# Spatial Survival Model for COVID-19 in México

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Area

^{2}) [32]. The INEGI categorizes the 32 Federal Entities as high, medium, or low population density (Figure 1).

#### 2.1. Database

#### 2.2. Spatial Autocorrelation

#### 2.3. Statistical Model

#### 2.3.1. Prior Distributions

#### 2.3.2. Posterior Distributions

## 3. Results

#### 3.1. State of México

#### 3.2. State of Guerrero

#### 3.3. State of Chihuahua

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Traceplots of the parameters in the proportional hazard spatial model for the State of México. $\alpha $: Transformed Bernstein polynomial parameter (Equation (10)); ${\tau}^{2}$: spatial variation across locations of GRF; $\phi $: range parameter.

**Figure A3.**Traceplots of the parameters in the proportional hazard spatial model for the state of Guerrero. $\alpha $: Transformed Bernstein polynomial parameter (Equation (10)); ${\tau}^{2}$: spatial variation across locations of GRF; $\phi $: range parameter.

**Figure A5.**Traceplots of the parameters in the proportional hazard spatial model for the state of Chihuahua. $\alpha $: Transformed Bernstein polynomial parameter (Equation (10)); ${\tau}^{2}$: spatial variation across locations of GRF; $\phi $: range parameter.

## References

- Maza-De La Torre, G.; Montelongo-Mercado, E.A.; Noyola-Villalobos, H.F.; García-Ruíz, A.; Hernández-Díaz, S.; Santiago-Torres, M.; Moreno-Delgado, L.F.; Carrera-Altamirano, R.; Muñoz-Monroy, E.; Martínez-Cuazitl, A.; et al. Epidemiología de los pacientes hospitalizados con COVID-19 en un hospital de tercer nivel. Gac. Médica México
**2021**, 157, 246–254. [Google Scholar] [CrossRef] - Escobedo-de la Peña, J.; Rascón-Pacheco, R.A.; de Jesús Ascencio-Montiel, I.; González-Figueroa, E.; Fernández-Gárate, J.E.; Medina-Gómez, O.S.; Borja-Bustamante, P.; Santillán-Oropeza, J.A.; Borja-Aburto, V.H. Hypertension, diabetes and obesity, major risk factors for death in patients with COVID-19 in Mexico. Arch. Med. Res.
**2021**, 52, 443–449. [Google Scholar] [CrossRef] [PubMed] - Burki, T. COVID-19 in Latin America. Lancet Infect. Dis.
**2020**, 20, 547–548. [Google Scholar] [CrossRef] - Piovani, D.; Nikolopoulos, G.K.; Bonovas, S. Pitfalls and perils of survival analysis under incorrect assumptions: The case of COVID-19 data. Biomedica
**2021**, 41, 21–28. [Google Scholar] [CrossRef] - Salinas-Aguirre, J.E.; Sánchez-García, C.; Rodríguez-Sanchez, R.; Rodríguez-Muñoz, L.; Díaz-Castaño, A.; Bernal-Gómez, R. Clinical characteristics and comorbidities associated with mortality in patients with COVID-19 in Coahuila (Mexico). Rev. Clin. Esp.
