# Risk Perception Influence on Vaccination Program on COVID-19 in Chile: A Mathematical Model

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PPEs | Personal Protective Elements |

## Appendix A

**Table A1.**Parameter values related to behavior. D = days, UN = unitless. $N={N}_{u}+{N}_{v}$. In order to not discriminate between people with different beliefs and attitudes towards vaccination, we used the same numeric values for both ${P}^{*}$ and ${P}_{v}^{*}$.

Parameters | Units | Baseline | Reference |
---|---|---|---|

b (d) | D${}^{-1}$ | 0 | Author chosen |

f | D${}^{-1}$ | 0.012 | [20] |

${f}_{v}$ | D${}^{-1}$ | 1/360 | Author chosen |

${\beta}_{a}^{*}$ (${\beta}_{i}^{*}$) | D${}^{-1}$ | $3.08*0.75*0.3$ (${\beta}_{a}*0.5$) | [17,35,36] |

$\sigma $ ($\delta $) | UN | 0.35 | [20,21,35] |

$\alpha $ (${\alpha}_{v}$) | D${}^{-1}$ | 1/5 | [17,21,35] |

${\lambda}_{a}$ (${\lambda}_{a}^{v}$) | UN | 0.2 (0.653) | [17,21] |

${\lambda}_{i}$ (${\lambda}_{i}^{v}$) | UN | 0.8 (0.347) | [17,21] |

$\varphi $ (${\varphi}_{v}$) | D${}^{-1}$ | 1/7 (1/14) | [20,35] |

$\mu $ (${\mu}_{v}$) | D${}^{-1}$ | 0.1 ($\mu \phantom{\rule{0.166667em}{0ex}}*$ 0.13) | [17,21,35] |

m (${m}_{v}$) | D${}^{-1}$ | 0.001 ($m\phantom{\rule{0.166667em}{0ex}}*$ 0.14) | [20,35] |

${\gamma}_{a}$ (${\gamma}_{a}^{v}$) | D${}^{-1}$ | 1/10 | [17,20,21,35] |

${\gamma}_{i}$ (${\gamma}_{i}^{v}$) | D${}^{-1}$ | 1/10 | [17,20,21,35] |

${\gamma}_{T}$ (${\gamma}_{T}^{v}$) | D${}^{-1}$ | 1/14 | [17,20,21,35] |

$\psi $ (${\psi}_{v}$) | D${}^{-1}$ | 1/90 | Author chosen |

${\mathsf{\Lambda}}_{1}$ (${\mathsf{\Lambda}}_{1}^{v}$) | D${}^{-1}$ | $\left[0,1\right]$ | Author chosen |

${\mathsf{\Lambda}}_{2}$ (${\mathsf{\Lambda}}_{2}^{v}$) | D${}^{-1}$ | $\left[0,1\right]$ | Author chosen |

${P}^{*}$ (${P}_{v}^{*}$) | UN | $\left[0.5,2\right]$ | Author chosen |

${N}_{u}$ (${N}_{v}$) | UN | $N={10}^{5}$ | Author chosen |

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**Figure 1.**The model is divided into two main groups: (i) the unvaccinated and (ii) the vaccinated, differentiating (ii) from (i) by their v index. The general dynamics of the two groups are similar, except for the associated values at the respective rates. It is assumed that all entering the model are unvaccinated susceptible. Susceptible people (S and ${S}_{V}$) are infected after encountering an infectious individual, either with symptoms (I and ${I}_{v}$) or without them (A and ${A}_{v}$), differing mainly by the respective transmission rates. Once the infection is contracted, it has an incubation period for some time (E or ${E}_{v}$) to become later infectious (A, I, ${A}_{v}$, or ${I}_{v}$). Asymptomatic people, after a while, may have symptoms. There are individuals with symptoms that need medical intervention (T or ${T}_{v}$), and some of them might die from the disease (m or ${m}_{v}$). The recovery of people, whether asymptomatic, with symptoms and of those who need medical intervention, are directed to the R or ${R}_{v}$ compartment depending on whether (i) or (ii), whose immunity is temporary, becoming susceptible again. Note that $\mathsf{\Omega}=({\beta}_{a}A+{\beta}_{i}I)/{N}_{u}+\delta ({\beta}_{a}{A}_{v}+{\beta}_{i}{I}_{v})/{N}_{v}$, ${\mathsf{\Omega}}_{v}=({\beta}_{a}^{v}A+{\beta}_{i}^{v}I)/{N}_{u}+\delta ({\beta}_{a}^{v}{A}_{v}+{\beta}_{i}^{v}{I}_{v})/{N}_{u}$, where ${N}_{v}$ and ${N}_{u}$ correspond to the vaccinated and unvaccinated populations, respectively. The transmission rate is dependent on the risk perception (P) so that ${\beta}_{x}={\beta}_{x}^{*}{\displaystyle \frac{{P}^{*}}{P}}$ and ${\beta}_{x}^{v}={\beta}_{x}^{*}{\displaystyle \frac{{P}_{v}^{*}}{{P}_{v}}}$, with $x\in \{a,i\}$. $N={N}_{u}+{N}_{v}$.

