# A Modified SIRD Model to Study the Evolution of the COVID-19 Pandemic in Spain

## Abstract

**:**

## 1. Introduction

## 2. Model Description

- (S)
- Susceptible—the people who could become infected.
- (I)
- Infected—the people who are infected at that moment.
- (R)
- Recovered—the people who have had the disease and are now healthy.

- (D)
- Deceased—the people who have died of the disease.

#### 2.1. Equations Governing the Model

#### 2.2. Estimation of Parameters $\alpha $ and $\gamma $

#### 2.3. Estimation of Parameter $\beta $

## 3. Analysis of the COVID-19 Pandemic in Spain during the Course of the Disease

#### 3.1. First Stage: 11–21 March 2020

- The discussion of the above section allows us to consider the Recovered/Deceased fraction as $\frac{55}{183}\approx 0.3$, in accordance with the figures from 11 March 2020 (see Table 1). Taking 0.3 as the Recovered/Deceased fraction, we can estimate that $E\left[X\right]\approx 0.7\times 14+0.3\times 42=22.4$ days.
- From Equation (5) and considering $\gamma \approx 0.3\phantom{\rule{0.166667em}{0ex}}\alpha $, we obtain $\alpha =0.0343\phantom{\rule{0.222222em}{0ex}}$ and $\gamma =0.0103$.
- From Equation (7), we estimate $\beta $ once $\alpha \phantom{\rule{0.222222em}{0ex}}$ and $\phantom{\rule{0.222222em}{0ex}}\gamma $ are known. So, $\beta =1.6554\times {10}^{-7}$.

#### 3.2. Second Stage: 22–31 March 2020

- We observed in the data that the Recovered/Deceased fraction follows the sequence tending towards 0.3 and $E\left[X\right]\approx $ 20 days.
- From Equation (5) and $\gamma \approx 0.3\phantom{\rule{0.166667em}{0ex}}\alpha $, we obtain $\alpha =0.0385\phantom{\rule{0.222222em}{0ex}}$ and $\gamma =0.0155$. From Equation (7), we estimate the value of $\beta $ once we know $\alpha \phantom{\rule{0.222222em}{0ex}}$ and $\phantom{\rule{0.222222em}{0ex}}\gamma $. So, $\beta =3.171\times {10}^{-7}$.

#### 3.3. Third Stage: 1–13 April 2020

- The Recovered/Deceased fraction has been observed to follow the sequence:$$0.65,\dots ,0.58,\dots ,0.31,\dots ,0.27,\dots .$$Consequently, we have estimated the value 0.22, which produces good approximations for the data that are already known. Taking 0.22 as the Recovered/Deceased fraction, we can estimate that $E\left[X\right]\approx $ 20 days.
- From Equation (5) and $\gamma \approx 0.22\phantom{\rule{0.166667em}{0ex}}\alpha $, we obtain $\alpha =0.041\phantom{\rule{0.222222em}{0ex}}$ and $\gamma =0.009$. From Equation (7) and the values of $\alpha \phantom{\rule{0.222222em}{0ex}}$ and $\phantom{\rule{0.222222em}{0ex}}\gamma $, we estimate $\beta $. So, $\beta =4.7878\times {10}^{-7}$.

#### 3.4. Fourth Stage: 14–23 April 2020

#### 3.5. Types of COVID-19 Screening Tests

- The Polymerase Chain Reaction technique (PCR test) amplifies the genetic material of the virus (RNA and DNA) to detect it. So far, it is the most reliable method, but it requires analysis in a laboratory.
- The test for coronavirus antibodies (antibody test) does not detect the virus, but it detects the antibodies that kill it. It is done with a blood test, which can be performed quickly and on a massive scale.

#### 3.6. Fifth Stage: 23 April to 2 May 2020

## 4. Discussion

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Comparison of real data with the predictions of the SIRD model on 21 March 2020. (

**b**) Zoomed-in graph of real data.

**Figure 3.**Comparison of the real data with the estimations of the SIRD model during the second stage.

**Figure 5.**(

**a**) Comparison of the state of the real data with the predictions of the SIRD model on 23 April 2020. (

**b**) Zoomed-in graph of real data.

**Figure 6.**Types of tests carried out and the % increase in the number of infected people on 23 April 2020. Source: Spanish Government [1] on 24 April 2020.

**Figure 8.**Values of the parameter ${\rho}_{0}$ (reproduction number) over time in Spain during the first wave of the COVID-19 Pandemic.

**Figure 9.**Comparison of real data with the estimates of the piecewise SIRD model during the evolution of the epidemic.