**2022**, 222, 288–292. [Google Scholar] [CrossRef] [PubMed] - Salinas-Escudero, G.; Carrillo-Vega, M.F.; Granados-García, V.; Martínez-Valverde, S.; Toledano-Toledano, F.; Garduño-Espinosa, J. A survival analysis of COVID-19 in the Mexican population. BMC Public. Health
**2020**, 20, 1616. [Google Scholar] - Márquez-González, H.; Méndez-Galván, J.F.; Reyes-López, A.; Klünder-Klünder, M.; Jiménez-Juárez, R.; Garduño-Espinosa, J.; Solórzano-Santos, F. Coronavirus disease-2019 survival in Mexico: A cohort study on the interaction of the associated factors. Front. Public. Health
**2021**, 9, 1018. [Google Scholar] [CrossRef] - Millán-Guerrero, R.O.; Caballero-Hoyos, R.; Monárrez-Espino, J. Poverty and survival from COVID-19 in Mexico. J. Public Health
**2021**, 43, 437–444. [Google Scholar] [CrossRef] - Ahmed, F.E.; Vos, P.W.; Holbert, D. Modeling survival in colon cancer: A methodological review. Mol. Cancer
**2007**, 6, 15. [Google Scholar] [CrossRef] - Kwon, J.; Sung, K.R.; Jo, J.; Yang, S.H. Glaucoma progression and its relationship with corrected and uncorrected intraocular pressure in eyes with history of refractive corneal surgery. Curr. Eye Res.
**2018**, 43, 1136–1144. [Google Scholar] [CrossRef] - Guzmán Martínez, M.; Pérez-Castro, E.; Reyes-Carreto, R.; Acosta-Pech, R. Spatial Modeling in Epidemiology. In Recent Advances in Medical Statistics; IntechOpen: London, UK, 2022. [Google Scholar]
- Allotey, P.A.; Harel, O. Modeling geostatistical incomplete spatially correlated survival data with applications to COVID-19 mortality in Ghana. Spat. Stat.
**2023**, 54, 100730. [Google Scholar] [CrossRef] [PubMed] - Taylor, B.M.; Rowlingson, B.S. Spatsurv: An R package for bayesian inference with spatial survival models. J. Stat. Softw.
**2017**, 77, 1–32. [Google Scholar] [CrossRef] - Louzada, F.; Do Nascimento, D.C.; Egbon, O.A. Spatial statistical models: An overview under the bayesian approach. Axioms
**2021**, 10, 307. [Google Scholar] [CrossRef] - Kirby, R.S.; Delmelle, E.; Eberth, J.M. Advances in spatial epidemiology and geographic information systems. Ann. Epidemiol.
**2017**, 27, 1–9. [Google Scholar] [CrossRef] [PubMed] - Diggle, P.J.; Ribeiro, P.J. Model-Based Geostatistics; Springer: New York, NY, USA, 2007; p. 228. [Google Scholar]
- Xu, H. Comparing spatial and multilevel regression models for binary outcomes in neighborhood studies. Sociol. Methodol.
**2014**, 44, 229–272. [Google Scholar] [CrossRef] [PubMed] - Elliott, P.; Wartenberg, D. Spatial epidemiology: Current approaches and future challenges. Environ. Health Perspect.
**2004**, 112, 998–1006. [Google Scholar] [CrossRef] - Lin, C.H.; Wen, T.H. How Spatial Epidemiology Helps Understand Infectious Human Disease Transmission. Trop. Med. Infect. Dis.
**2022**, 7, 164. [Google Scholar] [CrossRef] - Thamrin, S.A.; Jaya, A.K.; Mengersen, K. Bayesian spatial survival modelling for dengue fever in Makassar, Indonesia. Gac. Sanit.
**2021**, 35, S59–S63. [Google Scholar] [CrossRef] - Lawson, A.B.; Banerjee, S.; Haining, R.; Ugarte, L. Handbook of Spatial Epidemiology; CRC Press: New York, NY, USA, 2016. [Google Scholar]
- Nunes, C.; Taylor, B.M. Modelling the time to detection of urban tuberculosis in two big cities in Portugal: A spatial survival analysis. Int. J. Tuberc. Lung Dis.
