從2019 年底起，COVID19 迅速的演變成一場席捲全球的大災難。當我們認為疾病快要平息時，總是會有新的變種病毒再次造成確診人數創新高，如2020 年底的Alpha，2021 年中的Delta，2021 年底持續到現在的Omicron 及他的各種變種BA.1、BA.5 等等，及今年在印度及其他國家造成大規模爆發的Arcturus (XBB.1.16)。我們提出一個簡易的方法，利用SIR傳染病模型透過最小化實際資料和模擬數據之間的平方距離之和來估計參數，計算出參數β、γ 及推導基本再生數(R0)，再利用基本再生數和其他參數偵測其變化點，目的是為了找出基本再生數的劇烈變化點，以達到觀測基本再生數變化的過程。我們將這個方法應用於數個國家進行分析變種病毒出現的時間，可以了解這些國家變種病毒爆發的情況。在未來我們的方法可以應用於新疾病的發生，通過確認和清除的數量來檢測是否有變化點，從而在早期判斷是否存在新的變異病毒或社區爆發問題。

Since late 2019, the coronavirus pandemic (COVID-19) has rapidly escalated into a global catastrophe. Every time we think the disease is about to die down, there are always new variant appear, causing the confirmed cases to reach record highs. Such as the Alpha variant (B.1.1.7) in late 2020, the Delta variant (B.1.617.2) in mid-2021. Since late 2021 and up to the present, there has been a steady emergence of Omicron and its various sub-variants, such as BA.1, BA.5, and the Arcturus variant (XBB.1.16) that is responsible for the massive outbreak in India and other countries this year. We propose a simple method, using the compartmental model, the susceptible-infectious-removed (SIR) model to estimate the parameters, by minimizing the sum of the squared distances between the observed data and the simulated results. This allows us to calculate the parameters of the SIR model, β and γ. We then use the change point detection on the basic reproduction numbers(R0) and other parameters to identify the points of drastic change with the aim of observing the process of the basic reproduction number variation. We have applied our method to analyze the emergence time of variant strains in several countries, and gained insights into the outbreak of variant strains in these countries. In the future, our method can be applied to the emergence of new diseases by detecting change points through the number of confirmed and removed cases, thereby detecting the presence of new variant strains or community outbreaks at an early stage.

Acknowledgements
摘要 i
Abstract ii
1 Introduction 1
1.1 Motivation . . . . . . . .. . . . . . . . . . . . .. . . . . 1
1.2 Time-dependent SIR model . . . . . . . . . . . . . . . . . . 2
1.3 The basic reproduction number (R0) . . . . . . . . . . . . . 4
2 Method 6
2.1 Estimate the transmission rate (β(t)), the removal rate (γ(t)) and the basic reproduction number (R0(t)) . . . . . . . . . . . . .6
2.2 Detect the change points . . . . . . . . . . . . . . . . . . .10
3 Countries Analysis 14
3.1 Selection of target dates . . . . . . . . . . . . . . . . . . 14
3.2 Countries analysis . . . . . . . . . . .. . . . . . . . . . . 15
4 Discussion 21
5 Conclusion and future work 25
References 27

