# Analyzing the Costs and Benefits of Utilizing a Mixed-Strategy Approach in Infectious Disease Control under a Voluntary Vaccination Policy

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## Abstract

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## 1. Introduction

## 2. Model Formulation

#### 2.1. Cost–Benefit Payoff Matrix

#### 2.2. Evolutionary Dynamical Equation

#### 2.3. IB-RA (Individual-Based Risk Assessment)

#### 2.4. SB-RA (Society-Based Risk Assessment)

#### 2.5. Mutual Strategy Selection Dynamics

## 3. Result and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the model in which the population is divided into four states: susceptible $\left(S\right)$, vaccinated $\left(V\right)$, infected $\left(I\right)$, and recovered $\left(R\right)$, which applies in the epidemic season on a local time scale. On the other hand, the evolutionary decision-making process based on the Fermi pairwise game occurs globally. An individual chooses whether to vaccinate at the onset of each epidemic season based on two updated dynamics: IB-RA (individual-based risk assessment) and SB-RA (society-based risk assessment). The vaccine efficiency (VE) models determine the fraction of vaccinated and corresponding immunity systems.

**Figure 2.**The 2D heatmap of the final epidemic size (FES) is presented by varying two parameters: the $x$-axis contains the vaccination cost (${C}_{v}$), and the $y$-axis shows the vaccination efficacy $\left(\eta \right)$. In this figure, the first, second, and third rows display the result of varying the selection intensity rate (a-*) n = 0.0, (b-*) n = 0.5, and (c-*) n = 1.0. Also, the first, second, and third columns show the result of varying the benefit rate: (*-i) $B=0.0$, (*-ii) $B=0.5$, and (*-iii) $B=1.0$. Other parameters are $\beta =0.8333$, $\gamma =0.333$ [13,14].

**Figure 3.**The 2D heatmap representing the fraction of vaccination (FOV) illustrates variations in two parameters: the $x$-axis represents the vaccination cost (${C}_{v}$), while the $y$-axis represents vaccination efficacy ($\eta $). In this figure, we present results for different combinations of selection intensity rates—specifically, (a-*) $n=0.0$ in the first row, (b-*) $n=0.5$ in the second row, and (c-*) $n=1.0$ in the third row. Additionally, the first, second, and third columns depict results for different benefit rates: (*-i) $B=0.0$, (*-ii) $B=0.5$, and (*-iii) $B=1.0$. It is important to note that we have kept other parameters constant, with $\beta =0.8333$ and $\gamma =0.333$ [13,14].

**Figure 4.**The 2D heatmap, which represents the average social payoff $\left(ASP\right)$, demonstrates variations in two key parameters: the $x$-axis signifies the vaccination cost (${C}_{v}$), while the $y$-axis represents vaccination efficacy ($\eta $). In this visual representation, we showcase outcomes for various combinations of selection intensity rates, denoted as $n$, with (a-*) $n=0.0$ in the first row, (b-*) $n=0.5$ in the second row, and (c-*) $n=1.0$ in the third row. Furthermore, the first, second, and third columns display results for different benefit rates, identified as (*-i) $B=0.0$, (*-ii) $B=0.5$, and (*-iii) $B=1.0$, respectively. It is important to emphasize that we have maintained the stability of other parameters throughout, with $\beta =0.8333$and $\gamma =0.333$ [13,14].

**Figure 5.**The 2D heatmap depicting the social efficiency deficit (SED) is generated by manipulating two key parameters: the $x$-axis represents the vaccination cost (${C}_{v}$), while the $y$-axis signifies vaccination efficacy ($\eta $). Within this graphical representation, the results are organized into three rows, each presenting variation in the selection intensity rate $n$: the first row corresponds to (a-*) $n=0.0$, the second to (b-*) $n=0.5$, and the third to (c-*) $n=1.0$. Similarly, the results are arranged into three columns, each reflecting change in the benefit rate: (*-i) $\mathrm{B}=0.0$ in the first column, (*-ii) $B=0.5$ in the second column, and (*-iii) $B=1.0$ in the third column. We must note that we have maintained the constancy of other parameters throughout, with $\beta =0.8333$ and $\gamma =0.333$ [13,14].

