfacts.htm. [37] Ottar N Bjørnstad, Bärbel F Finkenstädt,and Bryan T Grenfell. Dynamics of measles epidemics: estimating scaling of transmission rates using a time series sir model. Ecological monographs, 72(2):169–184, 2002. [38] Lars Lorch, William Trouleau, Stratis Tsirtsis, Aron Szanto, Bernhard Schölkopf, and Manuel Gomez-Rodriguez. A spatiotemporal epidemic model to quantify the effects of contact tracing, testing, and containment. arXiv preprint arXiv:2004.07641, 2020. [39] Juanjuan Zhang, Maria Litvinova, Wei Wang, Yan Wang, Xiaowei Deng, Xinghui Chen, Mei Li, Wen Zheng, Lan Yi, Xinhua Chen, et al. Evolving epidemiology and transmission dynamics of coronavirus disease 2019 out- side hubei province, china: a descriptive and modelling study. The Lancet Infectious Diseases, 2020. [40] Lei Luo, Dan Liu, Xin-long Liao, Xian-bo Wu, Qin-long Jing, Jia-zhen Zheng, Fang-hua Liu, Shi-gui Yang, Bi Bi, Zhi-hao Li, Jian-ping Liu, Wei- qi Song, Wei Zhu, Zheng-he Wang, Xi-ru Zhang, Pei-liang Chen, Hua-min Liu, Xin Cheng, Miao-chun Cai, Qing-mei Huang, Pei Yang, Xin-fen Yang, Zhi-gang Huang, Jin-ling Tang, Yu Ma, and Chen Mao. Modes of contact and risk of transmission in covid-19 among close contacts. medRxiv, 2020.

33 [41] Yan Bai, Lingsheng Yao, Tao Wei, Fei Tian, Dong-Yan Jin, Lijuan Chen, and Meiyun Wang. Presumed Asymptomatic Carrier Transmission of COVID-19. JAMA, 02 2020.

[42] Eran Segal, Feng Zhang, Xihong Lin, Gary King, Ophir Shalem, Smadar Shilo, William E. Allen, Yonatan H. Grad, Casey S. Greene, Faisal Alquad- doomi, Simon Anders, Ran Balicer, Tal Bauman, Ximena Bonilla, Gisel Booman, Andrew T. Chan, Ori Ori Cohen, Silvano Coletti, Natalie David- son, Yuval Dor, David A. Drew, Olivier Elemento, Georgina Evans, Phil Ewels, Joshua Gale, Amir Gavrieli, Benjamin Geiger, Iman Hajirasouliha, Roman Jerala, Andre Kahles, Olli Kallioniemi, Ayya Keshet, Gregory Lan- dua, Tomer Meir, Aline Muller, Long H. Nguyen, Matej Oresic, Svetlana Ovchinnikova, Hedi Peterson, Jay Rajagopal, Gunnar Ratsch, Hagai Ross- man, Johan Rung, Andrea Sboner, Alexandros Sigaras, Tim Spector, Ron Steinherz, Irene Stevens, Jaak Vilo, Paul Wilmes, and . Building an international consortium for tracking coronavirus health status. medRxiv, 2020. [43] Hagai Rossman, Ayya Keshet, Smadar Shilo, Amir Gavrieli, Tal Bauman, Ori Cohen, Esti Shelly, Ran Balicer, Benjamin Geiger, Yuval Dor, and Eran Segal. A framework for identifying regional outbreak and spread of covid-19 from one-minute population-wide surveys. Nature Medicine, 2020. [44] Finbarr O’Sullivan. A statistical perspective on ill-posed inverse problems. Statistical science, pages 502–518, 1986. [45] Martin Hanke and Otmar Scherzer. Inverse problems light: Numerical dif- ferentiation. The American Mathematical Monthly, 108(6):512–521, 2001. [46] Peter J Bickel, Bo Li, Alexandre B Tsybakov, Sara A van de Geer, Bin Yu, TeófiloValdés,Carlos Rivero, Jianqing Fan, and Aad van der Vaart. Regularization in statistics. Test, 15(2):271–344, 2006. [47] Vladimir Vapnik. The nature of statistical learning theory. Springer science & business media, 2013. [48] Ernesto De Vito, Lorenzo Rosasco, Andrea Caponnetto, Umberto De Gio- vannini, and Francesca Odone. Learning from examples as an inverse problem. Journal of Machine Learning Research, 6(May):883–904, 2005.

[49] Vladimir Vapnik and Rauf Izmailov. V-matrix method of solving statistical inference problems. Journal of Machine Learning Research, 16(51):1683– 1730, 2015. [50] Huili Xiang and Bin Liu. Solving the inverse problem of an sis epidemic reaction-diﬀusion model by optimal control methods. Computers & Math- ematics with Applications, 70(5):805–819, 2015. [51] Alexandra Smirnova and Gerardo Chowell. A primer on stable parameter estimation and forecasting in epidemiology by a problem-oriented regular- ized least squares algorithm. Infectious Disease Modelling, 2(2):268–275, May 2017. 29928741[pmid].

