# Mathematical Modelling of the Impact of Non-Pharmacological Strategies to Control the COVID-19 Epidemic in Portugal

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Description

#### 2.2. Model Analysis

#### 2.2.1. Homogeneous Model

#### 2.2.2. Heterogeneous Model

## 3. Data

## 4. Model Fit Strategy, Results and Simulations

- 1.
- Starts at the 10th of February 2020 (day zero) and encompasses the first exponential growth phase of the epidemic, which was curbed by the implementation of the closure of schools and state-of-emergency NPIs.
- 2.
- Covers the first descendent incidence phase. Schools were closed and state-of-emergency was in effect. Ends with the phase out of the state of emergency during May-June. Schools were closed during this period.
- 3.
- This period covers the transition from the state-of-emergency to the summer. This period ends as the new exponential growth phase beginning and schools opening.
- 4.
- Covers the second exponential growth phase after the summer is over. During this period several softer NPIs were imposed during October and November. Schools are opened during this period.
- 5.
- This period takes into account the short window between the reduction in infectious contacts caused by the NPI implemented during October and November to the end of the year.
- 6.
- This period starts with the third exponential phase of the Portuguese epidemic during the Christmas holidays and continues until the 15th of January.

`lsoda`function in the

`deSolve`package in the

`R`language and environment for statistical computing, version 4.0.3 [18]. Model was fitted to the ICU prevalence data (${H}_{IC{U}_{obs}}$) and fitting was performed via the differential evolution algorithm using the

`R`package

`DEoptim`[19] by minimising the square root of the sum of squares presented below:

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

NPI | Non Pharmaceutical Intervention |

ICU | Intensive Care Units |

## Appendix A

Parameter | Description | Value | Source |
---|---|---|---|

$\beta $ | transmission probability | $0.068$ (taken from ${R}_{0}=2.5$) | [13] |

$\epsilon $ | latent period | $1/3.8$ days${}^{-1}$ | [9] |

${r}_{s}$/${r}_{a}$ | infectious period | $1/3.4$ days${}^{-1}$ | [9] |

$\theta $ | probability of hospitalisation | 5.1% for 0–49 years; | [12] |

(age-dependent) | 10.11 % for 50–59 years; | ||

21.99 % for 60–69 years; | |||

40.00 % for 70+ years; | |||

p | proportion of asymptomatic | $44.5\%$ | [3] |

$\rho $ | rate of progression out of H | 1/9 days${}^{-1}$ | ACSS/SPMS |

$\pi $ | fraction of hospitalized | $12.6\%$ | ACSS/SPMS |

individuals progressing to ICU | $7.9\%$ from 11/25 to 12/31 | ||

(time dependent) | $11.2\%$ from 12/31 onwards | ||

$\tau $ | proportion of hospitalization deaths | $19.2\%$ | ACSS/SPMS |

$\omega $ | rate of progression out of ICU | 1/20 days${}^{-1}$ | ACSS/SPMS |

$\mu $ | proportion of ICU deaths | $26.7\%$ | ACSS/SPMS |

${\alpha}_{A}$, ${\alpha}_{Aq}$ | asymptomatic reduction in transmission | $50\%$ | [22] |

${\alpha}_{S}$, ${\alpha}_{Sq}$ | symptomatic reduction in transmission | $0\%$ | assumed |

$Schoo{l}_{r}$ | contact reduction in schools | $33\%$ | assumed |

$Schoo{l}_{c}$ | school mask use compliance | $90\%$ | assumed |

$Mas{k}_{e}ff$ | mask effectiveness | $47\%$ | [17] |

in reducing transmission | |||

${S}_{0}$ | initial conditions for S | [436,202455,843504,940545,322 | INE |

550,444 547,680 566,594 672,422 | |||

784,224 789,733 745,178 740,141 | |||

676,762 622,912 549,591 1,107,921]^{t} | |||

$E.n{t}_{0}$ | initial condition for E | ${\left[0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}71\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\right]}^{t}$ | estimated |

$b{r}_{{1}_{H;W;O}}$ | change in transmission 1 | 2020-03-18 (t = 37) | estimated |

$b{r}_{{2}_{H;W;O}}$ | change in transmission 2 | 2020-05-10 (t = 90) | estimated |

$b{r}_{{3}_{H;W;O}}$ | change in transmission 3 | 2020-08-18 (t = 190) | estimated |

$b{r}_{{4}_{H;W;O}}$ | change in transmission 4 | 2020-11-02 (t = 266) | estimated |

$b{r}_{{5}_{H;W;O}}$ | change in transmission 5 | 2020-12-23 (t = 317) | estimated |

