# Modeling the Influence of Vaccine Administration on COVID-19 Testing Strategies

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**Daily testing rate.**As before [24], we define a daily per capita testing rate, $\rho $, corresponding to an overall testing capacity of C tests per day at time t in a population of size $N\left(t\right)$ to be given by

**Model equations.**On the domain $t\ge 0$, $0\le \tau \le {\tau}_{q}$, and $\alpha \ge 0$, the between-host model equations under combined vaccination and testing are

#### 2.1. Cumulative Statistics

#### 2.2. Parameter Choices

## 3. Results

#### 3.1. Alpha Variant Dynamics in the Absence of Testing

#### 3.2. Alpha Variant Model Outcomes in the Presence of Testing

#### 3.3. Delta Variant Dynamics in the Absence of Testing

#### 3.4. Delta Variant Model Outcomes in the Presence of Testing

#### 3.5. The Role of Testing for Emerging Variant Dynamics

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Zhu, N.; Zhang, D.; Wang, W.; Li, X.; Yang, B.; Song, J.; Zhao, X.; Huang, B.; Shi, W.; Lu, R.; et al. A novel coronavirus from patients with pneumonia in China, 2019. N. Engl. J. Med.
**2020**, 382, 727–733. [Google Scholar] [CrossRef] [PubMed] - WHO Cases Dashboard. Available online: https://covid19.who.int/ (accessed on 29 August 2021).
- Legido-Quigley, H.; Asgari, N.; Teo, Y.Y.; Leung, G.M.; Oshitani, H.; Fukuda, K.; Cook, A.R.; Hsu, L.Y.; Shibuya, K.; Heymann, D. Are high-performing health systems resilient against the COVID-19 epidemic? Lancet
**2020**, 395, 848–850. [Google Scholar] [CrossRef] [Green Version] - Rocha, R.; Atun, R.; Massuda, A.; Rache, B.; Spinola, P.; Nunes, L.; Lago, M.; Castro, M.C. Effect of socioeconomic inequalities and vulnerabilities on health-system preparedness and response to COVID-19 in Brazil: A comprehensive analysis. Lancet Glob. Health
**2021**, 9, e782–e792. [Google Scholar] [CrossRef] - Haldane, V.; De Foo, C.; Abdalla, S.M.; Jung, A.S.; Tan, M.; Wu, S.; Chua, A.; Verma, M.; Shrestha, P.; Singh, S.; et al. Health systems resilience in managing the COVID-19 pandemic: Lessons from 28 countries. Nat. Med.
**2021**, 27, 964–980. [Google Scholar] [CrossRef] - Ingram, C.; Downey, V.; Roe, M.; Chen, Y.; Archibald, M.; Kallas, K.A.; Kumar, J.; Naughton, P.; Uteh, C.O.; Rojas-Chaves, A.; et al. COVID-19 prevention and control measures in workplace settings: A rapid review and meta-analysis. Int. J. Environ. Res. Public Health
**2021**, 18, 7847. [Google Scholar] [CrossRef] - Perra, N. Non-pharmaceutical interventions during the COVID-19 pandemic: A review. Phys. Rep.
**2021**, 913, 1–52. [Google Scholar] [CrossRef] - Matrajt, L.; Leung, T. Evaluating the effectiveness of social distancing interventions to delay or flatten the epidemic curve of coronavirus disease. Emerg. Infect. Dis.
**2020**, 26, 1740. [Google Scholar] [CrossRef] - Kumar, A.; Dowling, W.E.; Román, R.G.; Chaudhari, A.; Gurry, C.; Le, T.T.; Tollefson, S.; Clark, C.E.; Bernasconi, V.; Kristiansen, P.A. Status report on COVID-19 vaccines development. Curr. Infect. Dis. Rep.
**2021**, 23, 9. [Google Scholar] [CrossRef] - Callaway, E. The race for coronavirus vaccines: A graphical guide. Nature
**2020**, 580, 576–577. [Google Scholar] [CrossRef] - Polack, F.P.; Thomas, S.J.; Kitchin, N.; Absalon, J.; Gurtman, A.; Lockhart, S.; Perez, J.L.; Marc, G.P.; Moreira, E.D.; Zerbini, C.; et al. Safety and efficacy of the BNT162b2 mRNA Covid-19 vaccine. N. Engl. J. Med.
**2020**, 383, 2603–2615. [Google Scholar] [CrossRef] - U.S. COVID-19 Vaccine Product Information. Available online: https://www.cdc.gov/vaccines/covid-19/info-by-product/index.html (accessed on 29 August 2021).
- Tenforde, M.W. Sustained Effectiveness of Pfizer-BioNTech and Moderna Vaccines Against COVID-19 Associated Hospitalizations Among Adults—United States, March–July 2021. Morb. Mortal. Wkly. Rep.
