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Solutions of Ternary Problems of Conditional Probability with Applications to Mathematical Epidemiology and the COVID-19 Pandemic
International Journal of Mathematical, Engineering and Management Sciences Pub Date : 2020-10-01 , DOI: 10.33889/ijmems.2020.5.5.062
Ali Muhammad Ali Rushdi , Hamzah Abdul Majid Serag

A normalized version of the ubiquitous two-by-two contingency matrix is associated with a variety of marginal, conjunctive, and conditional probabilities that serve as appropriate indicators in diagnostic testing. If this matrix is enhanced by being interpreted as a probabilistic Universe of Discourse, it still suffers from two inter-related shortcomings, arising from lack of length/area proportionality and a potential misconception concerning a false assumption of independence between the two underlying events. This paper remedies these two shortcomings by modifying this matrix into a new Karnaugh-map-like diagram that resembles an eikosogram. Furthermore, the paper suggests the use of a pair of functionally complementary versions of this diagram to handle any ternary problem of conditional probability. The two diagrams split the unknowns and equations between themselves in a fashion that allows the use of a divide-and-conquer strategy to handle such a problem. The method of solution is demonstrated via four examples, in which the solution might be arithmetic or algebraic, and independently might be numerical or symbolic. In particular, we provide a symbolic arithmetic derivation of the well-known formulas that express the predictive values in terms of prevalence, sensitivity and specificity. Moreover, we prove a virtually unknown interdependence among the two predictive values, sensitivity, and specificity. In fact, we employ a method of symbolic algebraic derivation to express any one of these four indicators in terms of the other three. The contribution of this paper to the diagnostic testing aspects of mathematical epidemiology culminates in a timely application to the estimation of the true prevalence of the contemporary world-wide COVID-19 pandemic. It turns out that this estimation is hindered more by the lack of global testing world-wide rather than by the unavoidable imperfection of the available testing methods.

中文翻译:

条件概率三元问题的解决及其在数学流行病学和COVID-19大流行中的应用

普遍存在的2×2列联矩阵的规范化版本与各种边际,合取和有条件的概率相关联,这些概率在诊断测试中用作适当的指标。如果通过解释为概率性话语宇宙来增强此矩阵,它仍然会遭受两个相互关联的缺陷,这是由于缺少长度/区域比例和潜在的关于两个潜在事件之间的独立性的错误假设的误解。本文通过将矩阵修改为类似于电子地图的新的类似于Karnaugh-map的图表来弥补这两个缺点。此外,本文建议使用该图的一对功能互补的版本来处理条件概率的任何三元问题。这两张图以允许使用分而治之策略来处理此类问题的方式将未知数和方程式在它们之间进行拆分。通过四个示例演示了求解方法,其中求解可以是算术或代数的,而独立地可以是数字或符号的。特别是,我们提供了众所周知的公式的符号算术推导,这些公式在患病率,敏感性和特异性方面表达了预测值。此外,我们证明了这两个预测值,敏感性和特异性之间几乎未知的相互依赖性。实际上,我们采用符号代数推导的方法来用其他三个指标来表达这四个指标中的任何一个。本文对数学流行病学的诊断测试方面的贡献最终体现在对当代世界范围内COVID-19大流行的真实患病率的估计中的及时应用。事实证明,这种估计更多地受到全球缺乏全球测试的阻碍,而不是由于可用测试方法的不可避免的缺陷而受到阻碍。
更新日期:2020-10-01
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