Mathematical Model of the Central Lombok Regency People’s Interest towards the COVID-19 Vaccination

ABSTRACT

hesitancy can reduce vaccination coverage, inhibit the formation of herd immunity and increase the transmission of COVID-19 . Herd immunity will be formed only if vaccination coverage is high and evenly distributed throughout region (Or Caspi, Michael J. Smart, 2020).
Data from the Ministry of Health of the Republic of Indonesia as of January 6, 2022, shown that the number of recipients of the complete COVID-19 vaccine (first and second doses) in Central Lombok Regency is still at 60.95%. This number has not reached the minimum value for the formation of herd immunity, which is 67% (Randolph & Barreiro, 2020). According to (Kadkhoda, 2021), for a high level of security this figure must reach 84% to 90%. This gives an indication of the lack of interest in the people of Central Lombok Regency to be vaccinated against COVID-19. Therefore, it is important to investigate what factors influence the public's interest in the COVID-19 vaccination and how much influence it has.
Many factors could affect the interest of public to get COVID-19 vaccination. The study have assessed the role of perceived risk, efficacy of vaccine, amount of information about vaccine, and types of occupation on acceptance of vaccination (Harapan et al., 2016), , (Lazarus et al., 2021). In the research that has been conducted by , attitude factors, perceptions about the risk of the COVID-19 pandemic, and the perceived benefits of the vaccine have a positive effect on people's intentions to get the COVID-19 vaccine. In another study, the factors that influence a person's willingness to accept pandemic prevention efforts are group beliefs, self-efficacy, perceived risk, knowledge about the pandemic .
The results of several previous research indicate that public interest or acceptance for COVID-19 vaccination in several regions in Indonesia is still lacking. Sutriyawan & Hidayatullah (2021) in their research explained that most people (54.1%) did not want to be vaccinated. The public's willingness to be vaccinated against COVID-19 is influenced by public knowledge, attitudes, and perceptions. In another study, Olivina (2021) found several factors that caused a lack of interest in the Indonesian people to vaccinate against COVID-19, including the limited information received regarding the types of vaccines, the number of vaccines available and the safety of the vaccine itself.
In addition, research on the investigation of factors that influence the acceptance of COVID-19 vaccination using a mathematical model has been carried out by (Faturohman et al., 2021). In his research, the Technology Acceptance Model (TAM) was used to determine the factors that influence the acceptance of COVID-19 vaccination in Indonesia. The results showed that a high perceived benefit significantly increased the acceptance of the COVID-19 vaccine and a high perceived ease of use significantly increased the perceived benefit. Religiosity and the abundance of information about COVID-19 have no significant effect on vaccine acceptance. Other study analyzed factors that influence individual's intention to use COVID-19 vaccine with an adapted Cognitive-Affective-Normative (CAN) model. The result show that vaccine efficacy is the most important factor that influence individual's intention to have COVID-19 vaccination followed by social influence (Pelegrín-Borondo et al., 2021). Furthermore, with a different model approach, namely Partial Least Square Structural Equation Modelling (PLS-SEM), (Akther & Nur, 2022) found that belief in conspiracy theory decreases people's interest to being vaccinated of COVID-19. Meanwhile, individual awareness, perceived benefits of COVID-19 vaccination and perceived ease of obtaining COVID-19 vaccines have a positive impact on attitude and acceptance of COVID-19 vaccination.
These three studies used a mathematical model approach to analyze the factors of public acceptance of COVID-19 vaccination, but it was different with this study in terms of the model approach used and the factors analyzed. This study aims to obtain a mathematical model of the interest of the people of Central Lombok Regency towards COVID-19 vaccination and to investigate which of the factors of ease of getting vaccinated, trust in the effectiveness of COVID-19 vaccination and trust in the government factors have the most influence on the interests of the people of Central Lombok Tengah towards COVID-19 vaccination. The model proposed in this study uses multiple linear regression approach with stepwise method to represent the interest of Central Lombok Regency in the COVID-19 vaccination. Validity and reliability of instrument testing is done using R program, as well as regression analysis and assumptions testing.

B. METHODS
The method used in this research is quantitative descriptive and survey method. We used 332 sample size, this amount has exceeded the minimum recommended for sample size with the sample-to-variable ratio rule. Ratio 15:1 or 20:1 are preferred to a minimum observation to variable (Hair et al., 2018), (Memon et al., 2020). Furthermore, this sample included people in Central Lombok Regency aged 12-70 years. Taking this age range based on the data that since July 2021 the process of COVID-19 vaccinating children aged 12 years has started (Toharudin et al., 2021) and the elderly (60 years and older) is one of priorities of COVID-19 vaccination (Paloyo et al., 2022). The data collection technique used a questionnaire to determine the interest of the people of Central Lombok Regency towards the COVID-19 vaccination which was distributed offline and online.
The variables in this study consisted of three independent variables and one dependent variable. The independent variables include the ease of getting vaccinated (X1), the effectiveness of the vaccine (X2), and trust in the government (X3), while the dependent variable in this study is the variable of interest in COVID-19 vaccination (Y). The following shows these variables and the question indicators for each variable, as shown in Table 1. 1. There is a belief that COVID-19 vaccination can protect vaccine recipients from the risk of infection with COVID-19 2. There is a belief that COVID-19 vaccination can reduce the transmission rate of COVID-19 3 Trust in Government (X3) 1. There is a belief that the government is capable of overcoming the COVID-19 pandemic

