# Vaccination Schedule under Conditions of Limited Vaccine Production Rate

^{*}

## Abstract

**:**

## 1. Introduction and Background

- the problem of distributing protective actions (PDPA)–(pessimistic), which assumes high availability of vaccines and a constant probability of disease in each social and occupational group that does not decrease even if the number of vaccinated people increases;
- the problem of distributing protective actions with a herd immunity threshold(PDPAHIT)–(optimistic) which assumes high availability of vaccines and a variable probability of disease in each social and occupational group that changes as the threshold of herd immunity is reached.

#### Positioning of the Paper

## 2. Materials and Methods

- Direct level—lowering the probability of severe disease for a vaccinated individual. All vaccinated individuals reduce the overall probability of fatal cases in the region.
- Indirect level—lowering the probability of infection for vaccinated individuals. The vaccine minimizes the probability of infection so that a growing group of vaccinated individuals lowers the probability of infection.

- ${P}_{{S}_{zg}}$— the probability of severe course of the disease,
- ${P}_{{S}_{zk}}$— the probability of infection in the vaccinated individual,
- ${S}_{zg}$—the effectiveness of the vaccine against the severe course of the disease,
- ${S}_{zk}$—the effectiveness of the vaccine against the infection,
- ${P}_{zg}$—the probability of death of a given individual, e.g., depending on the age,
- ${P}_{zk}$—the probability of an infection in a given area during an active vaccination program,
- Z—the percentage of vaccinated individuals expressed as a fraction of the total population,
- ${P}_{z}$—the probability of infection in a given area, e.g., depending on the level of urbanization.

#### 2.1. The Problem of Distributing Protective Actions on the Example of Vaccination

#### 2.2. The Problem of Distributing Protective Actions with a Herd Immunity Threshold

## 3. Results

**Social groups**—the first issue is to identify social groups in a given territory with a constant probability of infection. In the presented example, the territorial division of the country was used, and the probability of infection was related to the population density, which is presented in Table 4.

**Available vaccines**—these are characterized only by the probability of preventing infection. This data for individual vaccines are presented in Table 5.

**Vaccine delivery schedule**—the planned delivery schedule for each vaccine type is shown in Table 6.

- “District 1” has a medium-sized population (2,901,225 residents) and one of the lowest probabilities of infection (0.01);
- “District 10” has a relatively small population (1,181,533.00 inhabitants) and one of the highest probabilities of infection (0.09).

- “District 1”—329,623 people
- “District 10”—232,946 people

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DMU | Decision Making Unit |

DSS | Decision Support System |

MIP | Mix-Integer Programming |

NPI | Non-Pharmaceutical Interventions |

PDPA | Problem of Distributing Protective Actions |

PDPAHIT | Problem of Distributing Protective Actions with a Herd Immunity Threshold |

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**Table 1.**The main factors underlying severe COVID-19 infection [10].

Age Range [Years] | Cases | Hospitalization | Death |
---|---|---|---|

0–4 | <1x | 2x | 2x |

5–17 | Reference group | Reference group | Reference group |

18–29 | 2x | 6x | 10x |

30–39 | 2x | 10x | 45x |

40–49 | 2x | 15x | 130x |

50–64 | 2x | 25x | 440x |

65–74 | 1x | 40x | 1300x |

75–84 | 1x | 65x | 3200x |

85+ | 2x | 95x | 8700x |

Characteristics | Vaccines | |||
---|---|---|---|---|

Pfizer | Moderna | AstraZeneca | JohnsonAndJohnson | |

Number of doses | 2 | 2 | 2 | 1 |

Vaccine efficacy against COVID-19 | 0.950 | 0.941 | 0.595 | 0.669 |

Vaccine efficacy against severe COVID-19 | no data | 1 | 1 | 0.854 |

Sets | ||

$\mathcal{T}$ | = | the set of identical consecutive planning periods; |

$\mathcal{V}$ | = | the set of available protective measures (e.g., different preparations) to prevent an adverse phenomenon (e.g., death, disease); |

$\mathcal{R}$ | = | the set of homogeneous social groups defined based on a selected criterion (e.g., age, occupation, place of residence); |

Parameters | ||

M | – | a sufficiently large constant; |

${d}_{r}$ | – | the size of the group r; |

${p}_{r}^{R}$ | – | the probability of occurrence of an adverse phenomenon in a person in a given group r not covered by protective measures; |

${p}_{v}^{V}$ | – | the probability of occurrence of an adverse phenomenon in an individual covered by protective action v; |

${b}_{vt}$ | – | maximum available number of units of protective action v in period t; |

${C}_{t}$ | – | maximum available number of units of all protective actions in period t; |

f | – | the percentage of applied protective actions ensuring collective immunity of the community to the adverse phenomenon; |

