Analysis of the Spread of COVID-19 in the USA with a Spatio-Temporal Multivariate Time Series Model
Abstract
:1. Introduction
2. Development of the Model
Models
3. Results
3.1. Data of Interest
3.2. Weather and Environmental Factors’ Selection
3.3. Optimal Parameters’ Determination
3.4. Components Analysis
4. Discussion
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Materials for Parameter Estimation
Appendix A.1. Inference
Appendix A.2. First-Order Partial Gradients of the Penalized Log-Likelihood Function
Appendix A.3. Pseudo Algorithm
Algorithm A1 Adam based method for parameter optimization. Good default settings for the analyzed COVID-19 dataset are learning rate , exponential decay rates and , and . Algorithm tolerance . All operations are element-wise. |
Initialization: maxit = 200 (maximum iteration steps), flag = 0 (convergence indicator), (first moment vector), (second moment vector), (iteration-step indicator), , , . |
Iteration process: |
while maxit and flag = 0 do |
(gradients of shown in (A6) in Appendix A.2) |
(bias-corrected first moment estimate) |
(bias-corrected second raw moment estimate) |
(temporarily updated parameters) |
(updated parameters) |
(averaged parameters for further iteration) |
if (convergence determination) |
flag = 1 |
end while |
return (optimal estimates) |
References
- Gorbalenya, A.E.; Baker, S.C. The species Severe acute respiratory syndrome-related coronavirus: Classifying 2019-nCoV and naming it SARS-CoV-2. Nat. Microbiol. 2020, 5, 536. [Google Scholar]
- Le, T.T.; Andreadakis, Z.; Kumar, A.; Roman, R.G.; Tollefsen, S.; Saville, M.; Mayhew, S. The COVID-19 vaccine development landscape. Nat. Rev. Drug Discov. 2020, 19, 305–306. [Google Scholar] [CrossRef] [PubMed]
- Graham, B.S. Rapid COVID-19 vaccine development. Science 2020, 368, 945–946. [Google Scholar] [CrossRef] [PubMed]
- Suganya, S.; Divya, S.; Parani, M. Severe acute respiratory syndrome-coronavirus-2: Current advances in therapeutic targets and drug development. Rev. Med Virol. 2020. [Google Scholar] [CrossRef] [PubMed]
- Chowell, G.; Mizumoto, K. The COVID-19 pandemic in the USA: What might we expect? Lancet 2020, 395, 1093–1094. [Google Scholar] [CrossRef]
- Rui, R.; Tian, M. Joint Estimation of Case Fatality Rate of COVID-19 and Power of Quarantine Strategy Performed in Wuhan, China. Biom. J. 2020, 63, 46–58. [Google Scholar] [CrossRef] [PubMed]
- Kermack, W.O.; McKendrick, A.G. A contribution to the mathematical theory of epidemics. Proc. R. Soc. London. Ser. A Contain. Pap. Math. Phys. Character 1927, 115, 700–721. [Google Scholar]
- Helfenstein, U. Box-Jenkins modeling of some viral infectious diseases. Stat. Med. 1986, 5, 37–47. [Google Scholar] [CrossRef]
- Trottier, H.; Philippe, P.; Roy, R. Stochastic modeling of empirical time series of childhood infectious diseases data before and after mass vaccination. Emerg. Themes Epidemiol. 2006, 3, 9. [Google Scholar] [CrossRef] [Green Version]
- Papastefanopoulos, V.; Linardatos, P.; Kotsiantis, S. COVID-19: A Comparison of Time Series Methods to Forecast Percentage of Active Cases per Population. Appl. Sci. 2020, 10, 3880. [Google Scholar] [CrossRef]
- Held, L.; Höhle, M.; Hofmann, M. A statistical framework for the analysis of multivariate infectious disease surveillance counts. Stat. Model. 2005, 5, 187–199. [Google Scholar] [CrossRef] [Green Version]
- Paul, M.; Held, L.; Toschke, A.M. Multivariate modeling of infectious disease surveillance data. Stat. Med. 2008, 27, 6250–6267. [Google Scholar] [CrossRef] [PubMed]
- Paul, M.; Held, L. Predictive assessment of a non-linear random effects model for multivariate time series of infectious disease counts. Stat. Med. 2011, 30, 1118–1136. [Google Scholar] [CrossRef] [PubMed]
