JDS Journal of Data Science 1683-86021680-743X1680-743X School of Statistics, Renmin University of China JDS1009 10.6339/21-JDS1009 Statistical Data Science BayesSMILES: Bayesian Segmentation Modeling for Longitudinal Epidemiological Studies JiangShuang12 ZhouQuan3 ZhanXiaoweiXiaowei.Zhan@UTSouthwestern.edu2∗ https://orcid.org/0000-0002-1020-3050 LiQiweiQiwei.Li@UTDallas.edu4∗ Department of Statistical Science, Southern Methodist University, Dallas, TX 75205, USA Quantitative Biomedical Research Center, Department of Population and Data Sciences, The University of Texas Southwestern Medical Center, Dallas, TX 75390, USA Department of Statistics, Texas A&M University, College Station, TX 77843, USA Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USA Corresponding authors. Email: Xiaowei.Zhan@UTSouthwestern.edu or Qiwei.Li@UTDallas.edu. 20212422021 193365389 Supplementary Material

Supplement A: Software.

We provide software in the form of R/C++ codes on GitHub https://github.com/shuangj00/BayesSMILES. We have designed a website https://shuangj00.github.io/BayesSMILES/ to summarize the inference results for the 50 U.S. states, as a supplement to Section 6. The website shows that 1) the detected change points for each U.S. state; and 2) the COVID-19 transmission dynamics based on the segment-varying basic reproduction numbers R 0 ’s, including their posterior means and 95 % credible intervals.

Supplement B: Supplementary document.

The supplementary file encloses the detailed Markov chain Monte Carlo algorithms, additional simulation study and real data analysis results, and key notation tables.

7102020422021 2021 The Author(s). Published by the School of Statistics and the Center for Applied Statistics, Renmin University of China.2021 Open access article under the CC BY license.

The coronavirus disease of 2019 (COVID-19) is a pandemic. To characterize its disease transmissibility, we propose a Bayesian change point detection model using daily actively infectious cases. Our model builds on a Bayesian Poisson segmented regression model that 1) capture the epidemiological dynamics under the changing conditions caused by external or internal factors; 2) provide uncertainty estimates of both the number and locations of change points; and 3) has the potential to adjust for any time-varying covariate effects. Our model can be used to evaluate public health interventions, identify latent events associated with spreading rates, and yield better short-term forecasts.