Decision Variables | ||

${x}_{rvt}$ | – | the number of individuals subjected to the protective action v in group r in period t; |

${I}_{rt}$ | – | the expected value of the number of individuals in group r subject to the adverse phenomenon in period t; |

${y}_{rt}$ | = | 1 if group r achieved herd protection in period t, else 0 |

Name | Population Size | pR |
---|---|---|

District 1 | 2,901,225 | 0.01 |

District 2 | 2,077,775 | 0.06 |

District 3 | 2,117,619 | 0.05 |

District 4 | 1,014,548 | 0.08 |

District 5 | 2,466,322 | 0.03 |

District 6 | 3,400,577 | 0.03 |

District 7 | 5,403,412 | 0.07 |

District 8 | 986,506 | 0.04 |

District 9 | 2,129,015 | 0.09 |

District 10 | 1,181,533 | 0.09 |

District 11 | 2,333,523 | 0.07 |

District 12 | 4,533,565 | 0.04 |

District 13 | 1,241,546 | 0.06 |

District 14 | 1,428,983 | 0.09 |

District 15 | 3,493,969 | 0.06 |

District 16 | 1,701,030 | 0.09 |

Name | pR |
---|---|

Vaccine 1 | 0.950 |

Vaccine 2 | 0.941 |

Vaccine 3 | 0.595 |

Vaccine 4 | 0.669 |

Period | Vaccine 1 | Vaccine 2 | Vaccine 3 | Vaccine 4 |
---|---|---|---|---|

0 | 873,000 | 0 | 172,000 | 0 |

1 | 873,000 | 204,000 | 161,000 | 0 |

2 | 873,000 | 0 | 268,000 | 300,000 |

3 | 873,000 | 287,000 | 765,000 | 0 |

4 | 873,000 | 0 | 172,000 | 0 |

5 | 873,000 | 204,000 | 161,000 | 0 |

6 | 873,000 | 0 | 268,000 | 300,000 |

7 | 873,000 | 287,000 | 765,000 | 0 |

8 | 873,000 | 0 | 172,000 | 0 |

9 | 873,000 | 204,000 | 161,000 | 0 |

10 | 873,000 | 0 | 268,000 | 300,000 |

11 | 873,000 | 287,000 | 765,000 | 0 |

Period | Vaccine 1 | Vaccine 2 | Vaccine 3 | Vaccine 4 |
---|---|---|---|---|

0 | 172,000 | 0 | 0 | 0 |

1 | 0 | 0 | 204,000 | 0 |

2 | 0 | 0 | 0 | 0 |

3 | 92,911 | 0 | 0 | 630,527 |

4 | 0 | 0 | 0 | 0 |

5 | 0 | 0 | 0 | 0 |

6 | 0 | 0 | 0 | 0 |

7 | 0 | 0 | 0 | 0 |

8 | 0 | 0 | 0 | 0 |

9 | 0 | 0 | 0 | 0 |

10 | 0 | 0 | 0 | 0 |

11 | 0 | 0 | 0 | 0 |

Period | Vaccine 1 | Vaccine 2 | Vaccine 3 | Vaccine 4 |
---|---|---|---|---|

0 | 172,000 | 0 | 0 | 714,150 |

1 | 0 | 0 | 0 | 0 |

2 | 0 | 0 | 0 | 0 |

3 | 0 | 0 | 0 | 0 |

4 | 0 | 0 | 0 | 0 |

5 | 0 | 0 | 0 | 0 |

6 | 0 | 0 | 0 | 0 |

7 | 0 | 0 | 0 | 0 |

8 | 0 | 0 | 0 | 0 |

9 | 0 | 0 | 0 | 0 |

10 | 0 | 0 | 0 | 0 |

11 | 0 | 0 | 0 | 0 |

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**MDPI and ACS Style**

Książek, R.; Kapłan, R.; Gdowska, K.; Łebkowski, P.
Vaccination Schedule under Conditions of Limited Vaccine Production Rate. *Vaccines* **2022**, *10*, 116.
https://doi.org/10.3390/vaccines10010116

**AMA Style**

Książek R, Kapłan R, Gdowska K, Łebkowski P.
Vaccination Schedule under Conditions of Limited Vaccine Production Rate. *Vaccines*. 2022; 10(1):116.
https://doi.org/10.3390/vaccines10010116

**Chicago/Turabian Style**

Książek, Roger, Radosław Kapłan, Katarzyna Gdowska, and Piotr Łebkowski.
2022. "Vaccination Schedule under Conditions of Limited Vaccine Production Rate" *Vaccines* 10, no. 1: 116.
https://doi.org/10.3390/vaccines10010116