International Journal of Mathematical, Engineering and Management Sciences

ISSN: 2455-7749

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

Ali Muhammad Ali Rushdi
Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, P. O. Box 80204, Jeddah, 21589, Saudi Arabia.

Hamzah Abdul Majid Serag
Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, P. O. Box 80204, Jeddah, 21589, Saudi Arabia.

DOI https://doi.org/10.33889/IJMEMS.2020.5.5.062

Received on May 07, 2020
  ;
Accepted on May 28, 2020

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.

Keywords- Ternary problem, Conditional probability, Length/area proportionality, Probabilistic dependence, Divide and conquer, Imperfect testing, Mathematical epidemiology, COVID-19.

Citation

Rushdi, A. M. A., & Serag, H. A. M. (2020). 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, 5(5), 787-711. https://doi.org/10.33889/IJMEMS.2020.5.5.062.

Conflict of Interest

The authors assert that no conflict of interest exists.

Acknowledgements

This work is funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the authors acknowledge, with thanks, the DSR for their financial and technical support. The first-named author (AMAR) is gratefully indebted to Dr. Rufaidah Rushdi, of Kasr Al-Ainy Faculty of Medicine (Cairo University, Arab Republic of Egypt) for stimulating discussions concerning epidemiological testing and its various measures, and for proposing an application of our results to COVID-19.

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