**2021**, 70, 1156. [Google Scholar] [CrossRef] [PubMed] - Bernal, J.L.; Andrews, N.; Gower, C.; Robertson, C.; Stowe, J.; Tessier, E.; Simmons, R.; Cottrell, S.; Roberts, R.; O’Doherty, M.; et al. Effectiveness of the Pfizer-BioNTech and Oxford-AstraZeneca vaccines on covid-19 related symptoms, hospital admissions, and mortality in older adults in England: Test negative case-control study. BMJ
**2021**, 373, n1088. [Google Scholar] [CrossRef] [PubMed] - Haas, E.J.; Angulo, F.J.; McLaughlin, J.M.; Anis, E.; Singer, S.R.; Khan, F.; Brooks, N.; Smaja, M.; Mircus, G.; Pan, K.; et al. Impact and effectiveness of mRNA BNT162b2 vaccine against SARS-CoV-2 infections and COVID-19 cases, hospitalisations, and deaths following a nationwide vaccination campaign in Israel: An observational study using national surveillance data. Lancet
**2021**, 397, 1819–1829. [Google Scholar] [CrossRef] - Tenforde, M.W.; Patel, M.M.; Ginde, A.A.; Douin, D.J.; Talbot, H.K.; Casey, J.D.; Mohr, N.M.; Zepeski, A.; Gaglani, M.; McNeal, T.; et al. Effectiveness of SARS-CoV-2 mRNA vaccines for preventing Covid-19 hospitalizations in the United States. Clin. Infect. Dis.
**2021**. [Google Scholar] [CrossRef] - CDC Covid-19 Variants Proportion in USA. Available online: https://covid.cdc.gov/covid-data-tracker/#variant-proportions (accessed on 23 September 2021).
- Fowlkes, A. Effectiveness of COVID-19 Vaccines in Preventing SARS-CoV-2 Infection Among Frontline Workers Before and During B. 1.617. 2 (Delta) Variant Predominance—Eight US Locations, December 2020–August 2021. MMWR. Morb. Mortal. Wkly. Rep.
**2021**, 70, 1167. [Google Scholar] [CrossRef] - Lopez Bernal, J.; Andrews, N.; Gower, C.; Gallagher, E.; Simmons, R.; Thelwall, S.; Stowe, J.; Tessier, E.; Groves, N.; Dabrera, G.; et al. Effectiveness of Covid-19 vaccines against the B.1.617.2 (delta) variant. N. Engl. J. Med.
**2021**, 385, 585–594. [Google Scholar] [CrossRef] - Liu, Y.; Rocklöv, J. The reproductive number of the Delta variant of SARS-CoV-2 is far higher compared to the ancestral SARS-CoV-2 virus. J. Travel Med.
**2021**, 28, taab124. [Google Scholar] [CrossRef] - Earnest, R.; Uddin, R.; Matluk, N.; Renzette, N.; Siddle, K.J.; Loreth, C.; Adams, G.; Tomkins-Tinch, C.; Petrone, M.E.; Rothman, J.E.; et al. Comparative transmissibility of SARS-CoV-2 variants Delta and Alpha in New England, USA. medRxiv
**2021**. [Google Scholar] [CrossRef] - Birhane, M.; Bressler, S.; Chang, G.; Clark, T.; Dorough, L.; Fischer, M.; Watkins, L.F.; Goldstein, J.M.; Kugeler, K.; Langley, G.; et al. COVID-19 Vaccine Breakthrough Infections Reported to CDC—United States, January 1–April 30, 2021. Morb. Mortal. Wkly. Rep.
**2021**, 70, 792. [Google Scholar] - Pfizer-BioNTech Booster Approval. Available online: https://www.fda.gov/news-events/press-announcements/fda-authorizes-booster-dose-pfizer-biontech-covid-19-vaccine-certain-populations (accessed on 3 September 2021).
- Forde, J.; Ciupe, S. Quantification of the tradeoff between test sensitivity and test frequency in a COVID-19 epidemic-a multi-scale modeling approach. Viruses
**2021**, 13, 457. [Google Scholar] [CrossRef] - Nikin-Beers, R.; Blackwood, J.C.; Childs, L.M.; Ciupe, S.M. Unraveling within-host signatures of dengue infection at the population level. J. Theor. Biol.
**2018**, 446, 79–86. [Google Scholar] [CrossRef] - Dorratoltaj, N.; Nikin-Beers, R.; Ciupe, S.M.; Eubank, S.G.; Abbas, K.M. Multi-scale immunoepidemiological modeling of within-host and between-host HIV dynamics: Systematic review of mathematical models. PeerJ
**2017**, 5, e3877. [Google Scholar] [CrossRef] - Levine-Tiefenbrun, M.; Yelin, I.; Katz, R.; Herzel, E.; Golan, Z.; Schreiber, L.; Wolf, T.; Nadler, V.; Ben-Tov, A.; Kuint, J.; et al. Initial report of decreased SARS-CoV-2 viral load after inoculation with the BNT162b2 vaccine. Nat. Med.