**Figure 6.**We present a 2D heatmap illustrating two aspects: (

**A**) the deficiency of final epidemic size (${D}_{FES}$) and (

**B**) the deficiency of the fraction of vaccination (${D}_{FOV}$). These visualizations involve the manipulation of two key parameters: the $x$-axis represents the vaccination cost (${C}_{v}$), while the $y$-axis represents vaccination efficacy ($\eta $). Within this figure, you will find three rows, each showcasing the outcomes of varying the selection intensity rate (a-*) with values of $n=0.0$ in the first row, $n=0.5$ in the second row, and $n=1.0$ in the third row. Additionally, the figure features three columns, each presenting the results of varying the benefit rate: (*-i) $B=0.0$ in the first column, (*-ii) $B=0.5$ in the second column, and (*-iii) $B=1.0$ in the third column. It is important to note that we have kept other parameters constant throughout the analysis, specifically $\beta =0.8333$ and $\gamma =0.333$ [13,14].

**Figure 7.**We present a 2D heatmap representing (

**A**) individual-based deficiency of final epidemic size $\left({IB}_{D}^{FES}\right)$ and (

**B**) society-based deficiency of fraction of vaccination $\left({SB}_{D}^{FES}\right)$. These visualizations involve the manipulation of two key parameters: the $x$-axis denotes the vaccination cost (${C}_{V}$), while the $y$-axis represents vaccination efficacy ($\eta $). Within this figure, you will find three rows, each showcasing the outcomes of varying the process parameter (a-*) with values of $\theta =0.1$ in the first row, $\theta =0.5$ in the second row, and $\theta =0.9$ in the third row. Additionally, the figure features three columns, each presenting the results of varying the benefit rate: (*-i) $B=0.0$ in the first column, (*-ii) $B=0.5$ in the second column, and (*-iii) $B=1.0$ in the third column. It is important to note that we have maintained the constancy of other parameters throughout the analysis, specifically $\beta =0.8333$ and $\gamma =0.333$ [13,14].

**Figure 8.**We present a 2D heatmap that illustrates (

**A**) the deficiency of vaccination fraction at the individual level $\left({IB}_{D}^{FOV}\right)$ and (

**B**) the deficiency of vaccination fraction at the societal level $\left({SB}_{D}^{FOV}\right)$. These visualizations involve the manipulation of two essential parameters: the $x$-axis corresponds to the vaccination cost (${C}_{v}$), and the $y$-axis represents vaccination efficacy ($\eta $). Within this graphical representation, you will find three rows showcasing the outcomes of varying the process parameter (a-*) with values of $\theta =0.1$ in the first row, $\theta =0.5$ in the second row, and $\theta =0.9$ in the third row. Additionally, the figure includes three columns, each displaying the results of varying the benefit rate: (*-i)$B=0.0$ in the first column, (*-ii) $B=0.5$ in the second column, and (*-iii) $B=1.0$ in the third column. It is essential to emphasize that we have maintained the constancy of other parameters throughout the analysis, specifically $\beta =0.8333$ and $\gamma =0.333$ [13,14].