 Bayesian hierarchical modeling multiple change-point detection Poisson segmented regression stochastic SIR model National Institutes of Health1R56HG0110355P30CA1425435R01GM1264795R01HG008983This work was supported by the University of Texas at Dallas (UT Dallas) Office of Research [UT Dallas Center for Disease Dynamics and Statistics] and partially supported by the National Institutes of Health [1R56HG011035, 5P30CA142543, 5R01GM126479, 5R01HG008983]. References Akiyama MJ, Spaulding AC, Rich JD (2020). Flattening the curve for incarcerated populations—COVID-19 in jails and prisons. New England Journal of Medicine, 382(22): 2075–2077. Allen LJ (2008). An introduction to stochastic epidemic models. In: Mathematical Epidemiology, (F Brauer, P van den Driessche, J Wu, eds.), 81–130. Springer. Alvarez FE, Argente D, Lippi F (2020). A simple planning problem for COVID-19 lockdown, Technical report, National Bureau of Economic Research. Andersson H, Britton T (2012). Stochastic Epidemic Models and Their Statistical Analysis, volume 151. Springer Science & Business Media. Bailey NT, et al. (1975). The Mathematical Theory of Infectious Diseases and Its Applications. Charles Griffin & Company Ltd, 5a Crendon Street, High Wycombe, Bucks HP13 6LE. Becker NG, Britton T (1999). Statistical studies of infectious disease incidence. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(2): 287–307. Chen YC, Lu PE, Chang CS (2020). A time-dependent SIR model for COVID-19. arXiv preprint: https://arxiv.org/abs/2003.00122. Chowell G, Castillo-Chavez C, Fenimore PW, Kribs-Zaleta CM, Arriola L, Hyman JM (2004). Model parameters and outbreak control for SARS. Emerging Infectious Diseases, 10(7): 1258. Cooper I, Mondal A, Antonopoulos CG (2020). A SIR model assumption for the spread of COVID-19 in different communities. Chaos, Solitons & Fractals, 139: 110057. Dehning J, Zierenberg J, Spitzner FP, Wibral M, Neto JP, Wilczek M, et al. (2020). Inferring change points in the spread of COVID-19 reveals the effectiveness of interventions. Science, 369(6500): eabb9789. Dong E, Du H, Gardner L (2020). An interactive web-based dashboard to track COVID-19 in real time. The Lancet Infectious Diseases, 20(5): 533–534. Edwards AW, Cavalli-Sforza LL (1965). A method for cluster analysis. Biometrics, 21(2): 362–375. Flaxman S, Mishra S, Gandy A, Unwin HJT, Mellan TA, Coupland H, et al. (2020). Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe. Nature, 584(7820): 257–261. Gelman A, Lee D, Guo J (2015). Stan: A probabilistic programming language for Bayesian inference and optimization. Journal of Educational and Behavioral Statistics, 40(5): 530–543. Gelman A, Rubin DB, et al. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7(4): 457–472. Giordano G, Blanchini F, Bruno R, Colaneri P, Di Filippo A, Di Matteo A, et al. (2020). Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nature Medicine, 26: 855–860. 2020. Gostic KM, McGough L, Baskerville EB, Abbott S, Joshi K, Tedijanto C, et al. (2020 Dec 10). Practical considerations for measuring the effective reproductive number, R t . PLoS Computational Biology, 16(12): e1008409. Haario H, Laine M, Mira A, Saksman E (2006). DRAM: efficient adaptive MCMC. Statistics and Computing, 16(4): 339–354. Hinkley DV (1970). Inference about the change-point in a sequence of random variables. Biometrika, 57(1): 1–17. Hubert L, Arabie P (1985). Comparing partitions. Journal of Classification, 2(1): 193–218. Jen T, Gupta AK (1987). On testing homogeneity of variances for Gaussian models. Journal of Statistical Computation and Simulation, 27(2): 155–173. Kantner M, Koprucki T (2020). Beyond just “flattening the curve”: Optimal control of epidemics with purely non-pharmaceutical interventions. Journal of Mathematics in Industry, 10(1): 1–23. Kermack WO, McKendrick AG (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, 115(772): 700–721. Containing papers of a mathematical and physical character. Killick R, Eckley I (2014). changepoint: An R package for changepoint analysis. Journal of Statistical Software, 58(3): 1–19. Killick R, Fearnhead P, Eckley IA (2012). Optimal detection of changepoints with a linear computational cost. Journal of the American Statistical Association, 107(500): 1590–1598. Liang F, Liu C, Carroll R (2011). Advanced Markov Chain Monte Carlo Methods: Learning from Past Samples, volume 714. John Wiley & Sons. Liu JS, Liang F, Wong WH (2000). The multiple-try method and local optimization in Metropolis sampling. Journal of the American Statistical Association, 95(449): 121–134. Lloyd-Smith JO, Galvani AP, Getz WM (2003). Curtailing transmission of severe acute respiratory syndrome within a community and its hospital. Proceedings of the Royal Society of London. Series B: Biological Sciences, 270: 1979–1989. 1528. Martino L (2018). A review of multiple try MCMC algorithms for signal processing. Digital Signal Processing, 75: 134–152. Mira A, et al. (2001). On Metropolis-Hastings algorithms with delayed rejection. Metron, 59(3–4): 231–241. Pedersen MG, Meneghini M (2020). Quantifying undetected COVID-19 cases and effects of containment measures in Italy. ResearchGate Preprint (online 21 March 2020). DOI: https://doi.org/10.13140/RG.2.2.11753.85600. Rand WM (1971). Objective criteria for the evaluation of clustering methods. Journal of the American Statistical Association, 66(336): 846–850. Riley S, Fraser C, Donnelly CA, Ghani AC, Abu-Raddad LJ, Hedley AJ, et al. (2003). Transmission dynamics of the etiological agent of SARS in Hong Kong: impact of public health interventions. Science, 300(5627): 1961–1966. Riou J, Hauser A, Counotte MJ, Althaus CL (2020). Adjusted age-specific case fatality ratio during the COVID-19 epidemic in Hubei, China, January and February 2020. medRxiv. Roberts GO, Rosenthal JS (1998). Optimal scaling of discrete approximations to Langevin diffusions. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60(1): 255–268. Salvatier J, Wiecki TV, Fonnesbeck C (2016). Probabilistic programming in Python using PyMC3. PeerJ Computer Science, 2: e55. Santos JM, Embrechts M (2009). On the use of the adjusted rand index as a metric for evaluating supervised classification. In: International Conference on Artificial Neural Networks, (C Alippi, MM Polycarpou, CG Panayiotou, G Ellinas, eds.), 175–184. Springer. Sen A, Srivastava MS (1975). On tests for detecting change in mean. The Annals of Statistics, 3(1): 98–108. Steuer R, Kurths J, Daub CO, Weise J, Selbig J (2002). The mutual information: detecting and evaluating dependencies between variables. Bioinformatics, 18(suppl 2): S231–S240. Tadesse MG, Sha N, Vannucci M (2005). Bayesian variable selection in clustering high-dimensional data. Journal of the American Statistical Association, 100(470): 602–617. Talts S, Betancourt M, Simpson D, Vehtari A, Gelman A (2018). Validating bayesian inference algorithms with simulation-based calibration. arXiv preprint: https://arxiv.org/abs/1804.06788. Toda AA (2020). Susceptible-infected-recovered (SIR) dynamics of COVID-19 and economic impact. arXiv preprint: https://arxiv.org/abs/2003.11221. Vehtari A, Gelman A, Gabry J (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27(5): 1413–1432. Wang L, Zhou Y, He J, Zhu B, Wang F, Tang L, et al. (2020 Jul 1). An epidemiological forecast model and software assessing interventions on the COVID-19 epidemic in China. Journal of Data Science, 18(3): 409–432. Waqas M, Farooq M, Ahmad R, Ahmad A (2020). Analysis and prediction of COVID-19 pandemic in Pakistan using time-dependent SIR model. arXiv preprint: https://arxiv.org/abs/2005.02353. Weitz JS, Beckett SJ, Coenen AR, Demory D, Dominguez-Mirazo M, Dushoff J, et al. (2020). Modeling shield immunity to reduce COVID-19 epidemic spread. Nature Medicine, 26: 849–854. Zhou T, Ji Y (2020). Semiparametric Bayesian inference for the transmission dynamics of COVID-19 with a state-space model. arXiv preprint: https://arxiv.org/abs/2006.05581.