**2016**, 20, 1219–1225. [Google Scholar] [CrossRef] - Henderson, R.; Shimakura, S.; Gorst, D. Modeling spatial variation in leukemia survival data. J. Am. Stat. Assoc.
**2002**, 97, 965–972. [Google Scholar] [CrossRef] - Li, Y.; Lin, X. Semiparametric Normal Transformation Models for Spatially Correlated Survival Data. J. Am. Stat. Assoc.
**2006**, 101, 591–603. [Google Scholar] [CrossRef] - Aswi, A.; Cramb, S.; Duncan, E.; Hu, W.; White, G.; Mengersen, K. Bayesian spatial survival models for hospitalisation of Dengue: A case study of Wahidin hospital in Makassar, Indonesia. Int. J. Environ. Res. Public Health
**2020**, 17, 878. [Google Scholar] [CrossRef] - Mahanta, K.K.; Hazarika, J.; Barman, M.P.; Rahman, T. An application of spatial frailty models to recovery times of COVID-19 patients in India under Bayesian approach. J. Sci. Res.
**2021**, 65, 150–155. [Google Scholar] [CrossRef] - Liu, Y.; Sun, D.; He, C.Z. A hierarchical conditional autoregressive model for colorectal cancer survival data. Wiley Interdiscip. Rev. Comput. Stat.
**2014**, 6, 37–44. [Google Scholar] [CrossRef] - Lu, W.; Ying, Z. On semiparametric transformation cure models. Biometrika
**2004**, 91, 331–343. [Google Scholar] [CrossRef] - Daniel, K.; Onyango, N.O.; Sarguta, R.J. A spatial survival model for risk factors of Under-Five Child Mortality in Kenya. Int. J. Environ. Res. Public Health
**2021**, 19, 399. [Google Scholar] [CrossRef] - Martins, R.; Silva, G.L.; Andreozzi, V. Bayesian joint modeling of longitudinal and spatial survival AIDS data. Stat. Med.
**2016**, 35, 3368–3384. [Google Scholar] [CrossRef] - Schnell, P.; Bandyopadhyay, D.; Reich, B.J.; Nunn, M. A marginal cure rate proportional hazards model for spatial survival data. J. R. Stat. Soc. Ser. C Appl. Stat.
**2015**, 64, 673–691. [Google Scholar] [CrossRef] - Instituto Nacional de Estadística e Informática. Censo de Población y Vivienda 2020; INEGI: Lima, Peru, 2020. [Google Scholar]
- R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2023; Available online: https://www.R-project.org/ (accessed on 15 December 2021).
- Ministry of Health. Datos Abiertos Bases Históricas. Dirección General de Epidemiología. 2020. (Secretaría de Salud). Available online: https://www.gob.mx/salud/documentos/datosabiertos-bases-historicas-direccion-general-de-epidemiologia (accessed on 25 November 2021).
- Zhou, H.; Hanson, T. A Unified Framework for fitting Bayesian semiparametric models to arbitrarily censored survival data, including spatially referenced data. J. Am. Stat. Assoc.
**2018**, 113, 571–581. [Google Scholar] [CrossRef] - Zhou, H.; Hanson, T.; Zhang, J. SpBayesSurv: Fitting bayesian spatial survival models using R. J. Stat. Softw.
**2020**, 92. [Google Scholar] [CrossRef] - Brooks, S.P.; Gelman, A. General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Stat.
**1998**, 7, 434–455. [Google Scholar] - Gelman, A.; Rubin, D.B. Inference from iterative simulation using multiple sequences. Stat. Sci.
**1992**, 7, 457–511. [Google Scholar] [CrossRef] - Plummer, M.; Best, N.; Cowles, K.; Vines, K. Coda: Convergence diagnosis and output analysis for MCMC. R News
**2006**, 6, 7–11. [Google Scholar] - Lee, E.T.; Wang, J. Statistical Methods for Survival Data Analysis; John Wiley & Sons: Hoboken, NJ, USA, 2003; Volume 476. [Google Scholar]
- Klein, J.P.; Moeschberger, M.L. Survival Analysis: Techniques for Censored and Truncated Data; Springer: New York, NY, USA, 1997. [Google Scholar]