**2021**, 75, 790–792. [Google Scholar] [CrossRef] [PubMed] - Pfizer-BioNTech COVID-19 Vaccine EUA. Available online: https://www.fda.gov/media/144414/download (accessed on 3 September 2021).
- 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] - 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] [PubMed] - 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] - 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] [PubMed] [Green Version] - CDC Guidelines for Vaccinated People. Available online: https://www.cdc.gov/coronavirus/2019-ncov/vaccines/fully-vaccinated-guidance.html (accessed on 3 September 2021).
- Motta, F.C.; McGoff, K.A.; Deckard, A.; Wolfe, C.R.; Moody, M.A.; Cavanaugh, K.; Denny, T.N.; Harer, J.; Haase, S.B. Benefits of Surveillance Testing and Quarantine in a SARS-CoV-2 Vaccinated Population of Students on a University Campus. medRxiv
**2021**. [Google Scholar] [CrossRef] - Hacisuleyman, E.; Hale, C.; Saito, Y.; Blachere, N.E.; Bergh, M.; Conlon, E.G.; Schaefer-Babajew, D.J.; DaSilva, J.; Muecksch, F.; Gaebler, C.; et al. Vaccine breakthrough infections with SARS-CoV-2 variants. N. Engl. J. Med.
**2021**, 384, 2212–2218. [Google Scholar] [CrossRef] - Bergwerk, M.; Gonen, T.; Lustig, Y.; Amit, S.; Lipsitch, M.; Cohen, C.; Mandelboim, M.; Gal Levin, E.; Rubin, C.; Indenbaum, V.; et al. COVID-19 breakthrough infections in vaccinated health care workers. N. Engl. J. Med.
**2021**, 385, 1474–1484. [Google Scholar] [CrossRef] [PubMed] - Duerr, R.; Dimartino, D.; Marier, C.; Zappile, P.; Wang, G.; Lighter, J.; Elbel, B.; Troxel, A.B.; Heguy, A. Dominance of Alpha and Iota variants in SARS-CoV-2 vaccine breakthrough infections in New York City. J. Clin. Investig.
**2021**, 131, e152702. [Google Scholar] [CrossRef] [PubMed] - Musser, J.M.; Christensen, P.A.; Olsen, R.J.; Long, S.W.; Subedi, S.; Davis, J.J.; Hodjat, P.; Walley, D.R.; Kinskey, J.C.; Gollihar, J.D. Delta variants of SARS-CoV-2 cause significantly increased vaccine breakthrough COVID-19 cases in Houston, Texas. medRxiv
**2021**. [Google Scholar] [CrossRef] - Luo, C.H.; Morris, C.P.; Sachithanandham, J.; Amadi, A.; Gaston, D.; Li, M.; Swanson, N.J.; Schwartz, M.; Klein, E.Y.; Pekosz, A.; et al. Infection with the SARS-CoV-2 Delta Variant is Associated with Higher Infectious Virus Loads Compared to the Alpha Variant in both Unvaccinated and Vaccinated Individuals. medRxiv
**2021**. [Google Scholar] [CrossRef] - Moline, H.L.; Whitaker, M.; Deng, L.; Rhodes, J.C.; Milucky, J.; Pham, H.; Patel, K.; Anglin, O.; Reingold, A.; Chai, S.J.; et al. Effectiveness of COVID-19 vaccines in preventing hospitalization among adults aged ≥ 65 years—COVID-NET, 13 states, February–April 2021. Morb. Mortal. Wkly. Rep.
**2021**, 70, 1088. [Google Scholar] [CrossRef] - Rella, S.A.; Kulikova, Y.A.; Dermitzakis, E.T.; Kondrashov, F.A. Rates of SARS-CoV-2 transmission and vaccination impact the fate of vaccine-resistant strains. Sci. Rep.
**2021**, 11, 15729. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Virus profiles in non-vaccinated and vaccinated individuals. ${log}_{10}$ virus load per swab over time during Alpha variant natural infection (red line) and vaccination (green line) as given by the within-host model in [29]. Non-vaccinated patients are assumed to be infectious from $t=2.5$ days till $t=10.5$ days (shaded pink region). Vaccinated patients are assumed to be infectious from $t=2.85$ days till $t=9.8$ days (shaded purple region). Black horizontal lines correspond to RT-PCR test detection threshold (LOD) of ${log}_{10}\left(V\right)=2$ per swab and antigen test detection threshold (LOD) of ${log}_{10}\left(V\right)=5$ per swab.

