Association of Population Density and Distance to the City with the Risks of COVID-19: A Bayesian Spatial Analysis

The outbreak of Coronavirus disease-2019 (Covid-19) poses a severe threat around the world. Although several studies of modelling Covid-19 cases have been done, there appears to have been limited research into modelling Covid-19 using Bayesian hierarchical spatial models. This study aims to examine the most suitable Bayesian spatial CAR Leroux models in modelling the number of confirmed Covid-19 cases without and with covariates namely distance to the capital city and population density. Data on the number of confirmed positive cases of Covid-19 (March 20, 2020 - August 30, 2021) in 15 sub-districts in Makassar City, the number of populations, population density, and distance to the city are used. The best model selection is based on several criteria, namely Deviance Information Criteria (DIC), Watanabe Akaike Information Criteria (WAIC), residuals from Moran’s I Modification (MMI), and the 95% credible interval does not contain zero. The results showed that the best model in modelling Covid-19 is spatial CAR Leroux with hyperprior Inverse-Gamma (0.5, 0.05) model with the incorporation of distance to the capital city. It is found that there was a negative correlation between the distance to the capital city and Covid-19 risk, but the association between population density and the relative risk of Covid-19 was not statistically significant. Ujung Pandang district and Sangkarrang Island have the highest and the lowest relative risk respectively.


Introduction
The first case of coronavirus disease 2019   Although a number of studies of modelling Covid-19 cases have been done, there appears to have been limited research into modelling Covid-19 using Bayesian hierarchical spatial models. Hierarchical models include the random effect model, and generalized linear (mixed) model (GLM/GLMM) [3]. One of the advantages of Bayesian hierarchical models is that they enable for sharing of information across adjacent responses and allow the inclusion of covariates in the model [4]. Recently [5]. They found that there is a statistically significant correlation between the number of Covid-19 cases and population density, race, household income and children under 18 years old.
Bayesian hierarchical spatial BYM models using Poisson for count data was also used to analyze the relationship between positive covid-19 and socioeconomic and health [6]. They found that the proportion of populations older than 65 years, the proportion of the population with heart disease, and housing density were positively associated with Covid-19 cases. They also conclude that the proportion of black/African American populations is associated with the number of positive Covid-19 cases. The Bayesian hierarchical spatial using re-parametrization of the BYM models was also used to investigate the relationship between Covid-19 mortality and air pollution (NO2 and PM2·5) in England [7].
The association between the distance from the virus epicenter, population density, the ratio of the population older than 65 years, and the incidence of Covid-19 in Iranian provinces have been evaluated by using linear regression analysis [8]. The results show that the incidence of Covid-19 was strongly negatively associated with the distance from the virus epicenter. They also found that the ratio of the population older than 65 years was positively associated with the incidence of Covid-19 incidence. However, the association between population density and the Covid-19 incidence was not statistically significant [8].
The performance of different Bayesian spatial models such as conditional autoregressive BYM [9], CAR Leroux [10], CAR localised has been investigated and found that overall, the Leroux model performed better than the CAR BYM model [11]. The association of the population density, the distance from the virus epicenter, and the Covid-19 incidence have been done, but they used linear regression [8]. To our knowledge, the Bayesian hierarchical spatial CAR Leroux model has not been investigated in modeling Covid-19 risk. The association of the population density, the distance to the capital city, and the Covid-19 risk have not been explored yet. This paper aims to examine the most suitable Bayesian spatial CAR Leroux models in modelling the number of confirmed Cocid-19 cases without and with covariates (distance to the capital city and population density).

Covid-19 data
The number of confirmed cases of Covid-19 data from 19 March 2020 to 30 August 2021 for each district was obtained from the official website of Makassar Health Office (https://infocorona.makassar.go.id/) [1]. Population data in each district were gathered from the Badan Pusat Statistik (BPS) [12]. Population density and the distance to the capital (Ujung Pandang) which were also used in this study were gathered from the Badan Pusat Statistik (BPS) [12].