[1] C. Wang, L. Liu, X. Hao, H. Guo, Q. Wang, J. Huang, N. He, H. Yu, X. Lin, A. Pan, et al., “Evolving epidemiology and impact of non-pharmaceutical interventions on the outbreak of coronavirus disease 2019 in wuhan, china,” MedRxiv, pp. 2020–03, 2020.
[2] M. Alazab, A. Awajan, A. Mesleh, A. Abraham, V. Jatana, and S. Alhyari, “Covid-19 prediction and detection using deep learning,” International Journal of Computer Information Systems and Industrial Management Applications, vol. 12, no. June, pp. 168–181, 2020.
[3] L. Li, Z. Yang, Z. Dang, C. Meng, J. Huang, H. Meng, D. Wang, G. Chen, J. Zhang,
H. Peng, et al., “Propagation analysis and prediction of the covid-19,” Infectious Disease Modelling, vol. 5, pp. 282–292, 2020.
[4] “Johns hopkins cornonavirus resource center,” 2020. https://coronavirus.jhu.edu.
[5] R. Ross, “An application of the theory of probabilities to the study of a priori pathometry.—part i,” Proceedings of the Royal Society of London. Series A, Containing papers of a mathematical and physical character, vol. 92, no. 638, pp. 204–230, 1916.
[6] W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epidemics,” Proceedings of the royal society of london. Series A, Containing papers of a mathematical and physical character, vol. 115, no. 772, pp. 700–721, 1927.
[7] C.-L. Sung and Y. Hung, “Efficient calibration for imperfect epidemic models with applications to the analysis of covid-19,” arXiv preprint arXiv:2009.12523, 2020.
[8] S. Moein, N. Nickaeen, A. Roointan, N. Borhani, Z. Heidary, S. H. Javanmard, J. Ghaisari, and Y. Gheisari, “Inefficiency of sir models in forecasting covid-19 epidemic: a case study of isfahan,” Scientific reports, vol. 11, no. 1, p. 4725, 2021.
[9] R. M. May, “Simple mathematical models with very complicated dynamics,” Nature, vol. 261, no. 5560, pp. 459–467, 1976.
[10] H. G. Hong and Y. Li, “Estimation of time-varying reproduction numbers underlying epidemiological processes: A new statistical tool for the covid-19 pandemic,” PloS one, vol. 15, no. 7, p. e0236464, 2020.
[11] Y.-C. Chen, P.-E. Lu, C.-S. Chang, and T.-H. Liu, “A time-dependent sir model for covid-19 with undetectable infected persons,” Ieee transactions on network science and engineering, vol. 7, no. 4, pp. 3279–3294, 2020.
[12] M. Newman, Networks: An Introduction. Oxford university press, 2010.
[13] J. H. Jones, “Notes on r0. department of anthropological sciences,” 2007.
[14] G. Chowell, C. Castillo-Chavez, P. W. Fenimore, C. M. Kribs-Zaleta, L. Arriola, and J. M. Hyman, “Model parameters and outbreak control for sars,” Emerging infectious diseases, vol. 10, no. 7, p. 1258, 2004.
[15] A. Capaldi, S. Behrend, B. Berman, J. Smith, J. Wright, and A. L. Lloyd, “Parameter estimation and uncertainty quantication for an epidemic model,” Mathematical biosciences and engineering, p. 553, 2012.
[16] E. Levy, “Complex-curve fitting,” IRE transactions on automatic control, no. 1, pp. 37–43, 1959.
[17] A. Agrawal, B. Amos, S. Barratt, S. Boyd, S. Diamond, and J. Z. Kolter, “Differentiable convex optimization layers,” Advances in neural information processing systems, vol. 32, 2019.
[18] V. Cevher, S. Becker, and M. Schmidt, “Convex optimization for big data: Scalable, randomized, and parallel algorithms for big data analytics,” IEEE Signal Processing Magazine, vol. 31, no. 5, pp. 32–43, 2014.
[19] J. Edward Jackson, “Multivariate quality control,” Communications in Statistics-Theory and Methods, vol. 14, no. 11, pp. 2657–2688, 1985.
[20] F. Aparisi and C. L. Haro, “Hotelling’s t2 control chart with variable sampling intervals,” International Journal of Production Research, vol. 39, no. 14, pp. 3127–3140, 2001.
[21] W. Gani, H. Taleb, and M. Limam, “An assessment of the kernel-distance-based multivariate control chart through an industrial application,” Quality and Reliability Engineering International, vol. 27, no. 4, pp. 391–401, 2011.
[22] J. H. Sullivan and W. H. Woodall, “A comparison of multivariate control charts for individual observations,” Journal of Quality Technology, vol. 28, no. 4, pp. 398–408, 1996.
[23] D. S. Holmes and A. E. Mergen, “Improving the performance of the t2 control chart,” Quality Engineering, vol. 5, no. 4, pp. 619–625, 1993.
[24] N. J. A. Vargas, “Robust estimation in multivariate control charts for individual observations,” Journal of Quality Technology, vol. 35, no. 4, pp. 367–376, 2003.
[25] N. D. Tracy, J. C. Young, and R. L. Mason, “Multivariate control charts for individual observations,” Journal of quality technology, vol. 24, no. 2, pp. 88–95, 1992.
[26] “Our world in data,” 2020. https://ourworldindata.org/.
[27] G. C. Runger, J. B. Keats, D. C. Montgomery, and R. D. Scranton, “Improving the performance of the multivariate exponentially weighted moving average control chart,” Quality and reliability engineering international, vol. 15, no. 3, pp. 161–166, 1999.