Date | Infected | Recovered | Deceased | Total Cases |
---|---|---|---|---|

11 March | 2039 | 183 | 55 | 2277 |

12 March | 4906 | 193 | 133 | 5232 |

13 March | 5679 | 517 | 195 | 6391 |

14 March | 6992 | 517 | 289 | 7798 |

15 March | 9070 | 530 | 342 | 9942 |

16 March | 10,187 | 1028 | 533 | 11,748 |

17 March | 12,206 | 1081 | 623 | 13,910 |

18 March | 16,026 | 1107 | 830 | 17,963 |

19 March | 17,779 | 1588 | 1043 | 20,410 |

20 March | 21,874 | 2125 | 1375 | 25,374 |

21 March | 24,421 | 2575 | 1772 | 28,768 |

Date | Infected | Recovered | Deceased | Total Cases |
---|---|---|---|---|

22 March | 27,552 | 3355 | 2182 | 33,089 |

23 March | 33,183 | 3794 | 2696 | 39,673 |

24 March | 38,809 | 5367 | 3434 | 47,610 |

25 March | 45,084 | 7015 | 4089 | 56,188 |

26 March | 49,844 | 9357 | 4858 | 64,059 |

27 March | 54,273 | 12,285 | 5690 | 72,248 |

28 March | 57,560 | 14,709 | 6528 | 78,797 |

29 March | 61,075 | 16,780 | 7340 | 85,195 |

30 March | 66,969 | 19,259 | 8189 | 94,417 |

31 March | 70,436 | 22,647 | 9053 | 102,136 |

Date | Infected | Recovered | Deceased | Total Cases |
---|---|---|---|---|

1 April | 73,492 | 26,743 | 10,003 | 110,238 |

2 April | 76,262 | 30,513 | 10,935 | 117,710 |

3 April | 78,773 | 34,219 | 11,744 | 124,736 |

4 April | 80,261 | 38,080 | 12,418 | 130,759 |

5 April | 81,540 | 40,437 | 13,055 | 135,032 |

6 April | 83,504 | 43,208 | 13,798 | 140,510 |

7 April | 84,114 | 48,021 | 14,555 | 146,690 |

8 April | 85,043 | 52,165 | 15,238 | 152,446 |

9 April | 85,511 | 55,668 | 15,843 | 157,022 |

10 April | 86,390 | 59,109 | 16,353 | 161,852 |

11 April | 86,656 | 62,391 | 16,972 | 166,019 |

12 April | 87,280 | 64,727 | 17,489 | 169,496 |

13 April | 86,981 | 67,504 | 18,056 | 172,541 |

Date | Infected | Recovered | Deceased | Total Cases |
---|---|---|---|---|

14 April | 70,853 | 88,201 | 18,579 | 177,633 |

15 April | 74,797 | 88,889 | 19,130 | 182,816 |

16 April | 72,963 | 95,627 | 19,478 | 188,068 |

17 April | 74,662 | 97,021 | 20,043 | 191,726 |

18 April | 77,357 | 98,134 | 20,453 | 195,944 |

19 April | 80,587 | 98,771 | 20,852 | 200,210 |

20 April | 82,514 | 100,382 | 21,282 | 204,178 |

21 April | 85,915 | 100,757 | 21,717 | 208,389 |

22 April | 89,250 | 101,617 | 22,157 | 213,024 |

23 April | 92,355 | 88,111 | 22,524 | 202,990 |

Date | Infected | Recovered | Deceased | Total Cases |
---|---|---|---|---|

23 April | 88,111 | 92,355 | 22,524 | 202,990 |

24 April | 87,295 | 95,708 | 22,902 | 205,905 |

25 April | 85,712 | 98,732 | 23,190 | 207,634 |

26 April | 85,069 | 100,875 | 23,521 | 209,465 |

27 April | 84,403 | 102,548 | 23,822 | 210,773 |

28 April | 79,695 | 108,947 | 24,275 | 212,917 |

29 April | 76,842 | 112,050 | 24,543 | 213,435 |

30 April | 75,714 | 114,678 | 24,824 | 215,216 |

1 May | 74,234 | 117,248 | 25,100 | 216,582 |

2 May | 73,300 | 118,902 | 25,264 | 217,466 |

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**MDPI and ACS Style**

Martínez, V.
A Modified SIRD Model to Study the Evolution of the COVID-19 Pandemic in Spain. *Symmetry* **2021**, *13*, 723.
https://doi.org/10.3390/sym13040723

**AMA Style**

Martínez V.
A Modified SIRD Model to Study the Evolution of the COVID-19 Pandemic in Spain. *Symmetry*. 2021; 13(4):723.
https://doi.org/10.3390/sym13040723

**Chicago/Turabian Style**

Martínez, Vicente.
2021. "A Modified SIRD Model to Study the Evolution of the COVID-19 Pandemic in Spain" *Symmetry* 13, no. 4: 723.
https://doi.org/10.3390/sym13040723