No. Variables Question Indicators
2. There is a support for any government policy to overcome the COVID-19 pandemic 3. There is a support for government's COVID-19 vaccination program to overcome the COVID-19 pandemic 4. There is a willingness to accept COVID-19 vaccine if provided by the government 4 Interest in COVID-19 vaccination (Y) 1. There is a willingness to accept the complete COVID-19 vaccine voluntarily 2. There is a willingness to invite family/friends who have not received the vaccine to be willing to receive the complete COVID-19 vaccine Testing the validity of each item of the questionnaire statement uses a significance value, while testing the reliability of the questionnaire is based on the Cronbach's Alpha coefficient value ( = 0.05). Cronbach's Alpha of 0.6-0.7 indicates an acceptable level of reliability, and 0.8 or greater a very good level (Ursachi et al., 2015). The intervals of reliability level completely is described by (Taber, 2018), as shown in Table 2. Mathematical modeling using multiple linear regression analysis approach. The data analysis was done using R software and it was carried out in descriptive statistics, analytical statistics (validity and reliability test, assumptions test), and multiple regression linear to determine impact of independent variables on the variances of dependent variable.

C. RESULT AND DISCUSSION 1. Description of the Respondents Characteristic
Characteristics of the respondents in this study collected through questionnaires include age group, gender, occupation, last education and history of comorbid disease. Respondent data by age group are presented in Table 3. Based on Table 1, it can be seen that the respondents are dominated by the age group of 12-17 years. Furthermore, the respondent's data by gender can be seen in Table 4. The characteristics of respondents by occupation can be seen in Table 5. Based on the last education, the characteristics of the respondents can be seen in Table 6. All respondents in this study came from 8 districts in Central Lombok Regency as for the distribution is presented in Table 7. In addition, as many as nine out of 332 respondents had a history of comorbid or congenital diseases that could worsen health conditions if exposed to COVID-19.

Validity and Reliability Test
The results of validity test to each item of questionaire by using R is significant on P =0.000 which less than = 0.05, it means that each item of questionare is valid. Cronbach's Alpha value of 0.88 as shown in Table 8 is overall test reliability score. Furthermore, we obtain the reliability interval for 95% confidence ie 0.86 -0.9, The following can be concluded that overall this instrument is reliable, as shown in Table 8. In addition, Cronbach's Alpha value for each item of questionaire is displayed on the Table 9. These reliability scores indicating that each item of questionaire is reliable, as shown in Table 9.

