# Quantification of the Tradeoff between Test Sensitivity and Test Frequency in a COVID-19 Epidemic—A Multi-Scale Modeling Approach

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Within-Host Model

#### 2.2. Between-Host Model

#### 2.3. Daily Testing Rate

#### 2.4. Between-Host Model with Testing

## 3. Results

#### 3.1. The Relationship between Test Sensitivity and Virus Titers

#### 3.2. Mathematical Model of Testing during SARS-CoV-2 Transmission

#### 3.3. Quantifying the Tradeoff between Test Sensitivity and Return Delay

#### 3.4. Quantifying the Tradeoff between Test Sensitivity and Test Frequency

#### 3.5. Transmission According to Infection Status

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Numerical Scheme

#### Appendix A.1. Initialization

#### Appendix A.2. Discretized Functions

#### Appendix A.3. Updating State Variables

**Figure A1.**

**Cumulative positive cases (as proportion of the total population) at half a year.**RT-PCR with return delay $\ell =1$ days (dark blue), $\ell =2$ days (light blue) and testing capacity $C=0.1$; Ag test with return delay $\ell =0.5$ days and testing capacity $C=1$ (red), $C=0.5$ (orange), $C=0.33$ (maroon), and $C=0.1429$ (magenta). All other parameters and initial conditions are given in Table 1 and Table 2.

**Figure A2.**

**Cumulative cases (as proportion of the total population) at half a year, when testing $1\%$ of the population daily.**Heatmaps for the cumulative cases (as proportion of the total population) at half a year after the outbreak (% of the total population) as given by model Equation (5) versus test sensitivity and test return delay. Panel (

**A**): fixed testing capacity per day, $C=0.01$. Panel (

**B**): fixed testing budget per day. All other parameters and initial conditions are given in Table 1 and Table 2.

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**Figure 1.**

**RT-PCR versus rapid testing practices.**${log}_{10}$ virus load per swab over time as given by model (1) (grey curves) for values in [26]. Patients are assumed to be infectious from $t=2.5$ days (IS) till $t=10.5$ days (IE) (shaded region) and symptomatic beginning on day $t=4$ (SO). Panels (

**A**) and (

**B**) depict testing with a high-sensitivity RT-PCR test with detection threshold ${log}_{10}\left(V\right)=2$ per swab (red line) and test return delay of five days. In panel A, the test occurs immediately following symptoms onset, and in panel B, the test occurs before symptoms onset (red circles). Panel

**C**depicts frequent testing (yellow circles) with a low-sensitivity test with detection threshold ${log}_{10}\left(V\right)=5$ per swab (yellow line) and test return delay of one half day. TR shows the time of positive test result.

**Figure 2.**

**Epidemic dynamics over time**. Sample epidemic dynamics results from varying testing regimes, as given by model Equation (5) for fixed testing capacity. Panel (

**A**): RT-PCR, detection threshold ${log}_{10}\left(V\right)=2$, test return delay 5 days; Panel (

**B**): antigen test, detection threshold ${log}_{10}\left(V\right)=5$, test return delay 0.5 days; Panel (

**C**): paper-strip test, detection threshold ${log}_{10}\left(V\right)=6$, test return delay 0.1 days. Upper left figures: asymptomatic (blue), symptomatic (red) case (as proportion of the total population) over time. Upper right figures: cumulative positive cases (magenta) and cumulative detected cases (green) (as proportion of the total population) over time. Lower figures: daily new cases (yellow bars) and daily new case detections (blue bars).

**Figure 3.**

**Cumulative cases (as proportion of the total population) at half a year.**Heatmaps for the cumulative cases (as proportion of the total population) at half a year after the outbreak (% of the total population) as given by model Equation (5) versus test sensitivity and test return delay. Panel (

**A**): fixed testing capacity per day, $C=0.1$. Panel (

**B**): relationship between capacity and cost. Panel (

**C**): fixed testing budget per day. Parameters and initial conditions are given in Table 1 and Table 2.

**Figure 4.**

**Asymptomatic, presymptomatic and symptomatic transmissions (as proportion of the total population)**. (

**Upper figures**): daily cases (yellow bars) and daily detections (blue bars); (

**Lower figures**): daily cases due to asymptomatic transmission (blue bars), presymptomatic transmission (red bars) and symptomatic transmission (orange bars), as given by model Equation (5) for fixed testing capacity. Panel (

**A**): RT-PCR, detection threshold ${log}_{10}\left(V\right)=2$, test return delay 5 days; Panel (

**B**): antigen test, detection threshold ${log}_{10}\left(V\right)=5$, test return delay 0.5 days; Panel (

**C**): paper-strip test, detection threshold ${log}_{10}\left(V\right)=6$, test return delay 0.1 days.

**Table 1.**Parameter values and initial conditions used in model (1).

Fixed Parameters | Description | Value | Source |
---|---|---|---|

${k}_{j}$ | Eclipse phase duration | 4/day | [36] |

c | Viral clearance | 10/day | [28] |

${g}_{21}$ | Transport between tracts | 0 | [26] |

Estimated Parameters | Description | Value | Source |

${\beta}_{1T}$ | Infection rate in URT | 5.1$\times {10}^{-7}$/swab× day | [26] |