**Figure 9.**We present a 2D heatmap illustrating (

**A**) the individual-based average social payoff $\left({IB}_{D}^{ASP}\right)$ and (

**B**) the society-based average social payoff $\left({SB}_{D}^{ASP}\right)$. These visualizations involve the variation of two key parameters: the $x$-axis represents the vaccination cost (${C}_{V}$), and the $y$-axis denotes vaccination efficacy ($\eta $). The first, second, and third rows in this figure delineate the results of altering the process parameter (a-*) with values of $\theta =0.1$ in the first row, $\theta =0.5$ in the second row, and $\theta =0.9$ in the third row. Similarly, the first, second, and third columns portray the outcomes of adjusting the benefit rate: (*-i) $B=0.0$ in the first column, (*-ii) $\mathrm{B}=0.5$ in the second column, and (*-iii) $B=1.0$ in the third column. Notably, we have maintained the constancy of other parameters throughout the analysis, specifically $\beta =0.8333$ and $\gamma =0.333$ [13,14].

**Figure 10.**We present a 2D heatmap displaying (

**A**) the final epidemic size (FES), (

**B**) the fraction of vaccination (FOV), and (

**C**) the average social payoff (ASP) while varying two critical parameters: the $x$-axis represents the vaccination benefit ($B$) and the $y$-axis signifies vaccination efficacy ($\eta $). Within this graphical representation, the first, second, and third rows showcase the outcomes of altering the process parameter (a-*), with values of $\theta =0.1$ in the first row, $\theta =0.5$ in the second row, and $\theta =0.9$ in the third row. Correspondingly, the first, second, and third columns reveal the results of modifying the vaccination cost: (*-i) ${C}_{v}=0.1$ in the first column, (*-ii) ${\mathrm{C}}_{\mathrm{v}}=0.5$ in the second column, and (*-iii) ${C}_{v}=0.9$ in the third column. It is noteworthy that we have kept other parameters constant throughout the analysis, specifically $\beta =0.8333$ and $\gamma =0.333$ [13,14].

Strategy | Healthy | Infected |
---|---|---|

Vaccinated | HV (Healthy and vaccinators) | IV (Infected and vaccinators) |

$xexp[-\left(1-\eta \right){R}_{0}R\left(x,\infty \right)]$ | $x(1-exp\left[-\left(1-\eta \right){R}_{0}R\left(x,\infty \right)\right])$ | |

Non-vaccinated | SFR (Successful Free-Rider) | FFR (Failed Free-Rider) |

$(1-x)exp[-{R}_{0}R\left(x,\infty \right)]$ | $(1-x)(1-exp\left[-{R}_{0}R\left(x,\infty \right)\right])$ |

SFR | HV | FFR | IV |
---|---|---|---|

$B$ | $B-{C}_{v}$ | $-{C}_{d}$ | $-{C}_{v}-{C}_{d}$ |

SFR | HV | FFR | IV |
---|---|---|---|

$0$ | $-{C}_{v}$ | $-{C}_{d}-B$ | $-{C}_{v}-{C}_{d}-B$ |

SFR | HV | FFR | IV |
---|---|---|---|

$0$ | $-{C}_{v}/({C}_{d}+B)$ | $-1$ | $-{C}_{v}/({C}_{d}+B)-1$ |

SFR (Healthy) | HV (Healthy) | FFR (Infected) | IV (Infected) | |
---|---|---|---|---|

Payoff | $0$ | $-{C}_{r}$ | $-1$ | $-{C}_{r}-1$ |

Payoff gap | ${C}_{r}$ | |||

$1-{C}_{r}$ | ||||

${C}_{r}$ |

SFR (Healthy) | HV (Healthy) | FFR (Infected) | IV (Infected) | |
---|---|---|---|---|

Payoff | $0$ | $-\frac{{C}_{v}}{{C}_{d}+B}$ | $-1$ | $-\frac{{C}_{v}}{{C}_{d}+B}-1$ |

Payoff gap | $\frac{{C}_{v}}{{C}_{d}+B}$ | |||

$1-\frac{{C}_{v}}{{C}_{d}+B}$ | ||||

$\frac{{C}_{v}}{{C}_{d}+B}$ |

Transition Probability |
---|

Original form of PW-Fermi considering each agent; $Pr\left({s}_{i}\leftarrow {s}_{j}\right)=\frac{1}{1+\mathrm{exp}[-({\pi}_{j}-{\pi}_{i})/\kappa ]}$ |

