# Optimal Control Applied to Vaccination and Testing Policies for COVID-19

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model

#### 2.2. Vaccination Scenarios

- Scenario 1: During the time period under consideration, the total number of administered vaccines is assumed to be fixed. Thus, in this scenario, all vaccine supplies must be used.
- Scenario 2: During the time period under consideration, the total number of administered vaccines is limited. Thus, in this scenario, the use of all vaccine supplies is not mandatory.
- Scenario 3: The number of vaccines administered daily is limited.

#### 2.2.1. Scenario 1

#### 2.2.2. Scenario 2

#### 2.2.3. Scenario 3

#### 2.3. Numerical Resolution of the Optimal Control Problem

## 3. Results

#### 3.1. Optimal Vaccination and Testing Policies

#### 3.2. Sensitivity of the Optimal Policies to the Vaccine Efficacy

#### 3.3. Sensitivity of the Optimal Policies to the Weighting Parameters of the Objective Functional

#### 3.4. Sensitivity of the Optimal Policies to the Lower Bound of the Testing Rate

#### 3.5. Sensitivity of the Optimal Policies to the Immunity Loss Rate

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of the SIDARTHE compartmental model with vaccination and testing policies and immunity loss [10].

**Figure 2.**Optimal policies for scenario 1 (blue), scenario 2 (orange), and scenario 3 (green): (

**a**) Optimal vaccination rates. (

**b**) Optimal testing rates.

**Figure 3.**Proportions of people in each compartment for scenario 1 (blue), scenario 2 (orange), and scenario 3 (green): (

**a**) Susceptible people. (

**b**) Asymptomatic undetected people. (

**c**) Asymptomatic detected people. (

**d**) Symptomatic undetected people. (

**e**) Symptomatic detected people. (

**f**) Acutely symptomatic people. (

**g**) Healed people. (

**h**) Extinct people.

**Figure 4.**Proportions of people in each compartment for scenario 1 (blue), scenario 2 (orange), and scenario 3 (green). Proportions in dashed red lines were obtained while assuming that the vaccination term is omitted and considering the testing rate as a parameter: (

**a**) Symptomatic detected people. (

**b**) Extinct people.

**Figure 5.**Optimal proportions of vaccinated and tested people for scenario 1 (blue), scenario 2 (orange), and scenario 3 (green): (

**a**) Optimal proportion of susceptible vaccinated people. (

**b**) Optimal proportion of infected tested people.

**Figure 6.**Optimal policies obtained assuming vaccine efficacy levels $\varphi =0.50$ (red), $\varphi =0.75$ (blue), and $\varphi =0.95$ (green), with ${C}_{1}=1,{C}_{2}=0.1$, and ${C}_{3}=0.0001$: (

**a**) Optimal vaccination rates. (

**b**) Optimal testing rates.

**Figure 7.**Proportions of people in each compartment obtained assuming vaccine efficacy levels $\varphi =0.50$ (red), $\varphi =0.75$ (blue), and $\varphi =0.95$ (green), with ${C}_{1}=1,{C}_{2}=0.1$, and ${C}_{3}=0.0001$: (

**a**) Susceptible people. (

**b**) Asymptomatic undetected people. (

**c**) Asymptomatic detected people. (

**d**) Symptomatic undetected people. (

**e**) Symptomatic detected people. (

**f**) Acutely symptomatic people. (

**g**) Healed people. (

**h**) Extinct people.

**Figure 8.**Optimal policies obtained assuming ${C}_{1}=0.001$ (red), ${C}_{1}=0.01$ (orange), ${C}_{1}=0.1$ (blue), and ${C}_{1}=1$ (green), with fixed parameters ${C}_{2}=0.1$ and ${C}_{3}=0.0001$: (

**a**) Optimal vaccination rates. (

**b**) Optimal testing rates.

**Figure 9.**Proportions of people in each compartment obtained assuming ${C}_{1}=0.001$ (red), ${C}_{1}=0.01$ (orange), ${C}_{1}=0.1$ (blue), and ${C}_{1}=1$ (green), while fixing parameters ${C}_{2}=0.1$ and ${C}_{3}=0.0001$: (

**a**) Susceptible people. (

**b**) Asymptomatic undetected people. (

**c**) Asymptomatic detected people. (

**d**) Symptomatic undetected people. (

**e**) Symptomatic detected people. (

**f**) Acutely symptomatic people. (

**g**) Healed people. (

**h**) Extinct people.

**Figure 10.**Optimal policies and proportions of people in each compartment obtained assuming ${C}_{2}=0.001$ (red), ${C}_{2}=0.01$ (orange), ${C}_{2}=0.1$ (blue), and ${C}_{2}=1$ (green), with fixed parameters ${C}_{1}=1$ and ${C}_{3}=0.0001$: (

