# A Review of Bayesian Spatiotemporal Models in Spatial Epidemiology

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## Abstract

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## 1. Introduction

- To assess the anticipated value and uncertainty of a specific outcome variable at a defined spatial point throughout the observation period.
- To forecast the expected value of an outcome variable at a specific location.
- To identify evolving patterns of diseases that persist or undergo predictable changes over time and across various spatial units.
- To analyse the influence of environmental factors on the spatiotemporal dynamics of disease.

## 2. Literature Review

## 3. Bayesian Spatiotemporal Models

#### 3.1. Probability Distribution in Spatial Epidemiology

#### 3.2. Models

#### 3.2.1. General Effect Model

- Intercept overall rate $\mu $;
- Spatial group ${A}_{i}$;
- Temporal group ${B}_{j}$;
- Space-time interaction group ${C}_{ij}$.

#### 3.2.2. Infectious Disease Model

#### 3.2.3. Environmental Covariates

#### 3.2.4. Prior Distribution

#### 3.3. Bayesian Inference

#### 3.4. Computational Algorithms

#### 3.4.1. Markov Chain Monte Carlo

#### 3.4.2. Integrated Nested Laplace Approximation

#### 3.5. Model Fit Criteria

## 4. Early-Stage Modelling Work

#### 4.1. Data Collection

#### 4.1.1. Epidemic Data

#### 4.1.2. Geographical and Socioeconomic Data

#### 4.2. Data Preprocessing

#### 4.2.1. Preprocessing for Data Integrity

#### 4.2.2. Preprocessing for Data Statistics

#### 4.3. Statistical Software

## 5. Applications

#### 5.1. Viral Infection

#### 5.1.1. COVID-19

#### 5.1.2. Influenza

#### 5.1.3. Haemorrhagic Fever

#### 5.1.4. Ebola

#### 5.1.5. Dengue

#### 5.1.6. Rabies

#### 5.2. Bacterial Infection

#### 5.2.1. Salmonellosis

#### 5.2.2. Tuberculosis

#### 5.2.3. Brucellosis

#### 5.2.4. Anthrax

#### 5.3. Parasitic Infection

#### 5.3.1. Malaria

#### 5.3.2. Toxoplasmosis

#### 5.4. Other Infections

## 6. Discussion

#### 6.1. Difficulties

- (1)
- Computational complexity: Bayesian spatiotemporal models, particularly for large datasets, can be computationally demanding due to integrations over uncertain parameters, often requiring intensive numerical methods like MCMC algorithms. Despite this, the Bayesian hierarchical framework is widely used for its flexibility, allowing the construction of complex models through a hierarchical structure that combines data and prior information using prior distributions for each parameter.
- (2)
- Model interpretability: Complex Bayesian spatiotemporal models can pose challenges for result interpretation, as their structures demand a prior distribution on the parameters of interest. The use of different priors can yield varied results, introducing subjectivity and controversy. Striking a balance between complexity and interpretability is crucial in model development to enhance decision making.
- (3)
- Incorporating dynamic factors: enhancements in modelling dynamic factors influencing disease spread, such as human mobility or climate changes, are areas for future research.
- (4)
- Validation and comparison: while there have been reviews on Bayesian models, there is a continued need for standardized validation procedures and comparisons among various Bayesian spatiotemporal models to evaluate their performance and generalizability.

#### 6.2. Advantages

- (1)
- Bayesian spatiotemporal models yield more reasonable results than traditional methods. Unlike traditional approaches that rely on p values, Bayesian methods extract the mean, mode, confidence intervals, and other indicators from the posterior distribution of unknown parameters, providing a more comprehensive and interpretable representation of uncertainty. Bayesian methods can naturally handle missing data through their probabilistic framework, providing a more robust analysis in situations where traditional methods may struggle. Bayesian spatiotemporal models often involve hierarchical structures that allow for borrowing strength across space and time. This helps in improving parameter estimation, especially in regions or time periods with limited data. In situations where new data become available, Bayesian models can be easily updated to incorporate this information, allowing for dynamic and adaptive modelling. Traditional methods may require more extensive modifications to accommodate new data.
- (2)
- The Bayesian spatiotemporal model integrates prior information, treating unknown parameters as random variables influenced by both sample and prior information. By incorporating information about spatial and temporal structures as initial assumptions in the model, it becomes more realistic and gains valuable insights from past experiences, leading to a more accurate representation of the data.
- (3)
- Bayesian spatiotemporal models benefit from computational advantages, enabled using the MCMC algorithm and improved calculation speed. MCMC iteratively generates parameter samples, simplifying the estimation of posterior distributions. This approach facilitates the Bayesian analysis of complex datasets, addressing challenges such as missing observations and multidimensional outcomes.
- (4)
- The Bayesian spatiotemporal model adeptly handles statistical challenges in disease research, including small areas, low case counts, and significant regional differences. By incorporating prior information and adjacent spatiotemporal data, these models improve the realism of model estimation in spatial analyses, offering dependable solutions for addressing statistical issues in disease research.