- Cheng, Q.; Lu, X.; Wu, J.T.; Liu, Z.; Huang, J. Analysis of heterogeneous dengue transmission in Guangdong in 2014 with multivariate time series model. Sci. Rep. 2016, 6, 33755. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Adegboye, O.; Al-Saghir, M.; LEUNG, D.H. Joint spatial time-series epidemiological analysis of malaria and cutaneous leishmaniasis infection. Epidemiol. Infect. 2017, 145, 685–700. [Google Scholar] [CrossRef] [Green Version]
- Wu, H.; Wang, X.; Xue, M.; Wu, C.; Lu, Q.; Ding, Z.; Zhai, Y.; Lin, J. Spatial-temporal characteristics and the epidemiology of hemorrhagic fever with renal syndrome from 2007 to 2016 in Zhejiang Province, China. Sci. Rep. 2018, 8, 1–14. [Google Scholar]
- Dickson, M.M.; Espa, G.; Giuliani, D.; Santi, F.; Savadori, L. Assessing the effect of containment measures on the spatio-temporal dynamic of COVID-19 in Italy. Nonlinear Dyn. 2020, 101, 1833–1846. [Google Scholar] [CrossRef]
- Qi, H.; Xiao, S.; Shi, R.; Ward, M.P.; Chen, Y.; Tu, W.; Su, Q.; Wang, W.; Wang, X.; Zhang, Z. COVID-19 transmission in Mainland China is associated with temperature and humidity: A time-series analysis. Sci. Total Environ. 2020, 728, 138778. [Google Scholar] [CrossRef]
- Tosepu, R.; Gunawan, J.; Effendy, D.S.; Lestari, H.; Bahar, H.; Asfian, P. Correlation between weather and Covid-19 pandemic in Jakarta, Indonesia. Sci. Total Environ. 2020, 725, 138436. [Google Scholar] [CrossRef]
- Hugh-Jones, M.; Wright, P. Studies on the 1967–8 foot-and-mouth disease epidemic: The relation of weather to the spread of disease. Epidemiol. Infect. 1970, 68, 253–271. [Google Scholar] [CrossRef]
- Tan, J.; Mu, L.; Huang, J.; Yu, S.; Chen, B.; Yin, J. An initial investigation of the association between the SARS outbreak and weather: With the view of the environmental temperature and its variation. J. Epidemiol. Community Health 2005, 59, 186–192. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Verdasca, J.; Da Gama, M.T.; Nunes, A.; Bernardino, N.; Pacheco, J.; Gomes, M. Recurrent epidemics in small world networks. J. Theor. Biol. 2005, 233, 553–561. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Hassen, H.B.; Elaoud, A.; Salah, N.B.; Masmoudi, A. A SIR-Poisson Model for COVID-19: Evolution and Transmission Inference in the Maghreb Central Regions. Arab. J. Sci. Eng. 2021, 46, 93–102. [Google Scholar] [CrossRef] [PubMed]
- Read, J.M.; Bridgen, J.R.; Cummings, D.A.; Ho, A.; Jewell, C.P. Novel coronavirus 2019-nCoV: Early estimation of epidemiological parameters and epidemic predictions. MedRxiv 2020. [Google Scholar] [CrossRef] [Green Version]
- Giuliani, D.; Dickson, M.M.; Espa, G.; Santi, F. Modelling and Predicting the Spatio-Temporal Spread of Coronavirus Disease 2019 (COVID-19) in Italy; University of Trento: Trento, Italy, 2020. [Google Scholar]
- Gao, X.; Dong, Q. A logistic model for age-specific COVID-19 case-fatality rates. JAMIA Open 2020, 3, 151–153. [Google Scholar] [CrossRef]
- Kass, D.A.; Duggal, P.; Cingolani, O. Obesity could shift severe COVID-19 disease to younger ages. Lancet 2020, 399, 1544–1545. [Google Scholar] [CrossRef]
- Harris, J.E. Data from the COVID-19 epidemic in Florida suggest that younger cohorts have been transmitting their infections to less socially mobile older adults. Rev. Econ. Househ. 2020, 18, 1019–1037. [Google Scholar] [CrossRef]
- Bashir, M.F.; Ma, B.; Komal, B.; Bashir, M.A.; Tan, D.; Bashir, M. Correlation between climate indicators and COVID-19 pandemic in New York, USA. Sci. Total Environ. 2020, 728, 138835. [Google Scholar] [CrossRef]
