论文标题
与未检测到的感染者相关的时间依赖于期限的SIR模型
A Time-dependent SIR model for COVID-19 with Undetectable Infected Persons
论文作者
论文摘要
在本文中，我们进行数学和数值分析以解决COVID-19的以下关键问题：（Q1）是否可以包含COVID-19？ （Q2）什么时候是流行病的峰值和末端？ （Q3）无症状感染如何影响疾病的传播？ （Q4）需要感染以实现牛群免疫的人群的比率是多少？ （Q5）社会距离的方法有多有效？ （Q6）从长远来看，感染的人口比率是多少？对于（Q1）和（Q2），我们提出了一个时间依赖性的易感感染的（SIR）模型，该模型跟踪2个时间序列：（i）时间t和（ii）时间t的恢复速率在时间t和（ii）。这种方法比传统的静态SIR模型更具有适应性，并且比直接估计方法更健壮。使用中国提供的数据，我们表明确认案件数量的一日预测错误几乎为3％，并且确切地预测了确认案件的总数。同样，可以准确预测的转换点（定义为传输速率少于恢复速率的一天）。那天之后，基本的繁殖数$ r_0 $小于1。对于（Q3），我们通过考虑2种感染者的sir模型来扩展我们的SIR模型：可检测和无法检测到的感染者。这种模型中是否有爆发的特征是与$ r_0 $密切相关的2乘以2矩阵的光谱半径。对于（Q4），我们表明在至少1-1/$ r_0 $被感染的人中，可以实现牛群的免疫力。对于（Q5）和（Q6），我们分析了构型随机图中疾病传播的独立级联模型。通过将IC模型中的传播概率与SIR模型中的传输率和回收率联系起来，我们显示了2种社会距离的方法，这些方法可能导致$ r_0 $。
In this paper, we conduct mathematical and numerical analyses to address the following crucial questions for COVID-19: (Q1) Is it possible to contain COVID-19? (Q2) When will be the peak and the end of the epidemic? (Q3) How do the asymptomatic infections affect the spread of disease? (Q4) What is the ratio of the population that needs to be infected to achieve herd immunity? (Q5) How effective are the social distancing approaches? (Q6) 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 2 time series: (i) the transmission rate at time t and (ii) the recovering rate at time t. Such an approach is more adaptive than traditional static SIR models and more robust than direct estimation methods. Using the data provided by China, we show that the one-day prediction errors for the numbers of confirmed cases are almost in 3%, and the total number of confirmed cases is precisely predicted. Also, the turning point, defined as the day that the transmission rate is less than the recovering rate can be accurately predicted. After that day, the basic reproduction number $R_0$ is less than 1. For (Q3), we extend our SIR model by considering 2 types of infected persons: detectable and undetectable infected persons. Whether there is an outbreak in such a model is characterized by the spectral radius of a 2 by 2 matrix that is closely related to $R_0$. For (Q4), we show that herd immunity can be achieved after at least 1-1/$R_0$ fraction of individuals being infected. For (Q5) and (Q6), we analyze the independent cascade (IC) model for disease propagation in a configuration random graph. By relating the propagation probabilities in the IC model to the transmission rates and recovering rates in the SIR model, we show 2 approaches of social distancing that can lead to a reduction of $R_0$.