#### 6.3. Considerations

- (1)
- Spatial and temporal scale: Using Bayesian spatiotemporal models presents a challenge known as the scale effect, where varying spatial or time scales may produce inconsistent or contradictory results. Spatial scales generally refer to provinces, cities, counties, and time scales refers to years, quarters, months, weeks. Careful consideration of both spatial scales and time scales is essential when comparing conclusions from spatiotemporal analyses of the same disease. The stable correlation observed with certain factors in one space–time dimension may not hold in another.
- (2)
- Selection of prior information: Bayesian statistical methods benefit from the use of prior information, but its effectiveness depends on careful selection. Defining a Bayesian prior requires a thorough understanding of the relevant scientific literature. Extracting useful information from the literature and combining it with expert knowledge helps in choosing an appropriate prior distribution and setting parameters. In cases without support from the literature, opting for an uninformative prior ensures that results remain unbiased using uncertain prior information, preventing misleading conclusions.
- (3)
- Reasonable model selection: Conventional spatiotemporal model analysis methods may overlook mean regression bias and unobserved heterogeneity, leading to unstable and biased parameter estimates. Additionally, methods like spatial autoregression face challenges with multi-level data and population-level random effects. In contrast, the full Bayesian framework is more flexible, easily extending to models with random effects that serve as surrogates for unobserved or missing covariates with spatial or temporal structures.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Correction Statement

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**Figure 1.**Evolution of space–time in geographical epidemiology. (

**a**) Representation of space–time points. (

**b**) Dynamic points with temporal changes while maintaining a constant spatial location. (

**c**) Points constrained by static barriers that evolve over time. (

**d**) Temporal influence on the entry at point 0 and exit at point B (adapted from Cox and Isham (1980)).

**Figure 2.**Schematic diagram of spatially structured (

**a**), spatially unstructured (

**b**), temporally structured (

**c**), and spatially unstructured (

**d**) effect and their distribution.

**Figure 3.**Symbolic representation of the four possible types of interactions. Circles represent prior independence and ovals represent prior dependence (adopted from Knorr-Held [50]).

**Figure 4.**Bayesian workflow (modified from Martin et al. [73]).

**Figure 5.**(

**a**) The estimated overall spatial pattern derived from the posterior means of the spatial relative risks for COVID-19 within neighbourhoods of Toronto from January 2020 to October 2021. (

**b**) The spatiotemporal trend of the relative risks in Toronto from January 2020 to October 2021. (Adopted from Nazia et al. [111]).

Distribution | Parameterization | MLE and Comments | |
---|---|---|---|

1 | Binomial | ${Y}_{i}\left|{r}_{i}~\right.Bin({N}_{i},{r}_{i})$ | ${\widehat{r}}_{i}=\frac{{Y}_{i}}{{N}_{i}}$ |

2 | Poisson | ${Y}_{i}\left|{\theta}_{i}~\right.Poisson\left({E}_{i}{\theta}_{i}\right)$ | ${\theta}_{i}=\frac{{Y}_{i}}{{E}_{i}}$ |

3 | Negative binomial | ${Y}_{i}\left|{\theta}_{i}~\right.Poisson\left({E}_{i}{\theta}_{i}\right)$ ${\theta}_{i}~gamma(\alpha ,\beta )$ | Also known as Poisson–gamma mixture |

Space–Time Interaction | ${\mathit{R}}_{\mathit{\delta}}$ | Spatial Correlation | Temporal Correlation |
---|---|---|---|

Type I | ${R}_{v}\otimes {R}_{\gamma}$ | - | - |

Type II | ${R}_{v}\otimes {R}_{\beta}$ | - | ✓ |

Type III | ${R}_{\psi}\otimes {R}_{\gamma}$ | ✓ | - |

Type IV | ${R}_{\psi}\otimes {R}_{\beta}$ | ✓ | ✓ |

Name | Website | Data | |
---|---|---|---|

1 | World Health Organization (WHO) | https://www.who.int/ | global health data, including updates on ongoing epidemics and pandemics |

