# 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.

## References

- World Health Organization Coronavirus Disease (COVID-19) Dashboard. Available online: https://covid19.who.int/ (accessed on 12 February 2021).
- Lee, D.; Lee, J. Testing on the Move South Korea’s rapid response to the COVID-19 pandemic. Transp. Res. Interdiscip. Perspect.
**2020**, 5, 100111. [Google Scholar] [CrossRef] - Gudbjartsson, D.F.; Helgason, A.; Jonsson, H.; Magnusson, O.T.; Melsted, P.; Norddahl, G.L.; Saemundsdottir, J.; Sigurdsson, A.; Sulem, P.; Agustsdottir, A.B.; et al. Spread of SARS-CoV-2 in the Icelandic population. N. Engl. J. Med.
**2020**, 382, 2302–2315. [Google Scholar] [CrossRef] - Rosenberg, E.S.; Holtgrave, D.R. Widespread and frequent testing is essential to controlling COVID-19 in the United States. Clin. Infect. Dis. Off. Publ. Infect. Dis. Soc. Am.
**2020**. [Google Scholar] [CrossRef] [PubMed] - Oran, D.P.; Topol, E.J. Prevalence of asymptomatic SARS-CoV-2 infection: A narrative review. Ann. Intern. Med.
**2020**, 173, 362–367. [Google Scholar] [CrossRef] [PubMed] - Paltiel, A.D.; Zheng, A.; Walensky, R.P. Assessment of SARS-CoV-2 screening strategies to permit the safe reopening of college campuses in the United States. JAMA Netw. Open
**2020**, 3, e2016818. [Google Scholar] [CrossRef] - Zhang, K.; Shoukat, A.; Crystal, W.; Langley, J.M.; Galvani, A.P.; Moghadas, S.M. Routine saliva testing for the identification of silent COVID-19 infections in healthcare workers. Infect. Control. Hosp. Epidemiol.
**2020**, 1–17. [Google Scholar] [CrossRef] - Tang, Y.W.; Schmitz, J.E.; Persing, D.H.; Stratton, C.W. Laboratory diagnosis of COVID-19: Current issues and challenges. J. Clin. Microbiol.
**2020**, 58. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mina, M.J.; Parker, R.; Larremore, D.B. Rethinking Covid-19 test sensitivity—A strategy for containment. N. Engl. J. Med.
**2020**, 383, e120. [Google Scholar] [CrossRef] - He, D.; Zhao, S.; Lin, Q.; Zhuang, Z.; Cao, P.; Wang, M.H.; Yang, L. The relative transmissibility of asymptomatic COVID-19 infections among close contacts. Int. J. Infect. Dis.
**2020**, 94, 145–147. [Google Scholar] [CrossRef] - Huang, C.G.; Lee, K.M.; Hsiao, M.J.; Yang, S.L.; Huang, P.N.; Gong, Y.N.; Hsieh, T.H.; Huang, P.W.; Lin, Y.J.; Liu, Y.C.; et al. Culture-based virus isolation to evaluate potential infectivity of clinical specimens tested for COVID-19. J. Clin. Microbiol.
**2020**, 58. [Google Scholar] [CrossRef] [PubMed] - He, X.; Lau, E.H.; Wu, P.; Deng, X.; Wang, J.; Hao, X.; Lau, Y.C.; Wong, J.Y.; Guan, Y.; Tan, X.; et al. Temporal dynamics in viral shedding and transmissibility of COVID-19. Nat. Med.
**2020**, 26, 672–675. [Google Scholar] [CrossRef][Green Version] - McIntosh, K.; Hirsch, M.; Bloom, A. Coronavirus disease 2019 (COVID-19): Epidemiology, virology, and prevention. Lancet Infect. Dis.
**2020**, 1, 2019–2020. [Google Scholar] - Bullard, J.; Dust, K.; Funk, D.; Strong, J.E.; Alexander, D.; Garnett, L.; Boodman, C.; Bello, A.; Hedley, A.; Schiffman, Z.; et al. Predicting infectious severe acute respiratory syndrome coronavirus 2 from diagnostic samples. Clin. Infect. Dis.
**2020**, 71, 2663–2666. [Google Scholar] [CrossRef] [PubMed] - Lanser, L.; Bellmann-Weiler, R.; Öttl, K.W.; Huber, L.; Griesmacher, A.; Theurl, I.; Weiss, G. Evaluating the clinical utility and sensitivity of SARS-CoV-2 antigen testing in relation to RT-PCR Ct values. Infection
**2020**, 1–3. [Google Scholar] [CrossRef] - Vogels, C.B.; Brito, A.F.; Wyllie, A.L.; Fauver, J.R.; Ott, I.M.; Kalinich, C.C.; Petrone, M.E.; Casanovas-Massana, A.; Muenker, M.C.; Moore, A.J.; et al. Analytical sensitivity and efficiency comparisons of SARS-CoV-2 RT–qPCR primer–probe sets. Nat. Microbiol.
