# Vaccination Games with Peer Effects in a Heterogeneous Hospital Worker Population

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## Abstract

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## 1. Introduction

## 2. Example: A Simple Vaccination Game

Agent j | |||

Vac | No Vac | ||

Agent i | Vac | −c_{vac} | −c_{vac} |

No Vac | 0 | −π_{j}αc_{inf} |

**p**is the vector of vaccination probabilities for all agents in the population, except i. The no vaccination Nash equilibrium has a simple structure, even with many players; if the cost of the vaccine is too large relative to the probability of infection and the cost of infection, no one chooses to be vaccinated: ${c}_{vac}>{\pi}_{i}(\Gamma ,\eta ,0,\mathbf{0}){c}_{inf}$ for all i, where ${\pi}_{i}(\Gamma ,\eta ,0,\mathbf{0})$ indicates the probability of infection when agent i and all other agents in the population choose a vaccination probability of zero.

_{−i}## 3. Heterogeneous Contacts

#### 3.1. A Model of Heterogeneous Contacts

## 4. An Example Vaccination Game in a Heterogeneous Population

#### 4.1. Data

Category | Number of Employees |
---|---|

Floor Nurse | 804 |

Food Service | 456 |

Housekeeper | 356 |

IC Nurse | 190 |

Nurse Asst | 386 |

Patient | 482 |

Pharmacist | 276 |

Phlebotomist | 74 |

Physical/Occupational Therapist | 90 |

Residents/Fellows/Med Students | 666 |

Respiratory Therapist | 108 |

Social Worker | 114 |

Staff Physician | 760 |

Transporter | 108 |

Unit Clerk | 126 |

X-Ray Technician | 236 |

Total | 5,232 |

F N | Food | HK | IC N | N Asst | Pat | Pharm | Phleb | P Ther | Res | R Ther | SW | Stf Phys | Trans | UC | X-Ray | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

F N | 21 | 0 | 1 | 1 | 10 | 36 | 1 | 0 | 3 | 3 | 1 | 1 | 2 | 1 | 7 | 1 |

Food | 3 | 6 | 1 | 2 | 2 | 14 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 4 | 0 |

HK | 4 | 0 | 4 | 1 | 2 | 2 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 |

IC N | 1 | 0 | 1 | 21 | 1 | 23 | 1 | 0 | 1 | 3 | 3 | 1 | 4 | 0 | 1 | 1 |

N Asst | 11 | 1 | 1 | 5 | 4 | 25 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 5 | 0 |

Pharm | 7 | 0 | 0 | 2 | 1 | 3 | 2 | 0 | 0 | 6 | 0 | 1 | 3 | 0 | 2 | 0 |

Phleb | 3 | 0 | 0 | 0 | 1 | 40 | 1 | 2 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 |

P Ther | 9 | 1 | 1 | 4 | 3 | 15 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 3 | 0 |

Res | 5 | 1 | 0 | 9 | 1 | 11 | 1 | 0 | 0 | 21 | 2 | 1 | 8 | 0 | 1 | 1 |

R Ther | 5 | 1 | 1 | 16 | 1 | 12 | 1 | 0 | 1 | 6 | 16 | 0 | 2 | 0 | 2 | 2 |

SW | 17 | 0 | 1 | 1 | 1 | 4 | 1 | 0 | 2 | 6 | 1 | 2 | 3 | 0 | 3 | 0 |

Stf Phys | 3 | 0 | 0 | 3 | 1 | 11 | 1 | 0 | 1 | 26 | 0 | 1 | 3 | 0 | 1 | 1 |

Trans | 3 | 0 | 1 | 0 | 1 | 14 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 4 | 1 | 0 |

UC | 32 | 1 | 1 | 8 | 9 | 3 | 1 | 0 | 3 | 10 | 1 | 2 | 3 | 1 | 2 | 1 |

X-Ray | 4 | 0 | 1 | 4 | 1 | 19 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 18 |

#### 4.2. Simulating an Epidemic

**Table 3.**Regression coefficients. The values reported in the table are the regression coefficients of the OLS regression with the percentage infected of the group in the row as the dependent variable and the percentage vaccinated of the group listed in the column as independent variables. The first column represents the intercept coefficient from the specified regression.