2 | Centers for Disease Control and Prevention (CDC) | https://www.cdc.gov/ | the United States national public health agency on diverse infectious diseases |

3 | Johns Hopkins University (JHU)—Coronavirus Resource Center | https://coronavirus.jhu.edu/ | a global paltform for tracking the COVID-19 pandemic with statistics on cases, deaths, and vaccination |

4 | European Centre for Disease Prevention and Control (ECDC) | https://www.ecdc.europa.eu/ | data on infectious diseases in Europe with surveillance reports and epidemiological updates |

5 | Worldometer—COVID-19 Coronavirus Outbreak | https://www.worldometers.info/coronavirus/ | real-time statistics covering a range of topics with COVID-19 cases, deaths, and testing |

6 | Our World in Data | https://ourworldindata.org/ | visualisations and data on global health, including infectious diseases |

Name | Website | Data | |
---|---|---|---|

1 | United States Geological Survey (USGS) | https://www.usgs.gov/ | collection of geological and geospatial data with maps, satellite imagery, and geological information |

2 | National Centers for Environmental Information (NCEI) | https://www.ncei.noaa.gov/ | environmental data with climate, oceanography, and geophysical data |

3 | European Space Agency (ESA)—Earth Online | https://earth.esa.int/eogateway | satellite data on Earth observation missions with data related to climate, land cover, and environmental monitoring |

4 | National Aeronautics and Space Administration (NASA) Earth Observing System Data and Information System (EOSDIS) | https://www.earthdata.nasa.gov/eosdis | NASA’s Earth science data with satellite imagery, atmospheric data, and climate data |

5 | Global Biodiversity Information Facility (GBIF) | https://www.gbif.org/ | georeferenced species occurrence data related to biodiversity |

6 | OpenTopography | https://opentopography.org/ | high-resolution topographic data and lidar datasets |

Name | Website | Data | |
---|---|---|---|

1 | National Oceanic and Atmospheric Administration (NOAA) | https://www.noaa.gov/ | meteorological information with weather forecasts, satellite imagery, and climate data |

2 | National Weather Service (NWS) | https://www.weather.gov/ | weather forecasts, warnings, and other meteorological information for the United States |

3 | European Centre for Medium-Range Weather Forecasts (ECMWF) | https://www.ecmwf.int/ | global weather forecasts, climate reanalysis data, and meteorological products |

4 | World Meteorological Organization (WMO) | https://wmo.int/ | global meteorological information, reports, and publications |

5 | Weather Underground | https://www.wunderground.com/ | weather forecasts, radar imagery, and historical weather data |

Name | Website | Data | |
---|---|---|---|

1 | World Bank—World Development Indicators (WDI) | https://datatopics.worldbank.org/world-development-indicators/ (accessed on 9 March 2024) | socioeconomic data with population, poverty, education, and economic development |

2 | United Nations Development Programme (UNDP)—Human Development Indicators | https://hdr.undp.org/data-center/human-development-index (accessed on 9 March 2024) | human development indicators with life expectancy, education, and income |

3 | United Nations Statistics Division (UNSD) | https://unstats.un.org/UNSDWebsite/ | global statistical information with social, economic, and environmental indicators |

4 | U.S. Census Bureau—Data.census.gov | https://data.census.gov/ | socioeconomic data with population, housing, and economic indicators |

5 | Eurostat | https://ec.europa.eu/eurostat | socioeconomic data for EU member states |

6 | Statista | https://www.statista.com/ | statistics and data on various socioeconomic topics with industry, finance, and demographics |

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Wang, Y.; Chen, X.; Xue, F.
A Review of Bayesian Spatiotemporal Models in Spatial Epidemiology. *ISPRS Int. J. Geo-Inf.* **2024**, *13*, 97.
https://doi.org/10.3390/ijgi13030097

**AMA Style**

Wang Y, Chen X, Xue F.
A Review of Bayesian Spatiotemporal Models in Spatial Epidemiology. *ISPRS International Journal of Geo-Information*. 2024; 13(3):97.
https://doi.org/10.3390/ijgi13030097

**Chicago/Turabian Style**

Wang, Yufeng, Xue Chen, and Feng Xue.
2024. "A Review of Bayesian Spatiotemporal Models in Spatial Epidemiology" *ISPRS International Journal of Geo-Information* 13, no. 3: 97.
https://doi.org/10.3390/ijgi13030097