- Perera, M.; Tsokos, C. A statistical model with non-linear effects and non-proportional hazards for breast cancer survival analysis. Adv. Breast Cancer Res.
**2018**, 7, 65–89. [Google Scholar] [CrossRef] - Spiegelhalter, D.J.; Best, N.G.; Carlin, B.P.; Van Der Linde, A. Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B Stat. Methodol.
**2002**, 64, 583–639. [Google Scholar] [CrossRef] - Watanabe, S. Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. J. Mach. Learn Res.
**2010**, 11, 3571–3594. [Google Scholar] - Geisser, S.; Eddy, W.F. A predictive approach to model selection. J. Am. Stat. Assoc.
**1979**, 74, 153–160. [Google Scholar] [CrossRef] - Pérez-Sastré, M.A.; Valdés, J.; Ortiz-Hernández, L. Clinical characteristics and severity of COVID-19 among Mexican adults. Gac. Med. Mex.
**2020**, 156, 373–381. [Google Scholar] [CrossRef] - Plasencia Urizarri, T.M.; Aguilera Rodríguez, R.; Almaguer Mederos, L.E. Comorbilidades y gravedad clínica de la COVID-19: Revisión sistemática y meta-análisis. Rev. Habanera Cienc. Médicas
**2020**, 19, 1–17. [Google Scholar] - Instituto Nacional de Salud Pública. Encuesta Nacional de Salud y Nutrición (ENSANUT); Instituto Nacional de Salud Pública: Mexico City, Mexico, 2018. [Google Scholar]
- Kammar-Garcia, A.; Vidal-Mayo, J.D.J.; Vera-Zertuche, J.M.; Lazcano-Hernandez, M.; Vera-Lopez, O.; Segura-Badilla, O.; Aguilar-Alonso, P.; Navarro-Cruz, A.R. Impact of comorbidities in Mexican SARS-CoV-2-positive patients: A retrospective analysis in a national cohort. Rev. Investig. Clin.
**2020**, 72, 151–158. [Google Scholar] [CrossRef] - Muñoz, X.; Pilia, F.; Ojanguren, I.; Romero-Mesones, C.; Cruz, M.-J. Is asthma a risk factor for COVID-19? Are phenotypes important? ERJ Open Res.
**2021**, 7, 00216–02020. [Google Scholar] [CrossRef] [PubMed] - Farne, H.; Singanayagam, A. Why asthma might surprisingly protect against poor outcomes in COVID-19. Eur. Respir. J.
**2020**, 56, 2003045. [Google Scholar] [CrossRef] [PubMed] - Liuzzo Scorpo, M.; Ferrante, G.; La Grutta, S. An Overview of Asthma and COVID-19: Protective factors against SARS-CoV-2 in pediatric patients. Front. Pediatr.
**2021**, 9, 661206. [Google Scholar] [CrossRef] [PubMed] - Ortiz-Hernández, L.; Pérez-Sastré, M.A. Inequidades sociales en la progresión de la COVID-19 en población mexicana. Rev. Panam. Salud Publica
**2020**, 44, 1. [Google Scholar] [CrossRef] - Tan, T.Q.; Kullar, R.; Swartz, T.H.; Mathew, T.A.; Piggott, D.A.; Berthaud, V. Location matters: Geographic disparities and impact of Coronavirus disease 2019. J. Infect. Dis.
**2020**, 222, 1951–1954. [Google Scholar] [CrossRef] - Deb Nath, N.; Khan, M.M.; Schmidt, M.; Njau, G.; Odoi, A. Geographic disparities and temporal changes of COVID-19 incidence risks in North Dakota, United States. BMC Public Health
**2023**, 23, 720. [Google Scholar] [CrossRef]