${\beta}_{h}$ | Direct Care (${\beta}_{h,1}$) | SocWrk/Clrk(${\beta}_{h,2}$) | Remaining (${\beta}_{h,3}$) | |
---|---|---|---|---|

Direct Care | 0.434 | −0.542 | −0.022 | −0.047 |

SocWrk/Clrk | 0.422 | −0.308 | −0.305 | −0.045 |

Remaining | 0.225 | −0.146 | −0.012 | −0.175 |

**Table 4.**The decrease in the number of individuals infected of the row group from vaccinating one additional individual from the column group.

Direct Care | SocWrk/Clrk | Remaining | |
---|---|---|---|

Direct Care | 0.542 | 0.282 | 0.101 |

SocWrk/Clrk | 0.024 | 0.305 | 0.008 |

Remaining | 0.068 | 0.072 | 0.175 |

**Figure 1.**Plot of the number of individuals infected as a function of the total number of vaccinations performed in all groups. A trend line is plotted vs. the data.

**Table 5.**The decrease in the number of individuals infected of the row group from vaccinating one additional individual from the column group. Thirty percent of the population is immune at the start of the epidemic, resulting in a lower marginal effect of vaccinations for all groups.

Direct Care | SocWrk/Clrk | Remaining | |
---|---|---|---|

Direct Care | 0.123 | 0.051 | 0.032 |

SocWrk/Clrk | 0.005 | 0.070 | 0.002 |

Remaining | 0.012 | 0.000 | 0.046 |

#### 4.3. An Example Equilibrium

**Figure 2.**Mixed strategy equilibrium vaccination choices as a function of the ratio of vaccination and infection costs.

## 5. Peer Effects

Agent j | |||

Vac | No Vac | ||

Agent i | Vac | −c_{vac} + τ | −c_{vac} |

No Vac | 0 | −π_{j}αc_{inf} + τ |

#### 5.1. Nash Equilibrium with Peer Effects in a Two-Person Game

#### 5.2. Peer Effects in a Multi-Player Heterogeneous Contact Game

#### 5.3. Peer Effects Example

**Table 6.**Observed hospital worker contact shares. The values in the table represent the fraction of contacts of the row group that come from the column group.

Direct Care | SocWrk/Clrk | Remaining | |
---|---|---|---|

Direct Care | 0.68 | 0.27 | 0.05 |

SocWrk/Clrk | 0.66 | 0.29 | 0.05 |

Remaining | 0.34 | 0.27 | 0.39 |

**Figure 3.**Mixed strategy equilibrium vaccination choices as a function of the size of the peer effects. In this figure, λ is small, $\text{\lambda}=0.05$. All groups increase their equilibrium vaccination rate as peer effects increase.

**Figure 4.**Mixed strategy equilibrium vaccination choices as a function of the size of the peer effects. In this figure, λ is large, $\text{\lambda}=0.15$. All groups decrease their equilibrium vaccination rate as peer effects increase.

**Figure 5.**Mixed strategy equilibrium vaccination choices as a function of the size of the peer effects. In this figure, λ is an intermediate value, $\text{\lambda}=0.10$. At this level of λ, the direct care and social worker/clerk groups increase their equilibrium vaccination rate, and the “remaining” group decreases their equilibrium vaccination rate as peer effects increase.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Tassier, T.; Polgreen, P.; Segre, A.
Vaccination Games with Peer Effects in a Heterogeneous Hospital Worker Population. *Adm. Sci.* **2015**, *5*, 2-26.
https://doi.org/10.3390/admsci5010002

**AMA Style**

Tassier T, Polgreen P, Segre A.
Vaccination Games with Peer Effects in a Heterogeneous Hospital Worker Population. *Administrative Sciences*. 2015; 5(1):2-26.
https://doi.org/10.3390/admsci5010002

**Chicago/Turabian Style**

Tassier, Troy, Philip Polgreen, and Alberto Segre.
2015. "Vaccination Games with Peer Effects in a Heterogeneous Hospital Worker Population" *Administrative Sciences* 5, no. 1: 2-26.
https://doi.org/10.3390/admsci5010002