We propose a mathematical model based on probability theory to optimize COVID-19 testing by a multistep batch testing approach with variable batch sizes. This model and simulation tool dramatically increase the efficiency and efficacy of the tests in a large population at a low cost, particularly when the infection rate is low. The proposed method combines statistical modeling with numerical methods to solve nonlinear equations and obtain optimal batch sizes at each step of tests, with the flexibility to incorporate geographic and demographic information. In theory, this method substantially improves the false positive rate and positive predictive value as well. We also conducted a Monte Carlo simulation to verify this theory. Our simulation results show that our method significantly reduces the false negative rate. More accurate assessment can be made if the dilution effect or other practical factors are taken into consideration. The proposed method will be particularly useful for the early detection of infectious diseases and prevention of future pandemics. The proposed work will have broader impacts on medical testing for contagious diseases in general.

中文翻译：

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传染病多步批量检测的建模与计算

我们提出了一个基于概率论的数学模型，通过具有可变批量大小的多步批量测试方法来优化 COVID-19 测试。该模型和模拟工具以低成本显着提高了在大量人群中进行测试的效率和功效，尤其是在感染率较低的情况下。所提出的方法将统计建模与数值方法相结合，以求解非线性方程并在每个测试步骤中获得最佳批量大小，并且可以灵活地结合地理和人口统计信息。从理论上讲，这种方法也大大提高了假阳性率和阳性预测值。我们还进行了蒙特卡罗模拟来验证这一理论。我们的模拟结果表明，我们的方法显着降低了假阴性率。如果考虑稀释效应或其他实际因素，可以做出更准确的评估。所提出的方法对于早期发现传染病和预防未来的流行病特别有用。拟议的工作将对一般传染病的医学检测产生更广泛的影响。