具有因变量感染率的COVID-19流行动力学的数学模型：在大韩民国的应用
A mathematical model of the COVID-19 pandemic dynamics with dependent variable infection rate: Application to the Republic of Korea
摘要：
这项工作构建、分析和模拟了一个新的分室SEIR型模型，用于当前COVID-19大流行的动态和潜在控制。这项工作的新颖性是双重的。首先，根据对疾病控制指令的遵守情况（封锁、就地避难、口罩/面罩、物理距离等）对人口进行划分。这些人员分为完全遵守指示的人员和部分遵守指示的人员，或必须流动的人员（如医务人员）。这种分裂间接反映了这些措施的质量和一致性。这有助于评估控制措施的总体成效，以及放松或收紧这些措施对疾病传播的影响。其次，直接影响感染率的适当接触率是模型未知数之一，因为它通过微分包含跟踪人口行为的变化和各种疾病治疗方式的有效性。存在性，唯一性和积极的结果证明使用非标准的凸分析为基础的方法。作为一个案例研究，在大韩民国（韩国）的大流行疫情进行了模拟。模型参数是通过最小化模型预测与韩国大流行前100天报告数据的偏差来确定的。模拟结果表明，模型准确地捕捉了随后75天的大流行动态，这为模型预测及其未来使用提供了信心。特别是，该模型预测，约40%的感染没有记录，这意味着无症状感染默默地但实质性地促进了疾病的传播，这表明需要更广泛的无症状检测。
英文摘要：
This work constructs, analyzes, and simulates a new compartmental SEIR-type model for the dynamics and potential control of the current COVID-19 pandemic. The novelty in this work is two-fold. First, the population is divided according to its compliance with disease control directives (lockdown, shelter-in-place, masks/face coverings, physical distancing, etc.) into those who fully comply and those who follow the directives partially, or are necessarily mobile (such as medical staff). This split, indirectly, reflects on the quality and consistency of these measures. This allows the assessment of the overall effectiveness of the control measures and the impact of their relaxing or tightening on the disease spread. Second, the adequate contact rate, which directly affects the infection rate, is one of the model unknowns, as it keeps track of the changes in the population behavior and the effectiveness of various disease treatment modalities via a differential inclusion. Existence, uniqueness and positivity results are proved using a nonstandard convex analysis-based approach. As a case study, the pandemic outbreak in the Republic of Korea (South Korea) is simulated. The model parameters were found by minimizing the deviation of the model prediction from the reported data over the first 100 days of the pandemic in South Korea.The simulations show that the model captures accurately the pandemic dynamics in the subsequent 75 days, which provides confidence in the model predictions and its future use. In particular, the model predicts that about 40% of the infections were not documented, which implies that asymptomatic infections contribute silently but substantially to the spread of the disease indicating that more widespread asymptomatic testing is necessary.
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