34 [52] Anthony S. Fauci, H. Cliﬀord Lane, and Robert R. Redﬁeld. Covid-19 - navigating the uncharted. New England Journal of Medicine, 382(13):1268– 1269, 2020.

[53] Michael Day. Covid-19: four ﬁfths of cases are asymptomatic, china ﬁgures indicate. BMJ, 369, 2020. [54] Satyajith Amaran, Nikolaos V. Sahinidis, Bikram Sharda, and Scott J. Bury. Simulation optimization: a review of algorithms and applications. 4OR, 12(4):301–333, 2014. [55] Satyajith Amaran, Nikolaos V. Sahinidis, Bikram Sharda, and Scott J. Bury. Simulation optimization: a review of algorithms and applications. Annals of Operations Research, 240(1):351–380, 2016. [56] Corey M Peak, Rebecca Kahn, Yonatan H Grad, Lauren M Childs, Ruo- ran Li, Marc Lipsitch, and Caroline O Buckee. Modeling the comparative impact of individual quarantine vs. active monitoring of contacts for the mitigation of covid-19. medRxiv, 2020. [57] Stephen M Kissler, Christine Tedijanto, Marc Lipsitch, and Yonatan Grad. Social distancing strategies for curbing the covid-19 epidemic. medRxiv, 2020. [58] Martin McKee and David Stuckler. If the world fails to protect the economy, covid-19 will damage health not just now but also in the future. Nature Medicine, 2020.

[59] Ziyue Liu and Wensheng Guo. Government responses matter: Predicting covid-19 cases in us under an empirical bayesian time series framework. medRxiv, 2020. [60] Bobak Shahriari, Kevin Swersky, Ziyu Wang, Ryan P Adams, and Nando De Freitas. Taking the human out of the loop: A review of bayesian optimization. Proceedings of the IEEE, 104(1):148–175, 2015.

[61] Galit Shmueli, Thomas P. Minka, Joseph B. Kadane, Sharad Borle, and Peter Boatwright. A useful distribution for ﬁtting discrete data: revival of the conway-maxwell-poisson distribution. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54(1):127–142, 2005.

[62] Thomas House. Modelling epidemics on networks. Contemporary Physics, 53(3):213–225, 2012. [63] ZHAN Zhang and TN Krishnamurti. Ensemble forecasting of hurricane tracks. Bulletin of the American Meteorological Society, 78(12):2785–2796, 1997.

[64] Stephen A Lauer, Kyra H Grantz, Qifang Bi, Forrest K Jones, Qulu Zheng, Hannah R Meredith, Andrew S Azman, Nicholas G Reich, and Justin Lessler. The incubation period of coronavirus disease 2019 (covid-19) from publicly reported conﬁrmed cases: estimation and application. Annals of internal medicine, 2020.

35 [65] AG Abdel-Salam and M Mollazehi. Modeling survival time to recovery from covid-19: A case study on singapore. 2020.

[66] Coronavirus Pneumonia Emergency Response Epidemiology Novel et al. The epidemiological characteristics of an outbreak of 2019 novel coronavirus diseases (covid-19) in china. Zhonghua liu xing bing xue za zhi= Zhonghua liuxingbingxue zazhi, 41(2):145, 2020. [67] Robert Verity, Lucy C. Okell, Ilaria Dorigatti, Peter Winskill, Charles Whittaker, Natsuko Imai, Gina Cuomo-Dannenburg, Hayley Thompson, Patrick G. T. Walker, Han Fu, Amy Dighe, Jamie T. Griffin, Marc Baguelin, Sangeeta Bhatia, Adhiratha Boonyasiri, Anne Cori, Zulma Cu- cunubá,Rich FitzJohn, Katy Gaythorpe, Will Green, Arran Hamlet, Wes Hinsley, Daniel Laydon, Gemma Nedjati-Gilani, Steven Riley, Sabine van Elsland, Erik Volz, Haowei Wang, Yuanrong Wang, Xiaoyue Xi, Christl A. Donnelly, Azra C. Ghani, and Neil M. Ferguson. Estimates of the severity of coronavirus disease 2019: a model-based analysis. The Lancet Infectious Diseases, 2020/04/19 XXXX. [68] Henrik Salje, CécileTran Kiem, Noémie Lefrancq, NoémieCourtejoie, Paolo Bosetti, Juliette Paireau, Alessio Andronico, NathanaëlHoze, Je- hanne Richet, Claire-Lise Dubost, et al. Estimating the burden of sars- cov-2 in france. 2020. [69] Qifang Bi, Yongsheng Wu, Shujiang Mei, Chenfei Ye, Xuan Zou, Zhen Zhang, Xiaojian Liu, Lan Wei, Shaun A Truelove, Tong Zhang, et al. Epi- demiology and transmission of covid-19 in shenzhen china: Analysis of 391 cases and 1,286 of their close contacts. MedRxiv, 2020. [70] Katrin Zwirglmaier Ehmann, Christian Drosten, Clemens Wendtner, MD Zange, Patrick Vollmar, DVM Rosina Ehmann, Katrin Zwirglmaier, MD Guggemos, Michael Seilmaier, Daniela Niemeyer, et al. Virological assessment of hospitalized cases of coronavirus disease 2019.