Models
One measure of disease risk is the Standardised Incidence Ratio (SIR) which is defined as the ratio of the observed counts to the expected counts. However, the SIR may be inadequate when the diseases are rare and the population is small. Bayesian models are suggested to estimate disease risk because they enable the incorporation of information from neighboring areas and covariates in the model.
In this study, the Bayesian spatial CAR Leroux [10] was used in estimating the Covid-19 risk in Makassar, Indonesia as well as quantifying the risk associated between Covid-19 and covariates namely population density and the distance of each district to the capital city. Population density is defined as the number of populations divided by the area (the number of populations per square kilometer). Bayesian spatial CAR Leroux model consists of one component, namely spatial random effect ( ) which allows with different spatial autocorrelation ( ) between zero and one.
The confirmed Covid-19 counts ( ) were modelled using the Poisson distribution with mean is the expected count ( ) times the relative risk of Covid-19 in the ith area ( ). A logarithmic transformation enables a linear, additive model regression model along with a spatial random effect ( ). All model combinations were analysed using the CARBayes package version 5.2.3 [13] in the software R version 3.6.1 [14]. The model is explained as follows.
~Poisson( ) log( ) = + β + α and = ( 1, 2, . . ., ) are the overall levels of relative risk and the coefficients of the covariates respectively. is the relative risk of Covid-19 in the ith area and , the spatial structure random effect, is defined by a conditional autoregressive (CAR) prior as follows. The spatial adjacency matrix that quantify spatial between areas and is defined using the binary neighbourhood matrix [15,16] Modified Moran's I (MMI) [17,18] and Moran's I statistics [19] for the observed data as well as for the residuals from the model were computed to quantify the spatial autocorrelation. A sensitivity analysis was also performed to investigate the effect of the prior on the estimation of posterior quantities. Four different priors on the precision terms were used namely, Inverse-Gamma (1, 0.01), the default hyperprior in CARBayes, Inverse-Gamma (0.5, 0.05), Inverse-Gamma (0.1, 0.01), and Inverse-Gamma (1, 0.1).
Model formulations and combinations of covariates were compared using the 95% posterior credible interval (considered substantive when the interval does not contain zero), and the goodness of fit measures, Deviance Information Criterion (DIC) [20], Watanabe Akaike Information Criterion (WAIC) [21] and Modified Moran's I (MMI) [17,18] for the residual from the model.

Results
The first three highest number of confirmed Covid-19 cases in Makassar city is Rappocini district (6,602), Biringkanaya district (6,166), and Tamalate district (5,497), while the first three lowest number of confirmed Covid-19 cases is Sangkarrang Island (38), Ujung Tanah (639), and Wajo (1106). The first three highest population density is Makassar district (32566), Mariso (31553), and Bontoala district The results of Bayesian Spatial CAR models for all four different hyperpriors ( ) without and with covariates for confirmed Covid-19 cases from March 20, 2020, to August 30, 2021, with four distinct options for hyperprior ( ) are given in Table 1 and demonstrate insensitivity to the choice of hyperprior. All four models indicated that distance to the capital city was considered significant as the 95% posterior credible interval for the coefficient does not include zero. However, three models found that population density was not considered significant as the 95% credible interval contain zero. Only one model indicated that the population density was significant (model with hyperprior Inverse-Gamma (0.5, 0.05). Furthermore, Bayesian Spatial CAR Leroux with hyperprior Inverse-Gamma (1, 0.1) with the inclusion of both the distance to the capital city and population density was not significant. While using the other three hyperpriors suggested that the inclusion of both the distance to the capital city and population density were significant.
The number of confirmed Covid-19 cases, population density, distance to the capital city, and the relative risk (RR) value for each district based on the best model, namely Bayesian Spatial CAR Leroux with hyperprior Inverse-Gamma (0.5, 0.05) incorporate with covariate distance to the capital city are given in Table 2.