**2020**, 5, 1299–1305. [Google Scholar] [CrossRef] - Etievant, S.; Bal, A.; Escurret, V.; Brengel-Pesce, K.; Bouscambert, M.; Cheynet, V.; Generenaz, L.; Oriol, G.; Destras, G.; Billaud, G.; et al. Sensitivity assessment of SARS-CoV-2 PCR assays developed by WHO referral laboratories. medRxiv
**2020**. [Google Scholar] [CrossRef] - Böhmer, M.M.; Buchholz, U.; Corman, V.M.; Hoch, M.; Katz, K.; Marosevic, D.V.; Böhm, S.; Woudenberg, T.; Ackermann, N.; Konrad, R.; et al. Investigation of a COVID-19 outbreak in Germany resulting from a single travel-associated primary case: A case series. Lancet Infect. Dis.
**2020**, 20, 920–928. [Google Scholar] [CrossRef] - Singanayagam, A.; Patel, M.; Charlett, A.; Bernal, J.L.; Saliba, V.; Ellis, J.; Ladhani, S.; Zambon, M.; Gopal, R. Duration of infectiousness and correlation with RT-PCR cycle threshold values in cases of COVID-19, England, January to May 2020. Eurosurveillance
**2020**, 25, 2001483. [Google Scholar] [CrossRef] - Bryan, A.; Fink, S.L.; Gattuso, M.A.; Pepper, G.; Chaudhary, A.; Wener, M.H.; Morishima, C.; Jerome, K.R.; Mathias, P.C.; Greninger, A.L. SARS-CoV-2 viral load on admission is associated with 30-day mortality. In Open Forum Infectious Diseases; Oxford University Press: Oxford, UK, 2020; Volume 7, p. ofaa535. [Google Scholar]
- FDA. EUA Authorizations. Available online: https://www.fda.gov/medical-devices/coronavirus-disease-2019-covid-19-emergency-use-authorizations-medical-devices/vitro-diagnostics-euas individual-molecular (accessed on 30 November 2020).
- COVID-19 Update: FDA Authorizes First Diagnostic Test Where Results Can Be Read Directly From Testing Card. Available online: https://www.fda.gov/news-events/press-announcements/covid-19-update-fda-authorizes-first-diagnostic-test-where-results-can-be-read-directly-testing-card (accessed on 1 February 2021).
- Prince-Guerra, J.L.; Almendares, O.; Nolen, L.D.; Gunn, J.K.; Dale, A.P.; Buono, S.A.; Deutsch-Feldman, M.; Suppiah, S.; Hao, L.; Zeng, Y.; et al. Evaluation of Abbott BinaxNOW Rapid Antigen Test for SARS-CoV-2 Infection at Two Community-Based Testing Sites—Pima County, Arizona, 3–17 November 2020. Morb. Mortal. Wkly. Rep.
**2021**, 70, 100. [Google Scholar] [CrossRef] [PubMed] - Harritshoej, L.H.; Gybel-Brask, M.; Afzal, S.; Kamstrup, P.R.; Joergensen, C.S.; Thomsen, M.K.; Hilsted, L.M.; Friis-Hansen, L.J.; Szecsi, P.B.; Pedersen, L.; et al. Comparison of sixteen serological SARS-CoV-2 immunoassays in sixteen clinical laboratories. medRxiv
**2020**. [Google Scholar] [CrossRef] - Larremore, D.B.; Wilder, B.; Lester, E.; Shehata, S.; Burke, J.M.; Hay, J.A.; Tambe, M.; Mina, M.J.; Parker, R. Test sensitivity is secondary to frequency and turnaround time for COVID-19 screening. Sci. Adv.
**2021**, 7, eabd5393. [Google Scholar] [CrossRef] - Ke, R.; Zitzmann, C.; Ribeiro, R.M.; Perelson, A.S. Kinetics of SARS-CoV-2 infection in the human upper and lower respiratory tracts and their relationship with infectiousness. medRxiv
**2020**. [Google Scholar] [CrossRef] - Wölfel, R.; Corman, V.M.; Guggemos, W.; Seilmaier, M.; Zange, S.; Müller, M.A.; Niemeyer, D.; Jones, T.C.; Vollmar, P.; Rothe, C.; et al. Virological assessment of hospitalized patients with COVID-2019. Nature
**2020**, 581, 465–469. [Google Scholar] [CrossRef][Green Version] - Baccam, P.; Beauchemin, C.; Macken, C.A.; Hayden, F.G.; Perelson, A.S. Kinetics of influenza A virus infection in humans. J. Virol.