**Figure 2.**Dynamics of the vaccinated and unvaccinated infected, that is, adding asymptomatic patients, with symptoms, and who are undergoing health treatment. The initial values are $S\left(0\right)=$ 99,000, ${S}_{v}\left(0\right)=100$, $E\left(0\right)=500$, ${E}_{v}=100$, $A\left(0\right)=100$, ${A}_{v}\left(0\right)=85$, $I\left(0\right)=100$, ${I}_{v}\left(0\right)=10$, $T\left(0\right)=0$, ${T}_{v}\left(0\right)=0$, $R\left(0\right)=5$, ${R}_{v}\left(0\right)=0$, $P\left(0\right)={P}^{*}$, and ${P}_{v}\left(0\right)={P}_{v}^{*}$.

**Figure 3.**Dynamics of people who need some intervention after varying the vaccination rate. It is observed for the unvaccinated (T) and the vaccinated (${T}_{v}$). (

**a**) Base case. (

**b**) Application of the double dose of daily vaccines. (

**c**) Quadruplication of daily vaccine doses.

**Figure 4.**Dynamics of the behavior of the pandemic after varying the perception of risk baseline values. (

**a**) Base case. (

**b**) Halving of the rate of resistance to change (${\mathsf{\Lambda}}_{1}$), that is, people are less reluctant to behavioral change. A similar graph is obtained after doubling the speed of people’s reaction to the pandemic (${\mathsf{\Lambda}}_{2}$). (

**c**) Reduction of resistance to change by half and increase in reaction speed by double.

**Figure 5.**Dynamics of people who need some intervention after varying the baseline values of Risk Perception. (

**a**) Base case. (

**b**) Halving of the rate of resistance to change (${\mathsf{\Lambda}}_{1}$), that is, people are less reticent to change their behavior and start adopting PPE and reduce social distancing. A similar graph is obtained after doubling the speed of people’s reaction to the pandemic (${\mathsf{\Lambda}}_{2}$). (

**c**) Reduction of resistance to change by half and increase in reaction speed by double.

**Figure 6.**Accumulated deaths associated with SARS-CoV-2. (

**a**) Base case. (

**b**) Half the rate of resistance to change (${\mathsf{\Lambda}}_{1}$), that is, people are less reticent to change their behavior with PPE and reduce social distancing. A similar graph is obtained after doubling the speed of people’s reaction to the pandemic (${\mathsf{\Lambda}}_{2}$). (

**c**) Reduction of resistance to change by half and increase in reaction speed by double.

**Table 1.**Notation of compartments associated with the mathematical model (NVac. = Not vaccinated, Vac = Vaccinated).

Susceptible | Exposed | Asymptomatic | Infected | Treatment | Recovered | |
---|---|---|---|---|---|---|

NVac. | S | E | A | I | T | R |

Vac. | ${S}_{v}$ | ${E}_{v}$ | ${A}_{v}$ | ${I}_{v}$ | ${T}_{v}$ | ${R}_{v}$ |

**Table 2.**Description of parameters and parameter values related to behavior. D = days, UN = unitless. $N={N}_{u}+{N}_{v}$. In order to not discriminate between people with different beliefs and attitudes towards vaccination, we used the same numeric values for both ${P}^{*}$ and ${P}_{v}^{*}$. ${}^{*}$ The detail of the referenced values is presented in the Appendix A, Table A1.

