在本論文中，我們透過數學與數值上的分析來解決以下關於COVID-19的重要問題：(問題一)我們是否能夠控制COVID-19的疫情發展?(問題二)如果COVID-19的疫情能夠被控制，傳染病的高峰期何時會出現，何時會結束?(問題三)無症狀感染者會如何影響疾病的傳播?(問題四)如果COVID-19的疫情無法被控制，為了達到群體免疫(herd immunity)而需要被感染的人口比例是多少?(問題五)社交距離(social distancing)的防疫措施效果如何?(問題六)如果COVID-19的疫情無法被控制，從長遠來看，會有多少比例的人口被感染?對於(問題一和二)，我們提出了一個時間依持 可感染-已感染-已復原(SIR) 模型來追蹤兩個時間序列:(其一)在時間t時的傳染率(transmission rate)與(其二)在時間t時的復原率(recovering rate)。這種方法不僅比傳統的靜態SIR模型更具適應性，而且比直接估計的方法更強健(robust)。使用中華人民共和國國家衛生健康委員會(NHC)所提供的疫情資料，我們展示了對確診病例數的一日預測誤差幾乎不到3%。還有轉捩點(定義為傳染率小於復原率的那一天)預計是在2020年2月17日。在那天之後，基本再生數(即所謂時間t時的R0值)將小於1。在這種情況下，我們的模型預測中國的確診病例總數約為80,000例。對於(問題三)我們透過考慮兩種類型的感然者來延伸我們的SIR模型：可被偵測與不可被偵測的感染者。在這種模型下是否會有疫情爆發與一個2x2矩陣的譜半徑(spectral radius)有關，而這個譜半徑與基本再生數R0密切相關。我們繪製了疫情爆發的相變圖並顯示有多個國家在2020年3月2日時瀕臨COVID-19疫情爆發，包括南韓、義大利與伊朗。對於(問題四)我們展示了至少1-1/R0比例的個體感染COVID-19並復原後可達到群體免疫。對於(問題五和六)，我們分析指定度分佈(degree distribution)的隨機網路中疾病傳播的獨立串聯(independent cascade, IC)模型。透過將IC模型中的傳播機率與SIR模型中的傳染率與復原率相關聯，我們展示了兩種社交距離方式皆可使R0值下降。

In this thesis, we conduct mathematical and numerical analyses to address the following important questions for COVID-19: (Q1) Is it possible to contain COVID-19? (Q2) If COVID-19 can be contained, when will be the peak of the epidemic, and when will it end? (Q3) How do the asymptomatic infections affect the spread of disease? (Q4) If COVID-19 cannot be contained, what is the ratio of the population that needs to be infected in order to achieve herd immunity? (Q5) How effective are the social distancing approaches? (Q6) If COVID-19 cannot be contained, what is the ratio of the population infected in the long run? For (Q1) and (Q2), we propose a time-dependent susceptible-infected-recovered (SIR) model that tracks two time series: (i) the transmission rate at time t and (ii) the recovering rate at time t. Such an approach is not only more adaptive than traditional static SIR models, but also more robust than direct estimation methods. Using the data provided by the National Health Commission of the People's Republic of China (NHC), we show that the one-day prediction errors for the numbers of confirmed cases are almost less than 3%. Also, the turning point, defined as the day that the transmission rate is less than the recovering rate, is predicted to be Feb. 17, 2020. After that day, the basic reproduction number, known as the R0 value at time t, is less than 1. In that case, the total number of confirmed cases is predicted to be around 80,000 cases in China under our model. For (Q3), we extend our SIR model by considering two types of infected persons: detectable infected persons and undetectable infected persons. Whether there is an outbreak in such a model is characterized by the spectral radius of a 2x2 matrix that is closely related to the basic reproduction number R0. We plot the phase transition diagram of an outbreak and show that there are several countries, including South Korea, Italy, and Iran, that are on the verge of COVID-19 outbreaks on Mar. 2, 2020. For (Q4), we show that herd immunity can be achieved after at least 1-1/R0 fraction of individuals being infected and recovered from COVID-19. For (Q5) and (Q6), we analyze the independent cascade (IC) model for disease propagation in a random network specified by a degree distribution. By relating the propagation probabilities in the IC model to the transmission rates and recovering rates in the SIR model, we show two approaches of social distancing that can lead to a reduction of R0.