Mathematical Model Building
We used a multiple linear regression approach to build a mathematical model of Central Lombok Regency people's interest in COVID-19 vaccination. At current stage, stepwise method for multiple regression analysis has been used to evaluate the collective role of the independent variables on the dependent variables. Stepwise method is a procedure to build a model in successive steps, and predictors can be added or deleted at each step (Aljarrah et al., 2017). This method introduced the most powerful variables into the regression model one at a time and this continues until the significance test error reaches to 5% (Arayesh, 2015). Initially, we considered the correlation between independent variables and dependent variables that can be seen in Table 10. According to Table 6 can be said that there is a relationship between the ease of getting vaccines (X1), trust in the vaccine effectiveness (X2), trust in the government (X3) with interest in the COVID-19 vaccination (Y). After considering the correlation between independent variables and dependent variables, fourth variables were introduced into the multiple regression equation, respectively as follows: a.
Step 1: According to Table 6, variable of X3 has greatest correlation with variable of Y, that is r=0.704. So, the trust in the government (X3) is the first variable was entered into the regression equation. The value of F obtained from variation analysis is significant at P=0.000 level. It means that regression between variable of X3 and variabel of Y is significant. On the other hand, the coefficient of partial regression correlation is significant at p=0.000 that less than = 0.05. So, variable of X3 is fixed in the regression equation. In addition, the coefficient of determination and adjusted determination coefficient were calculated as equal to R 2 =0.496 and R 2 adj=0.494 respectively. It can be said that trust in the government variable alone contributes to about 49.4% of variations in the dependent variable of interest in the COVID-19 vaccination. The regression equation of the first step is: = 0.571 + 0.490 3 (1) b.
Step 2: According to Table 6, variable of X2 is the second greatest correlation with variable of Y, that is r=0.499. So, the trust in the vaccine effectiveness is the second variable was entered into the regression equation. The value of F obtained from variation analysis is significant at p=0.000 level. It means that regression between variable of X2, X3 and variabel of Y is significant. On the other hand, both of the coefficient of partial regression correlation for X2 and X3 is significant at p=0.000 that less than = 0.05. So, variable of X2 and X3 is fixed in the regression equation. In addition, the coefficient of determination and adjusted determination coefficient were calculated as equal to R 2 =0.520 and R 2 adj=0.517 respectively. It can be said that the trust of people in the vaccine effectiveness and the trust of people in the government variable contributes to about 51.7% of variations in the dependent variable of interest in the COVID-19 vaccination. The regression equation of the second step is: = −1.258 + 0.425 3 + 0.232 2 (2) c.
Step 3: According to Table 6, the last variable of X1 has a correlation with variable of Y, that is r=0.262. So, the trust in the ease of getting vaccines (X1) is the last variable was entered into the regression equation. The value of F obtained from variation analysis is significant at p=0.000 level. It means that regression between variable of X2, X3 and variabel of Y is significant. On the other hand, the coefficient of partial regression correlation for X1 is significant at p=0.694 that greater than = 0.05. It means that there is no effect on the ease of getting vaccines for the interest in the COVID-19 vaccination. In addition, the coefficient of determination and adjusted determination coefficient were calculated as equal to R 2 =0.520 and R 2 adj=0.516 respectively. It can be seen that the addition of variable X1 causes decrease in the value of adjusted determination coefficient by 0.01. So, the variable X1 should be removed from regression equation. The regression equation of the third step is: The following are the results of the regression analysis, as shown in Table 11 and Table  12.

a. Normality Test
In multiple linear regression, normality test on the data is performed on the residual variabel and determined distribution of the residuals whether normally distributed or not. Residuals are the difference between a predicted and observed quantity utilizing a particular mathematical model (Shamaan et al., 2015). Data is normally distributed if the significant value is greater or equal to = 0.05, otherwise data isn't normally distributed if the significant value is less than 0.05 (Himmah & Kaestria, 2022). This study used Kolmogorov-Smirnov as the proper residual variable normality test for data sample greater than 50 (Mishra et al., 2019). The distribution of residual variable described by histogram with normality curve and normal Q-Q plot as shown in Figure 1 and Figure 2. The results of normality test for residual variable with R is p-value=0.0658. It is greater than = 0.05, so we can conclude that the residual variable is normally distributed. b. Multicollinearity Test The indication of a linear correlation between independent variables can be seen by the results of multicollinearity test in the regression model. In this study, a multicollinearity test was carried out to determine the existence of multicollinearity between the variables of the trust in the vaccine effectiveness (X2) and trust in the government (X3). The existence of multicollinearity is determined by the value of Variance Inflation Factor (VIF). The regression model doesn't have a multicollinearity problem if the VIF less than 10 (Shrestha, 2020). The value of VIF for both of X2 and X3 as the results of multicollinearity test using R is 1.374732. All values are less than 10, no multicollinearity problem between independent variables in the regression model is decided. c. Heteroscedasticity Test Heteroscedasticity test was carried out to detect an inequality of variance from residuals in each observation on the regression model (Halunga et al., 2017). In this study, authors use Breusch Pagan Godfrey test. There is a heteroscedasticity if p-value is less than 0.05. By using R, we obtained p-value=0.41, it means that no variance inequality from residual is found.
This study suggested that Central Lombok Regency people's interest towards COVID-19 vaccination is influenced by their trust in the effectiveness of vaccination and trust in the government. It means that their interest to vaccinate against COVID-19 will increase if the government is able to maintain public trust especially to conquer pandemic of COVID-19 and make an effort to achieve the effectiveness of vaccination. This provides benefits for the government to determine the right strategy so that people's interest in receiving vaccines increases. The strategy includes administering the COVID-19 vaccine which has been recognized for its efficacy, such as Pfizer, BioNTech, and Moderna (Patel et al., 2022). Also, people are willing to accept COVID-19 vaccines only if they are provided by the government, so the government needs to ensure that the availability of these vaccines is sufficient and free of charge. Furthermore, the government should educate the public by mass campaign about the importance of the COVID-19 vaccination, their benefits for individual and community also.

D. CONCLUSION AND SUGGESTIONS
Our data suggest that the trust of people in the effectiveness of COVID-19 vaccination and the trust of people in the government contributes influence the people's interest for getting COVID-19 vaccination. Therefore, we recommend the government of Central Lombok Regency focus on providing good efficacious vaccines of COVID-19 and conduct mass campaigns on the importance of COVID-19 vaccination.