**2006**, 80, 7590–7599. [Google Scholar] [CrossRef] [PubMed][Green Version] - Beauchemin, C.A.; Handel, A. A review of mathematical models of influenza A infections within a host or cell culture: Lessons learned and challenges ahead. BMC Public Health
**2011**, 11, S7. [Google Scholar] [CrossRef][Green Version] - Smith, A.M.; Perelson, A.S. Influenza A virus infection kinetics: Quantitative data and models. Wiley Interdiscip. Rev. Syst. Biol. Med.
**2011**, 3, 429–445. [Google Scholar] [CrossRef][Green Version] - Nikin-Beers, R.; Ciupe, S.M. Modelling original antigenic sin in dengue viral infection. Math. Med. Biol. A J. IMA
**2017**, 35, 257–272. [Google Scholar] [CrossRef] [PubMed] - Nikin-Beers, R.; Ciupe, S.M. The role of antibody in enhancing dengue virus infection. Math. Biosci.
**2015**, 263, 83–92. [Google Scholar] [CrossRef] - Ben-Shachar, R.; Schmidler, S.; Koelle, K. Drivers of inter-individual variation in dengue viral load dynamics. PLoS Comput. Biol.
**2016**, 12, e1005194. [Google Scholar] [CrossRef] [PubMed] - Best, K.; Guedj, J.; Madelain, V.; de Lamballerie, X.; Lim, S.Y.; Osuna, C.E.; Whitney, J.B.; Perelson, A.S. Zika plasma viral dynamics in nonhuman primates provides insights into early infection and antiviral strategies. Proc. Natl. Acad. Sci. USA
**2017**, 114, 8847–8852. [Google Scholar] [CrossRef][Green Version] - Banerjee, S.; Guedj, J.; Ribeiro, R.M.; Moses, M.; Perelson, A.S. Estimating biologically relevant parameters under uncertainty for experimental within-host murine West Nile virus infection. J. R. Soc. Interface
**2016**, 13, 20160130. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ogando, N.S.; Dalebout, T.J.; Zevenhoven-Dobbe, J.C.; Limpens, R.W.; van der Meer, Y.; Caly, L.; Druce, J.; de Vries, J.J.; Kikkert, M.; Bárcena, M.; et al. SARS-coronavirus-2 replication in Vero E6 cells: Replication kinetics, rapid adaptation and cytopathology. J. Gen. Virol.
**2020**, 101, 925. [Google Scholar] [CrossRef] - Duke Covid Testing Tracker. Available online: https://coronavirus.duke.edu/covid-testing/ (accessed on 1 March 2021).
- Coronavirus Information. Available online: https://coronavirus.virginia.edu/covid-tracker (accessed on 1 March 2021).
- COVID-19 Tracking. Available online: https://covid.cornell.edu/testing/dashboard/ (accessed on 1 March 2021).
- Neilan, A.M.; Losina, E.; Bangs, A.C.; Flanagan, C.; Panella, C.; Eskibozkurt, G.E.; Mohareb, A.; Hyle, E.P.; Scott, J.A.; Weinstein, M.C.; et al. Clinical Impact, Costs, and Cost-Effectiveness of Expanded SARS-CoV-2 Testing in Massachusetts. medrxiv
**2020**. [Google Scholar] [CrossRef] - Wyllie, A.L.; Fournier, J.; Casanovas-Massana, A.; Campbell, M.; Tokuyama, M.; Vijayakumar, P.; Warren, J.L.; Geng, B.; Muenker, M.C.; Moore, A.J.; et al. Saliva or nasopharyngeal swab specimens for detection of SARS-CoV-2. N. Engl. J. Med.
**2020**, 383, 1283–1286. [Google Scholar] [CrossRef] [PubMed] - Brook, C.E.; Northrup, G.R.; Ehrenberg, A.J.; Doudna, J.A.; Boots, M.; IGI Testing Consortium. Optimizing COVID-19 control with asymptomatic surveillance testing in a university environment. medRxiv
**2020**. [Google Scholar] [CrossRef] - Bergstrom, T.; Bergstrom, C.T.; Li, H. Frequency and accuracy of proactive testing for COVID-19. medRxiv
**2020**. [Google Scholar] [CrossRef]

**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