# Estimation of Population Prevalence of COVID-19 Using Imperfect Tests

## Abstract

**:**

## 1. Introduction

## 2. Basic Assumptions

**A. Independence.**The events of current or past infection in all tested individuals are stochastically independent.

**B. Test Uniformity.**The sensitivity and specificity of a test are identical for all tested individuals.

**C. The Matching Principle.**Tested individuals are selected independently of each other, and the prevalence structure of the sample of these individuals matches that of the target population.

## 3. Distribution of the Number of True and False Positive Test Results

- (a)
- They are independent of the total number, $N,$ of tested individuals;
- (b)
- Parameters $\theta $ and $1-\theta $ specified in (9) represent the predictive positive and predictive negative values that can be obtained by applying Bayes theorem to prior probabilities p and $1-p,$ see, e.g., [17];
- (c)
- Distributions (7) and (8) depend on a single parameter$$\frac{\alpha p}{(1-\beta )(1-p)}$$

## 4. Conditional Expected Number of Infected Individuals for a Given Number of Positive Test Results

## 5. The Equivalence Principle for Heterogeneous Populations

**Equivalence Principle:**

- Under Assumption C, the distribution of the total number of infected individuals among N tested individuals selected from a heterogeneous population consisting of r homogeneous subpopulations with weights ${w}_{1},{w}_{2},\dots ,{w}_{r}$ and infection prevalences ${p}_{1},{p}_{2},\dots ,{p}_{r}$ is the same as for a homogeneous population with infection prevalence$$p=\sum _{i=1}^{r}{w}_{i}{p}_{i}.\phantom{\rule{2.em}{0ex}}$$

## 6. Prevalence Estimation

## 7. Discussion and Recommendations

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Hanin, L.
Estimation of Population Prevalence of COVID-19 Using Imperfect Tests. *Mathematics* **2020**, *8*, 1900.
https://doi.org/10.3390/math8111900

**AMA Style**

Hanin L.
Estimation of Population Prevalence of COVID-19 Using Imperfect Tests. *Mathematics*. 2020; 8(11):1900.
https://doi.org/10.3390/math8111900

**Chicago/Turabian Style**

Hanin, Leonid.
2020. "Estimation of Population Prevalence of COVID-19 Using Imperfect Tests" *Mathematics* 8, no. 11: 1900.
https://doi.org/10.3390/math8111900