${\alpha}_{{1}_{H;W;O}}$ | change in contacts after $b{r}_{{1}_{H;W;O}}$ | $69\%$ | estimated |

${\alpha}_{{2}_{H;W;O}}$ | change in contacts after $b{r}_{{2}_{H;W;O}}$ | $55\%$ | estimated |

${\alpha}_{{3}_{H;W;O}}$ | change in contacts after $b{r}_{{3}_{H;W;O}}$ | $43\%$ | estimated |

${\alpha}_{{4}_{H;W;O}}$ | change in contacts after $b{r}_{{4}_{H;W;O}}$ | $58\%$ | estimated |

${\alpha}_{{5}_{H;W;O}}$ | change in contacts after $b{r}_{{5}_{H;W;O}}$ | $35\%$ | estimated |

**Table A2.**Introduction and lifting of NPI adopted in Portugal, dates and descriptions. Lockdown refers to a mandatory stay-at-home order. In Portugal this refers to a declaration of “state-of-emergency” by the Portuguese government to provide a response to a national crisis. This state allows the implementation of severe measures to fight disease spread. “state of contingency” refers to the introduction of milder measures and “state of calamity” corresponds to a state in between contingency and emergency.

Date | Description |
---|---|

2020-03-12 | announcement of schools closure |

2020-03-16 | closure of schools |

2020-03-18 | lockdown (“state-of-emergency”) announcement |

2020-03-22 | lockdown goes into effect |

2020-04-28 | announcement of lockdown phase-out |

2020-05-04 | first wave of lockdown phase-out |

2020-05-18 | second wave of lockdown phase-out |

2020-06-01 | third wave of lockdown phase-out |

2020-09-15 | “state of contingency” goes into effect |

2020-10-15 | “state of calamity” goes into effect |

2020-10-28 | outdoor obligatory use of mask |

2020-11-04 | lockdown measures on weekends for counties above 480/100,000 incidence |

2020-11-09 | “state of emergency” |

2020-12-24 | relaxation of measures during Christmas |

2020-01-15 | lockdown (“state of emergency”) |

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**Figure 1.**Schematic diagram of the homogeneous COVID-19 transmission model. Disease transmission is $\lambda S=\beta ({\alpha}_{S}{C}_{S}{I}_{S}+{\alpha}_{A}{C}_{A}{I}_{A}+{\alpha}_{Sq}{C}_{Sq}{I}_{Sq}+{\alpha}_{Aq}{C}_{Aq}{I}_{Aq})\frac{S}{N}$, where ${\alpha}_{i}$ and ${C}_{i}$, for $i=S,\phantom{\rule{4pt}{0ex}}A,\phantom{\rule{4pt}{0ex}}Sq,\phantom{\rule{4pt}{0ex}}Aq$, are the relative transmissibility and contacts of infectious classes.

**Figure 3.**Model fit to ICU data since the start of the epidemic (

**top left**). Model fit to ICU data from the 1st of October 2020 to the 15th of January 2021 (

**top right**). Observed and model values for hospitalised individuals (

**bottom left**) and deaths (

**bottom right**). Solid lines represent the model, the dots represent observed values (data available in [14]).

**Figure 4.**Evolution of the effective reproduction number given by the model. The solid vertical lines correspond to the date of the implementation of a NPI and the dashed lines mark the relaxation of the NPIs in place.

**Figure 5.**School closure and lockdown simulations on the number of ICU cases. The dotted vertical lines depict the implementation of the lockdown and the closure of schools. The simulation in red represents the scenario with no interventions.

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Caetano, C.; Morgado, M.L.; Patrício, P.; Pereira, J.F.; Nunes, B.
Mathematical Modelling of the Impact of Non-Pharmacological Strategies to Control the COVID-19 Epidemic in Portugal. *Mathematics* **2021**, *9*, 1084.
https://doi.org/10.3390/math9101084

**AMA Style**

Caetano C, Morgado ML, Patrício P, Pereira JF, Nunes B.
Mathematical Modelling of the Impact of Non-Pharmacological Strategies to Control the COVID-19 Epidemic in Portugal. *Mathematics*. 2021; 9(10):1084.
https://doi.org/10.3390/math9101084

**Chicago/Turabian Style**

Caetano, Constantino, Maria Luísa Morgado, Paula Patrício, João F. Pereira, and Baltazar Nunes.
2021. "Mathematical Modelling of the Impact of Non-Pharmacological Strategies to Control the COVID-19 Epidemic in Portugal" *Mathematics* 9, no. 10: 1084.
https://doi.org/10.3390/math9101084