[71] Hiroshi Nishiura, Tetsuro Kobayashi, Takeshi Miyama, Ayako Suzuki, Sungmok Jung, Katsuma Hayashi, Ryo Kinoshita, Yichi Yang, Baoyin Yuan, Andrei R. Akhmetzhanov, and Natalie M Linton. Estimation of the asymptomatic ratio of novel coronavirus infections (covid-19). medRxiv, 2020.

[72] Sakiko Tabata, Kazuo Imai, Shuichi Kawano, Mayu Ikeda, Tatsuya Ko- dama, Kazuyasu Miyoshi, Hirofumi Obinata, Satoshi Mimura, Tsutomu Kodera, Manabu Kitagaki, Michiya Sato, Satoshi Suzuki, Toshimitsu Ito, Yasuhide Uwabe, and Kaku Tamura. Non-severe vs severe symptomatic covid-19: 104 cases from the outbreak on the cruise ship “diamond princess” in japan. medRxiv, 2020.

[73] Kenji Mizumoto, Katsushi Kagaya, Alexander Zarebski, and Gerardo Chowell. Estimating the asymptomatic proportion of coronavirus disease 2019 (covid-19) cases on board the diamond princess cruise ship, yokohama, japan, 2020. Eurosurveillance, 25(10), 2020.

36 [74] Dimple D Rajgor, Meng Har Lee, Sophia Archuleta, Natasha Bagdasarian, and Swee Chye Quek. The many estimates of the covid-19 case fatality rate. The Lancet Infectious Diseases, 2020.

[75] Robert Verity, Lucy C. Okell, Ilaria Dorigatti, Peter Winskill, Charles Whittaker, Natsuko Imai, Gina Cuomo-Dannenburg, Hayley Thompson, Patrick G. T. Walker, Han Fu, Amy Dighe, Jamie T. Griﬃn, Marc Baguelin, Sangeeta Bhatia, Adhiratha Boonyasiri, Anne Cori, Zulma Cu- cunub´a,Rich FitzJohn, Katy Gaythorpe, Will Green, Arran Hamlet, Wes Hinsley, Daniel Laydon, Gemma Nedjati-Gilani, Steven Riley, Sabine van Elsland, Erik Volz, Haowei Wang, Yuanrong Wang, Xiaoyue Xi, Christl A. Donnelly, Azra C. Ghani, and Neil M. Ferguson. Estimates of the severity of coronavirus disease 2019: a model-based analysis. The Lancet Infectious Diseases, 2020/04/06 XXXX.

[76] Joseph T. Wu, Kathy Leung, Mary Bushman, Nishant Kishore, Rene Niehus, Pablo M. de Salazar, Benjamin J. Cowling, Marc Lipsitch, and Gabriel M. Leung. Estimating clinical severity of covid-19 from the transmission dynamics in wuhan, china. Nature Medicine, 2020. [77] Elisabeth Mahase. Covid-19: death rate is 0.66% and increases with age, study estimates. BMJ, 369:m1327, Apr 2020. [78] Christian Dudel, Tim Riffe, Enrique Acosta, Alyson A van Raalte, and Mikko Myrskyla. Monitoring trends and differences in covid-19 case fatality rates using decomposition methods: Contributions of age structure and age-specific fatality. medRxiv, 2020.

[79] Nikolaus Hansen. The cma evolution strategy: A tutorial. arXiv preprint arXiv:1604.00772, 2016. [80] Nikolaus Hansen, Anne Auger, Raymond Ros, Steﬀen Finck, and Petr Povsik. Comparing results of 31 algorithms from the black-box optimization benchmarking bbob-2009. In Proceedings of the 12th Annual Confer- ence Companion on Genetic and Evolutionary Computation, GECCO 10, pages 1689–1696, New York, NY, USA, 2010. Association for Computing Machinery. [81] David H Brookes, Akosua Busia, Clara Fannjiang, Kevin Murphy, and Jennifer Listgarten. A view of estimation of distribution algorithms through the lens of expectation-maximization. arXiv preprint arXiv:1905.10474, 2019. [82] niko, Youhei Akimoto, yoshihikoueno, Dimo Brockhoﬀ, Matthew Chan, and ARF1. Cma-es/pycma: r3.0.3, April 2020.

[83] COVID-19 Rub´enFern´andezCasal. https://github.com/rubenfcasal/ COVID-19. [84] Datadista. https://github.com/datadista/datasets/tree/master/ COVID%2019.