**Figure 1.**States of México with number of municipalities (bars and left axis) and total population (lines and right axis).

**Figure 3.**Confirmed cases of COVID-19 in the municipalities of the State of México from 28 February 2020 to 24 November 2021.

**Figure 5.**Confirmed cases of COVID-19 in the municipalities of Guerrero from 28 February 2020 to 24 November 2021.

**Figure 7.**Confirmed cases of COVID-19 in the municipalities of Chihuahua from 28 February 2020 to 24 November 2021.

Variable | Code | Description |
---|---|---|

Age | Number of years | Patient’s age |

Gender | 1: Male; 2: Female | Identifies the gender of the patient |

Pneumonia | 1: Yes; 2: No | Indicates if the patient has pneumonia |

Diabetes | 1: Yes; 2: No | Indicates if the patient has diabetes |

Chronic obstructive pulmonary disease | 1: Yes; 2: No | Indicates if the patient has COPD |

Cardiovascular disease | 1: Yes; 2: No | Indicates if the patient has CVD |

Obesity | 1: Yes; 2: No | Indicates if the patient has obesity |

Asthma | 1: Yes; 2: No | Indicates if the patient has asthma |

Chronic renal disease | 1: Yes; 2: No | Indicates if the patient has CKD |

Hypertension | 1: Yes; 2: No | Indicates if the patient has hypertension |

Variable | Mean | Median | 95% CrI | $\widehat{\mathit{R}}$ | CrI-Upper |
---|---|---|---|---|---|

Age | 0.042 | 0.042 | (0.041, 0.043) ** | 1.01 | 1.02 |

Sex (male) | 0.341 | 0.341 | (0.308, 0.373) ** | 1.00 | 1.01 |

Pneumonia | 1.680 | 1.679 | (1.645, 1.719) ** | 1.00 | 1.02 |

Diabetes | 0.164 | 0.164 | (0.130, 0.197) ** | 1.01 | 1.03 |

COPD | 0.017 | 0.017 | (−0.066, 0.096) | 1.00 | 1.00 |

CVD | −0.096 | −0.094 | (−0.186, −0.011) ** | 1.01 | 1.04 |

Obesity | 0.129 | 0.129 | (0.085, 0.170) ** | 1.01 | 1.07 |

Asthma | −0.228 | −0.226 | (−0.389, −0.081) ** | 1.02 | 1.08 |

CKD | 0.463 | 0.463 | (0.394, 0.535) ** | 1.01 | 1.05 |

Hypertension | 0.063 | 0.064 | (0.026, 0.098) ** | 1.00 | 1.00 |

$\widehat{\alpha}$ | 0.068 | 0.066 | (0.036, 0.110) ** | 1.66 | 2.97 |

${\widehat{\tau}}^{2}$ | 1.551 | 1.464 | (0.901, 2.692) ** | 1.01 | 1.05 |

$\widehat{\varphi}$ | 0.273 | 0.263 | (0.133, 0.470) ** | 1.01 | 1.03 |

${\widehat{\theta}}_{1}$ | −4.167 | −4.173 | (−4.182, −4.131) ** | - | - |

${\widehat{\theta}}_{2}$ | −0.059 | -0.061 | (−0.071, −0.045) ** | - | - |

Variable | Mean | Median | 95% CrI | $\widehat{\mathit{R}}$ | CrI-Upper |
---|---|---|---|---|---|

Age | 0.035 | 0.035 | (0.033, 0.037) ** | 1.00 | 1.00 |

Sex (Male) | 0.191 | 0.192 | (0.127, 0.252) ** | 1.00 | 1.00 |

Pneumonia | 2.819 | 2.818 | (2.718, 2.913) ** | 1.00 | 1.01 |

Diabetes | 0.304 | 0.305 | (0.239, 0.368) ** | 1.00 | 1.00 |

COPD | 0.140 | 0.140 | (0.001, 0.277) ** | 1.00 | 1.00 |

CVD | 0.206 | 0.206 | (0.063, 0.354) ** | 1.00 | 1.00 |

Obesity | 0.219 | 0.218 | (0.146, 0.296) ** | 1.00 | 1.00 |

Asthma | −0.120 | −0.119 | (−0.343, 0.103) | 1.00 | 1.02 |

CKD | 0.196 | 0.197 | (0.073, 0.325) ** | 1.01 | 1.03 |

Hypertension | 0.070 | 0.071 | (0.004, 0.132) ** | 1.00 | 1.00 |

$\widehat{\alpha}$ | 0.076 | 0.072 | (0.041, 0.126) ** | 1.03 | 1.11 |

${\widehat{\tau}}^{2}$ | 1.838 | 1.731 | (1.072, 3.166) ** | 1.00 | 1.00 |

$\widehat{\varphi}$ | 0.083 | 0.077 | (0.032, 0.163) ** | 1.00 | 1.00 |

${\widehat{\theta}}_{1}$ | −4.315 | −4.280 | (−4.520, −4.198) ** | - | - |

${\widehat{\theta}}_{2}$ | 0.174 | 0.174 | (0.152, 0.208) ** | - | - |

Variable | Mean | Median | 95% CrI | $\widehat{\mathit{R}}$ | CrI-Upper |
---|---|---|---|---|---|