- Kimball, A.; Hatfield, K.M.; Arons, M.; James, A.; Taylor, J.; Spicer, K.; Bardossy, A.C.; Oakley, L.P.; Tanwar, S.; Chisty, Z.; et al. Asymptomatic and presymptomatic SARS-CoV-2 infections in residents of a long-term care skilled nursing facility—King County, Washington, March 2020. Morb. Mortal. Wkly. Rep. 2020, 69, 377. [Google Scholar] [CrossRef] [Green Version]
- Rader, B.; Scarpino, S.V.; Nande, A.; Hill, A.L.; Adlam, B.; Reiner, R.C.; Pigott, D.M.; Gutierrez, B.; Zarebski, A.E.; Shrestha, M.; et al. Crowding and the shape of COVID-19 epidemics. Nat. Med. 2020, 26, 1829–1834. [Google Scholar] [CrossRef]
- Kneib, T.; Fahrmeir, L. A mixed model approach for geoadditive hazard regression. Scand. J. Stat. 2007, 34, 207–228. [Google Scholar] [CrossRef]
- Pearson, K. Historical note on the origin of the normal curve of errors. Biometrika 1924, 16, 402–404. [Google Scholar] [CrossRef]
- Kingma, D.P.; Ba, J. Adam: A method for stochastic optimization. arXiv 2014, arXiv:1412.6980. [Google Scholar]
Lags | |||||||
---|---|---|---|---|---|---|---|
47512882.850 | 48510611.518 | 48188861.487 | 47895162.927 | 47783167.600 | 47998732.406 | 47976026.240 | |
1600987.450 | 301043.787 | 923652.391 | 849380.316 | 1297418.980 | 825285.208 | 1055666.220 | |
lags | |||||||
47747910.670 | 47946826.823 | 47493940.570 | 47630036.889 | 47306645.479 | 47309743.565 | 47115278.420 | |
1199672.290 | 643524.434 | 1288241.920 | 773951.669 | 920626.589 | 673304.655 | 1158747.060 |
Estimates | ||||||||
---|---|---|---|---|---|---|---|---|
mean | 0.851 | 0.836 | 0.816 | 0.853 | 0.220 | −0.136 | 0.173 | −0.233 |
sd | 0.121 | 0.113 | 0.107 | 0.124 | 0.911 | 0.929 | 0.958 | 0.991 |
estimates | ||||||||
mean | −0.300 | 0.232 | −0.064 | −0.119 | −0.076 | 0.731 | 0.774 | 0.764 |
sd | 0.873 | 1.068 | 1.004 | 1.180 | 1.085 | 0.285 | 0.286 | 0.269 |
estimates | ||||||||
mean | 0.739 | 0.697 | 0.754 | 0.776 | 0.744 | 0.732 | 0.730 | 0.693 |
sd | 0.286 | 0.296 | 0.284 | 0.270 | 0.285 | 0.301 | 0.275 | 0.297 |
estimates | ||||||||
mean | 0.707 | 0.673 | 0.781 | 0.715 | 0.696 | 0.751 | 0.816 | 0.784 |
sd | 0.253 | 0.278 | 0.284 | 0.285 | 0.292 | 0.281 | 0.296 | 0.290 |
estimates | ||||||||
mean | 0.780 | 0.658 | 0.740 | 0.724 | 0.758 | 0.708 | 0.858 | 0.780 |
sd | 0.301 | 0.291 | 0.296 | 0.289 | 0.308 | 0.291 | 0.273 | 0.306 |
estimates | ||||||||
mean | 0.716 | 0.757 | 0.732 | 0.780 | 0.776 | 0.793 | 0.769 | 0.710 |
sd | 0.299 | 0.281 | 0.298 | 0.303 | 0.284 | 0.273 | 0.284 | 0.301 |
estimates | ||||||||
mean | 0.667 | 0.732 | 0.742 | 0.763 | 0.812 | 0.765 | 0.746 | 0.729 |
sd | 0.294 | 0.277 | 0.306 | 0.283 | 0.296 | 0.279 | 0.273 | 0.290 |
estimates | ||||||||
mean | 0.736 | 0.742 | 0.687 | 0.757 | 0.805 | 0.779 | 0.727 | 0.791 |
sd | 0.291 | 0.287 | 0.327 | 0.283 | 0.287 | 0.300 | 0.289 | 0.296 |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 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 (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Rui, R.; Tian, M.; Tang, M.-L.; Ho, G.T.-S.; Wu, C.-H. Analysis of the Spread of COVID-19 in the USA with a Spatio-Temporal Multivariate Time Series Model. Int. J. Environ. Res. Public Health 2021, 18, 774. https://doi.org/10.3390/ijerph18020774
Rui R, Tian M, Tang M-L, Ho GT-S, Wu C-H. Analysis of the Spread of COVID-19 in the USA with a Spatio-Temporal Multivariate Time Series Model. International Journal of Environmental Research and Public Health. 2021; 18(2):774. https://doi.org/10.3390/ijerph18020774
Chicago/Turabian StyleRui, Rongxiang, Maozai Tian, Man-Lai Tang, George To-Sum Ho, and Chun-Ho Wu. 2021. "Analysis of the Spread of COVID-19 in the USA with a Spatio-Temporal Multivariate Time Series Model" International Journal of Environmental Research and Public Health 18, no. 2: 774. https://doi.org/10.3390/ijerph18020774