${\beta}_{2T}$ | Infection rate in LRT | $7\times {10}^{-7}$/mL× day | [26] |

${\pi}_{1}$ | URT virus production | 50/day | [26] |

${\pi}_{2}$ | LRT virus production | $0.34$/day | [26] |

${\delta}_{1}$ | URT cell death | 2/day | [26] |

${\delta}_{2}$ | LRT cell death | $0.53$/day | [26] |

$\mathrm{\Gamma}$ | - | 0.01 | [26] |

Initial Conditions | Description | Value | Source |

${T}_{1}\left(0\right)$ | Epithelial cells in URT | $4\times {10}^{6}$/mL | [26] |

${T}_{2}\left(0\right)$ | Epithelial cells in LRT | $4\times {10}^{8}$/mL | [28] |

${E}_{j}\left(0\right)$ | Exposed epithelial cells | 0 | [26] |

${I}_{1}\left(0\right)$ | Infectious epithelial cells in URT | 10 | [26] |

${I}_{2}\left(0\right)$ | Infectious epithelial cells in LRT | 1 | [26] |

${V}_{j}\left(0\right)$ | Virus | 0 | [26] |

**Table 2.**Parameter values and initial conditions used in model Equation (5).

Fixed Parameters | Description | Value | Source |
---|---|---|---|

$\beta $ | Transmission rate | $0.25$/day | |

b | Birth rate | $1/(70\times 365)$/day | |

$\mu $ | Death rate | $1/(70\times 365)$/day | |

${m}_{j}$ | Disease induced mortality rate | ${10}^{-4}$/day | |

f | Fraction of symptomatic infections | $0.7$ | [10] |

$\gamma $ | Relative asymp. infectiousness | 0.7 | |

ℓ | Test return delay | varied | |

${\tau}_{j}^{1}$ | Age of onset of virus detectability | varied (days) | |

${\tau}_{j}^{2}$ | Age of onset of infectiousness | $2.5$ days | [27] |

${\tau}_{j}^{3}$ | Age of end of infectiousness | $10.5$ days | [10,12] |

${\tau}_{j}^{4}$ | Age of loss of virus detectability | varied (days) | |

Initial Conditions | Description | Value | Source |

$S\left(0\right)$ | Susceptible population | 0.99 | |

${i}_{s}(\tau ,0)$ | Infected symptomatic population | $0.01f\delta \left(\tau \right)$ | |

${i}_{a}(\tau ,0)$ | Infected asymptomatic population | $0.01(1-f)\delta \left(\tau \right)$ |

**Table 3.**Percent reduction in daily incidence transmission for antigen and paper-strip tests at various daily testing capacities compared to daily incidence transmission for a RT-PCR test administered at $C=0.1$ testing capacity per day.

Test Type | Infectious Subgroup | $\mathit{C}=0.1$ | $\mathit{C}=0.3$ | $\mathit{C}=0.6$ | $\mathit{C}=1$ |
---|---|---|---|---|---|

Antigen | Symptomatic | $39.6\%$ | $70.5\%$ | $76.4\%$ | $76.7\%$ |

Presymptomatic | $32.8\%$ | $70.0\%$ | $75.3\%$ | $85.6\%$ | |

Asymptomatic | $37.9\%$ | $70.2\%$ | $76.0\%$ | $79.0\%$ | |

Total | $37.9\%$ | $70.2\%$ | $76.0\%$ | $78.9\%$ | |

Paper-strip | Symptomatic | $8.9\%$ | $68.2\%$ | $73.8\%$ | $76.7\%$ |

Presymptomatic | $9.2\%$ | $68.0\%$ | $70.3\%$ | $79.1\%$ | |

Asymptomatic | $9.0\%$ | $68.1\%$ | $73.0\%$ | $77.3\%$ | |

Total | $9.0\%$ | $68.1\%$ | $73.0\%$ | $77.3\%$ |

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**MDPI and ACS Style**

Forde, J.E.; Ciupe, S.M.
Quantification of the Tradeoff between Test Sensitivity and Test Frequency in a COVID-19 Epidemic—A Multi-Scale Modeling Approach. *Viruses* **2021**, *13*, 457.
https://doi.org/10.3390/v13030457

**AMA Style**

Forde JE, Ciupe SM.
Quantification of the Tradeoff between Test Sensitivity and Test Frequency in a COVID-19 Epidemic—A Multi-Scale Modeling Approach. *Viruses*. 2021; 13(3):457.
https://doi.org/10.3390/v13030457

**Chicago/Turabian Style**

Forde, Jonathan E., and Stanca M. Ciupe.
2021. "Quantification of the Tradeoff between Test Sensitivity and Test Frequency in a COVID-19 Epidemic—A Multi-Scale Modeling Approach" *Viruses* 13, no. 3: 457.
https://doi.org/10.3390/v13030457