Age | 0.048 | 0.048 | (0.046, 0.050) ** | 1.01 | 1.03 |

Sex (male) | 0.310 | 0.310 | (0.257, 0.365) ** | 1.00 | 1.00 |

Pneumonia | 1.689 | 1.688 | (1.630, 1.750) ** | 1.00 | 1.02 |

Diabetes | 0.341 | 0.341 | (0.283, 0.401) ** | 1.00 | 1.00 |

COPD | 0.068 | 0.070 | (−0.072, 0.209) | 1.00 | 1.00 |

CVD | −0.021 | −0.022 | (−0.130, 0.087) | 1.00 | 1.00 |

Obesity | 0.277 | 0.277 | (0.211, 0.342) ** | 1.00 | 1.00 |

Asthma | −0.141 | −0.144 | (−0.313, 0.029) | 1.00 | 1.00 |

CKD | 0.531 | 0.533 | (0.420, 0.633) ** | 1.00 | 1.01 |

Hypertension | 0.245 | 0.245 | (0.183, 0.301) ** | 1.00 | 1.00 |

$\widehat{\alpha}$ | 0.099 | 0.096 | (0.051, 0.160) ** | 1.00 | 1.01 |

${\widehat{\tau}}^{2}$ | 2.158 | 2.035 | (1.261, 3.724) ** | 1.00 | 1.01 |

$\widehat{\varphi}$ | 0.099 | 0.093 | (0.043, 0.185) ** | 1.00 | 1.01 |

${\widehat{\theta}}_{1}$ | −4.384 | −4.362 | (−4.593, −4.286) ** | ||

${\widehat{\theta}}_{2}$ | 0.155 | 0.159 | (0.133, 0.178) ** |

State | PH Model | DIC | WAIC | LPML |
---|---|---|---|---|

State of México | Spatial | 186,805.2 | 186,809.3 | −93,404.61 |

Cox | 193,904.9 | 194,385.3 | −97,192.46 | |

Guerrero | Spatial | 46,866.87 | 46,871.20 | −23,435.59 |

Cox | 48,396.29 | 48,395.89 | −24,197.94 | |

Chihuahua | Spatial | 63,225.16 | 63,232.3 | −31,616.2 |

Cox | 65,013.54 | 65,014.24 | −32,507.11 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pérez-Castro, E.; Guzmán-Martínez, M.; Godínez-Jaimes, F.; Reyes-Carreto, R.; Vargas-de-León, C.; Aguirre-Salado, A.I.
Spatial Survival Model for COVID-19 in México. *Healthcare* **2024**, *12*, 306.
https://doi.org/10.3390/healthcare12030306

**AMA Style**

Pérez-Castro E, Guzmán-Martínez M, Godínez-Jaimes F, Reyes-Carreto R, Vargas-de-León C, Aguirre-Salado AI.
Spatial Survival Model for COVID-19 in México. *Healthcare*. 2024; 12(3):306.
https://doi.org/10.3390/healthcare12030306

**Chicago/Turabian Style**

Pérez-Castro, Eduardo, María Guzmán-Martínez, Flaviano Godínez-Jaimes, Ramón Reyes-Carreto, Cruz Vargas-de-León, and Alejandro Iván Aguirre-Salado.
2024. "Spatial Survival Model for COVID-19 in México" *Healthcare* 12, no. 3: 306.
https://doi.org/10.3390/healthcare12030306