Contents-----1
List of Figures-----4
Introduction-----5
The Time-dependent SIR Model-----12
The SIR Model with Undetectable Infected Persons-----18
The Independent Cascade (IC) Model for Disease Propagation in Networks-----23
Numerical Results-----35
Discussions and Suggestions-----50
Conclusion and Future Work-----53

[1] “Outbreak notification,” Jan 2020. [Online]. Available: http://www.nhc.gov.cn/xcs/yqtb/list gzbd.shtml
[2] “Coronavirus disease (covid-19) outbreak,” Jan 2020. [Online]. Available: https://www.who.int/emergencies/diseases/novel-coronavirus-2019
[3] I. Nesteruk, “Statistics based predictions of coronavirus 2019-ncov spreading in mainland china,” MedRxiv, 2020.
[4] Y. Chen, J. Cheng, Y. Jiang, and K. Liu, “A time delay dynamical model for outbreak of 2019-ncov and the parameter identification,” arXiv preprint arXiv:2002.00418, 2020.
[5] L. Peng, W. Yang, D. Zhang, C. Zhuge, and L. Hong, “Epidemic analysis of covid-19 in china by dynamical modeling,” arXiv preprint arXiv:2002.06563, 2020.
[6] T. Zhou, Q. Liu, Z. Yang, J. Liao, K. Yang, W. Bai, X. Lu, and W. Zhang, “Preliminary prediction of the basic reproduction number of the wuhan novel coronavirus 2019-ncov,” Journal of Evidence-Based Medicine, 2020.
[7] B. F. Maier and D. Brockmann, “Effective containment explains subexponential growth in confirmed cases of recent covid-19 outbreak in mainland china,” arXiv preprint arXiv:2002.07572, 2020.
[8] S. Zhao, Q. Lin, J. Ran, S. S. Musa, G. Yang, W. Wang, Y. Lou, D. Gao, L. Yang, D. He et al., “Preliminary estimation of the basic reproduction number of novel coronavirus (2019-ncov) in china, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak,” International Journal of Infectious Diseases, 2020.
[9] T. Zeng, Y. Zhang, Z. Li, X. Liu, and B. Qiu, “Predictions of 2019- ncov transmission ending via comprehensive methods,” arXiv preprint arXiv:2002.04945, 2020.
[10] Z. Hu, Q. Ge, L. Jin, and M. Xiong, “Artificial intelligence forecasting of covid-19 in china,” arXiv preprint arXiv:2002.07112, 2020.
[11] M. Newman, Networks: An Introduction. Oxford University Press, 2010.
[12] CSSEGISandData and J. H. University, “Covid-19,” Feb 2020. [Online]. Available: https://github.com/CSSEGISandData/COVID-19
[13] D. Kempe, J. Kleinberg, and ´ E. Tardos, “Maximizing the spread of influence through a social network,” in Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, 2003, pp. 137–146.
[14] A. Y. Ng, A. X. Zheng, and M. I. Jordan, “Stable algorithms for link analysis,” in Proceedings of the 24th annual international ACM SIGIR conference on Research and development in information retrieval, 2001, pp. 258–266.
[15] B. S. Dayal and J. F. MacGregor, “Identification of finite impulse response models: methods and robustness issues,” Industrial & engineering chemistry research, vol. 35, no. 11, pp. 4078–4090, 1996.
[16] T. Ganyani, C. Kremer, D. Chen, A. Torneri, C. Faes, J. Wallinga, and N. Hens, “Estimating the generation interval for covid-19 based on symptom onset data,” medRxiv, 2020.
[17] C.-J. Chen, Epidemiology: Principles and Methods. Linking Publishing Company, 1999.
[18] N. G. Becker, Modeling to inform infectious disease control. CRC Press, 2015, vol. 74.
[19] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, “Scikit-learn: Machine learning in Python,” Journal of Machine Learning Research, vol. 12, pp. 2825–2830, 2011.
[20] “Real-time big data report of covid-19 outbreak.” [Online]. Available: https://voice.baidu.com/act/newpneumonia/newpneumonia
[21] W.-J. Guan, Z.-Y. Ni, Y. Hu, W.-H. Liang, C.-Q. Ou, J.-X. He, L. Liu, H. Shan, C.-L. Lei, D. S. Hui et al., “Clinical characteristics of 2019 novel coronavirus infection in china,” medRxiv, 2020.
[22] B. Rozemberczki, R. Davies, R. Sarkar, and C. Sutton, “Gemsec: Graph embedding with self clustering,” in Proceedings of the 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining 2019. ACM, 2019, pp. 65–72. [Online]. Available: https://snap.stanford.edu/data/gemsec-Facebook.html
[23] P. Erdos, “On random graphs,” Publicationes mathematicae, vol. 6, pp. 290–297, 1959.
[24] A.-L. Barab´asi and R. Albert, “Emergence of scaling in random networks,” science, vol. 286, no. 5439, pp. 509–512, 1999.