**a**) Optimal vaccination rates. (

**b**) Optimal testing rates. (

**c**) Susceptible people. (

**d**) Healed people.

**Figure 11.**Optimal policies obtained assuming ${C}_{3}=1$ (red), ${C}_{3}=0.1$ (orange), ${C}_{3}=0.01$ (blue), and ${C}_{3}=0.001$ (green), with fixed parameters ${C}_{1}=1$ and ${C}_{2}=0.1$: (

**a**) Optimal vaccination rates. (

**b**) Optimal testing rates.

**Figure 12.**Proportions of people in each compartment obtained assuming ${C}_{3}=1$ (red), ${C}_{3}=0.1$ (orange), ${C}_{3}=0.01$ (blue), and ${C}_{3}=0.001$ (green), with fixed parameters ${C}_{1}=1$ and ${C}_{2}=0.1$: (

**a**) Susceptible people. (

**b**) Asymptomatic undetected people. (

**c**) Asymptomatic detected people. (

**d**) Symptomatic undetected people. (

**e**) Symptomatic detected people. (

**f**) Acutely symptomatic people. (

**g**) Healed people. (

**h**) Extinct people.

**Figure 13.**Optimal policies and proportions of people in each compartment obtained assuming ${\kappa}_{L}=0$ (red), ${\kappa}_{L}=0.025$ (blue), and ${\kappa}_{L}=0.05$ (green), with fixed parameters ${C}_{1}=1$, ${C}_{2}=0.1$, and ${C}_{3}=0.1$: (

**a**) Optimal vaccination rates. (

**b**) Optimal testing rates. (

**c**) Asymptomatic undetected people. (

**d**) Symptomatic undetected people.

**Figure 14.**Optimal policies obtained assuming immunity loss rates $\iota =0.0056$ (red), $\iota =0.0028$ (orange), $\iota =0.0014$ (blue), and $\iota =0.0009$ (green), with ${C}_{1}=1,{C}_{2}=0.1$, and ${C}_{3}=0.0001$: (

**a**) Optimal vaccination rates. (

**b**) Optimal testing rates.

**Figure 15.**Proportions of people in each compartment obtained assuming immunity loss rates $\iota =0.0056$ (red), $\iota =0.0028$ (orange), $\iota =0.0014$ (blue), and $\iota =0.0009$ (green): (

**a**) Susceptible people. (

**b**) Asymptomatic undetected people. (

**c**) Asymptomatic detected people. (

**d**) Symptomatic undetected people. (

**e**) Symptomatic detected people. (

**f**) Acutely symptomatic people. (

**g**) Healed people. (

**h**) Extinct people.

Parameter | Value | Unit |
---|---|---|

$\alpha $ | $0.330$ | people per day |

$\beta $ | $0.005$ | people per day |

$\gamma $ | $0.110$ | people per day |

$\delta $ | $0.005$ | people per day |

$\theta $ | $0.371$ | dimensionless |

$\zeta $ | $0.025$ | days${}^{-1}$ |

$\eta $ | $0.025$ | days${}^{-1}$ |

$\mu $ | $0.008$ | days${}^{-1}$ |

$\nu $ | $0.015$ | days${}^{-1}$ |

$\tau $ | $0.010$ | days${}^{-1}$ |

$\lambda $ | $0.080$ | days${}^{-1}$ |

$\kappa $ | $0.020$ | days${}^{-1}$ |

$\xi $ | $0.020$ | days${}^{-1}$ |

$\rho $ | $0.020$ | days${}^{-1}$ |

$\sigma $ | $0.010$ | days${}^{-1}$ |

$\iota $ | $0.0009,0.0014,0.0028,0.0056$ | days${}^{-1}$ |

$\varphi $ | $0.50,0.75,0.95$ | dimensionless |

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**MDPI and ACS Style**

Olivares, A.; Staffetti, E.
Optimal Control Applied to Vaccination and Testing Policies for COVID-19. *Mathematics* **2021**, *9*, 3100.
https://doi.org/10.3390/math9233100

**AMA Style**

Olivares A, Staffetti E.
Optimal Control Applied to Vaccination and Testing Policies for COVID-19. *Mathematics*. 2021; 9(23):3100.
https://doi.org/10.3390/math9233100

**Chicago/Turabian Style**

Olivares, Alberto, and Ernesto Staffetti.
2021. "Optimal Control Applied to Vaccination and Testing Policies for COVID-19" *Mathematics* 9, no. 23: 3100.
https://doi.org/10.3390/math9233100