Solutions of Ternary Problems of Conditional Probability with Applications to Mathematical Epidemiology and the COVID-19 Pandemic

Abstract

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.

Has the Pandemic Triggered a ‘Paperdemic’? Towards an Assessment of Diagnostic Indicators for COVID-19

“…The world currently witnesses an ongoing epidemic of the novel coronavirus (SARS-CoV-2) that causes the disease COVID-19, now characterized as a pandemic by the World Health Organization (WHO) [1][2][3][4][5][6]. This pandemic seems to have expanded from the Wuhan province in China, but has definitely reached (in repetitive waves) almost every inhabited territory on the globe.…”

“…This pandemic seems to have expanded from the Wuhan province in China, but has definitely reached (in repetitive waves) almost every inhabited territory on the globe. This fatal disease, being a catastrophic threat of dramatic public-health and economic concerns proved to have diverse grave (and potentially irreversible) consequences [1][2][3][4]. For the past thirty years, a once-per-decade novel coronavirus has pushed the global public health system to the limit, with SARS-CoV-2 being the most severe.…”

“…In [40], they introduced a non-statistical method based on the premise that flawed data might be detected via the excessive inconsistencies it causes in a variant of Boolean Analysis called Qualitative Comparative Analysis (QCA) [40][41][42][43][44]. Rushdi and Serag [4,45] developed yet another nonstatistical checking method, which can be used for validating a certain category of bio-statistical data. This method is based on a newlydiscovered inter-relation among the four most prominent indicators of diagnostic testing (sensitivity, specificity, positive predictive value (PPV), and negative predictive value (NPV)) [4,40,[45][46][47][48][49].…”

“…Rushdi and Serag [4,45] developed yet another nonstatistical checking method, which can be used for validating a certain category of bio-statistical data. This method is based on a newlydiscovered inter-relation among the four most prominent indicators of diagnostic testing (sensitivity, specificity, positive predictive value (PPV), and negative predictive value (NPV)) [4,40,[45][46][47][48][49]. Rushdi and Serag [4,45] developed simple formulas that express any one of these four indicators in terms of the other three.…”

“…This method is based on a newlydiscovered inter-relation among the four most prominent indicators of diagnostic testing (sensitivity, specificity, positive predictive value (PPV), and negative predictive value (NPV)) [4,40,[45][46][47][48][49]. Rushdi and Serag [4,45] developed simple formulas that express any one of these four indicators in terms of the other three. They called a set of four values satisfying these formulas (to within permissible round-off errors) a consistent set.…”