$Pr\left(SFR\leftarrow HV\right)=\frac{1}{1+\mathrm{exp}[-(-{C}_{v}/({C}_{d}+B)-0)/\kappa ]}$ $Pr\left(FFR\leftarrow HV\right)=\frac{1}{1+\mathrm{exp}[-(-{C}_{v}/({C}_{d}+B)+1)/\kappa ]}$ $Pr\left(SFR\leftarrow IV\right)=\frac{1}{1+\mathrm{exp}[-(-{C}_{v}/({C}_{d}+B)-1-0)/\kappa ]}$ $Pr\left(FFR\leftarrow IV\right)=\frac{1}{1+\mathrm{exp}[-(-{C}_{v}/({C}_{d}+B)-1+1)/\kappa ]}$ $Pr\left(HV\leftarrow SRF\right)=\frac{1}{1+\mathrm{exp}[-(0+{C}_{v}/({C}_{d}+B))/\kappa ]}$ $Pr\left(HV\leftarrow FFR\right)=\frac{1}{1+\mathrm{exp}[-(-{1+C}_{v}/({C}_{d}+B))/\kappa ]}$ $Pr\left(IV\leftarrow SFR\right)=\frac{1}{1+\mathrm{exp}[-(0+{C}_{v}/({C}_{d}+B)+1)/\kappa ]}$ $Pr\left(IV\leftarrow FFR\right)=\frac{1}{1+\mathrm{exp}[-(1+{C}_{v}/({C}_{d}+B)+1)/\kappa ]}$ |

Transition Probability | |
---|---|

Fermi pairwise rules for SB-RA, $Pr\left({s}_{i}\leftarrow {<\pi}_{j}>\right)=\frac{1}{1+\mathrm{exp}[-({<\pi}_{j}>-{S}_{i})/\kappa ]}$ | |

$Pr\left(HV\leftarrow NV\right)=\frac{1}{1+\mathrm{exp}[-({\mathsf{\pi}}_{\mathrm{D}}+{C}_{v}/({C}_{d}+B))/\kappa ]}$ $Pr\left(IV\leftarrow NV\right)=\frac{1}{1+\mathrm{exp}[-({\mathsf{\pi}}_{\mathrm{D}}+{C}_{v}/({C}_{d}+B)+1)/\kappa ]}$ $Pr\left(SFR\leftarrow V\right)=\frac{1}{1+\mathrm{exp}[-({\mathsf{\pi}}_{C}-0)/\kappa ]}$ $Pr\left(FFR\leftarrow V\right)=\frac{1}{1+\mathrm{exp}[-({\mathsf{\pi}}_{\mathrm{C}}+1)/\kappa ]}$ |

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**MDPI and ACS Style**

Kabir, K.M.A.; Ullah, M.S.; Tanimoto, J.
Analyzing the Costs and Benefits of Utilizing a Mixed-Strategy Approach in Infectious Disease Control under a Voluntary Vaccination Policy. *Vaccines* **2023**, *11*, 1476.
https://doi.org/10.3390/vaccines11091476

**AMA Style**

Kabir KMA, Ullah MS, Tanimoto J.
Analyzing the Costs and Benefits of Utilizing a Mixed-Strategy Approach in Infectious Disease Control under a Voluntary Vaccination Policy. *Vaccines*. 2023; 11(9):1476.
https://doi.org/10.3390/vaccines11091476

**Chicago/Turabian Style**

Kabir, K. M. Ariful, Mohammad Sharif Ullah, and Jun Tanimoto.
2023. "Analyzing the Costs and Benefits of Utilizing a Mixed-Strategy Approach in Infectious Disease Control under a Voluntary Vaccination Policy" *Vaccines* 11, no. 9: 1476.
https://doi.org/10.3390/vaccines11091476