Parameters | Description | Units |
---|---|---|

b (d) | Birth (Mortality) rate | D${}^{-1}$ |

f | Vaccination rate | D${}^{-1}$ |

${f}_{v}$ | Loss of immunity of vaccinated | D${}^{-1}$ |

${\beta}_{a}^{*}$ (${\beta}_{i}^{*}$) | Transmission rate of asymptomatic (infectious) | D${}^{-1}$ |

$\sigma $ ($\delta $) | Susceptibility (Infectivity) reduction factor | UN |

$\alpha $ (${\alpha}_{v}$) | Exit rate from latent unvaccinated (vaccinated) | D${}^{-1}$ |

to infectious unvaccinated (vaccinated) | ||

${\lambda}_{a}$ (${\lambda}_{a}^{v}$) | Proportion of latent unvaccinated (vaccinated) | UN |

that transit to asymptomatic unvaccinated | ||

(vaccinated) | ||

${\lambda}_{i}$ (${\lambda}_{i}^{v}$) | Proportion of latent unvaccinated (vaccinated) | UN |

that transit to infectious unvaccinated | ||

(vaccinated) | ||

$\varphi $ (${\varphi}_{v}$) | Transition rate from asymptomatic unvaccinated | D${}^{-1}$ |

(vaccinated) to infectious unvaccinated (vaccinated) | ||

$\mu $ (${\mu}_{v}$) | Transition rate of people unvaccinated (vaccinated) | D${}^{-1}$ |

needing medical intervention | ||

m (${m}_{v}$) | Disease induced death rate of unvaccinated (vaccinated) | D${}^{-1}$ |

${\gamma}_{a}$ (${\gamma}_{a}^{v}$) | Recovered rate of asymptomatic unvaccinated (vaccinated) | D${}^{-1}$ |

${\gamma}_{i}$ (${\gamma}_{i}^{v}$) | Recovered rate of infectious unvaccinated (vaccinated) | D${}^{-1}$ |

${\gamma}_{T}$ (${\gamma}_{T}^{v}$) | Recovery rate from treatment of people | D${}^{-1}$ |

unvaccinated (vaccinated) | ||

$\psi $ (${\psi}_{v}$) | Natural immunity loss rate of people | D${}^{-1}$ |

unvaccinated (vaccinated) | ||

${\mathsf{\Lambda}}_{1}$ (${\mathsf{\Lambda}}_{1}^{v}$) | Rate of resistance to behavioral change those | D${}^{-1}$ |

unvaccinated (vaccinated) | ||

${\mathsf{\Lambda}}_{2}$ (${\mathsf{\Lambda}}_{2}^{v}$) | Reaction rate to behavior change those | D${}^{-1}$ |

unvaccinated (vaccinated) | ||

${P}^{*}$ (${P}_{v}^{*}$) | quantified average risk perception of those | UN |

unvaccinated (vaccinated) | ||

${N}_{u}$ (${N}_{v}$) | Unvaccinated (vaccinated) population | UN |

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**MDPI and ACS Style**

Gutiérrez-Jara, J.P.; Saracini, C.
Risk Perception Influence on Vaccination Program on COVID-19 in Chile: A Mathematical Model. *Int. J. Environ. Res. Public Health* **2022**, *19*, 2022.
https://doi.org/10.3390/ijerph19042022

**AMA Style**

Gutiérrez-Jara JP, Saracini C.
Risk Perception Influence on Vaccination Program on COVID-19 in Chile: A Mathematical Model. *International Journal of Environmental Research and Public Health*. 2022; 19(4):2022.
https://doi.org/10.3390/ijerph19042022

**Chicago/Turabian Style**

Gutiérrez-Jara, Juan Pablo, and Chiara Saracini.
2022. "Risk Perception Influence on Vaccination Program on COVID-19 in Chile: A Mathematical Model" *International Journal of Environmental Research and Public Health* 19, no. 4: 2022.
https://doi.org/10.3390/ijerph19042022