**Figure 2.**Dynamics of Alpha and Delta variants infection over time. Left panels: daily asymptomatic (blue), symptomatic (red), breakthrough asymptomatic (cyan), and breakthrough asymptomatic (magenta) infections over time; Middle panels: cumulative cases $\mathsf{\Sigma}\left(t\right)$ (magenta), cumulative breakthrough cases $B\left(t\right)$ (green) over time; Right panels: cumulative fully vaccinated $F\left(t\right)$ (cyan), cumulative vaccinated after infection $CVR\left(t\right)$ (magenta) and cumulative total vaccinated $T\left(t\right)$ (black) over time in the absence of testing. Panel (

**A**): Alpha variant; Panel (

**B**): Delta variant. The background vaccination is $30\%$ and the other parameters and initial conditions are given in Table 1.

**Figure 3.**Percent breakthrough cases at day 100. Heatmaps for the percent breakthrough cases in the vaccinated population at day 100 versus additional daily vaccines, $\nu $, and background vaccination levels, ${V}_{0}$. Panel (

**A**): Alpha variant; Panel (

**B**): Delta variant. Parameters and initial conditions are given in Table 1.

**Figure 4.**Reduction in Alpha variant cases at 100 days. Heatmaps for the reduction in cumulative cases at 100 days after an outbreak with an Alpha variant, ${\mathsf{\Sigma}}_{noTests}\left(100\right)-{\mathsf{\Sigma}}_{Tests}\left(100\right)$, as given by model Equation (1) versus RT-PCR testing capacity with a return delay of 1 day, C, and background vaccination levels, ${V}_{0}$. Panel (

**A**): Test non-vaccinated only; Panel (

**B**): Test everybody. The gray heatmaps represent the cumulative cases at day 100 in the absence of testing, ${\mathsf{\Sigma}}_{noTest}\left(100\right)$. Parameters and initial conditions are given in Table 1.

**Figure 5.**Reduction in Delta variant cases at 100 days. Heatmaps for the reduction in cumulative cases at 100 days after an outbreak with a Delta variant, ${\mathsf{\Sigma}}_{noTests}\left(100\right)-{\mathsf{\Sigma}}_{Tests}\left(100\right)$, as given by model Equation (1) versus RT-PCR testing capacity with a return delay of 1 day, C, and background vaccination levels, ${V}_{0}$. Panel (

**A**): Test non-vaccinated only; Panel (

**B**): Test everybody. The gray heatmaps represent the cumulative cases at day 100 in the absence of testing, ${\mathsf{\Sigma}}_{noTest}\left(100\right)$. Parameters and initial conditions are given in Table 1.

**Figure 6.**Percent reduction in variant cases at 100 days as vaccination increases. Percent reduction in cumulative cases at 100 days after an outbreak with a variants of different infectivity rate $\beta $, $({\mathsf{\Sigma}}_{noTests}\left(100\right)-{\mathsf{\Sigma}}_{Tests}\left(100\right))/{\mathsf{\Sigma}}_{noTests}\left(100\right)$, as given by model Equation (1) versus background vaccination levels, ${V}_{0}$. Panel (

**A**): Test non-vaccinated only; Panel (

**B**): Test everybody. Parameters and initial conditions are given in Table 1 and we assume RT-PCR testing capacity $C=0.1$ with a return delay of 1 day.

**Figure 7.**Percent reduction in variant cases at 100 days as vaccination efficacy decreases. Percent reduction in cumulative cases at 100 days after an outbreak with a variants of different infectivity rate $\beta $, $({\mathsf{\Sigma}}_{noTests}\left(100\right)-{\mathsf{\Sigma}}_{Tests}\left(100\right))/{\mathsf{\Sigma}}_{noTests}\left(100\right)$, as given by model Equation (1) versus efficacy of the second dose of the vaccine, ${\eta}_{2}$. Panel (

**A**): Test non-vaccinated only; Panel (

**B**): Test everybody. Parameters and initial conditions are given in Table 1, we assume background vaccination ${V}_{0}=0.5$ and RT-PCR testing capacity $C=0.1$ with a return delay of 1 day.

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

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

$\beta $ | Transmission rate | varied | |

b | Birth rate | $7\times {10}^{-4}$/day | [25] |

$\mu $ | Death rate | $7\times {10}^{-4}$/day | [25] |

m | Disease induced mortality rate | ${10}^{-3}$/day | |

f | Fraction of symptomatic infections | $0.7$ | [30] |

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

${\alpha}_{v}$ | Age of vaccination for reduced virus | 11 days | [27] |

${\alpha}_{f}$ | Age of vaccination corresponding to full protection | 35 days | [12] |

${\tau}_{q}$ | Recovery age | 14 days | [24] |

ℓ | PCR test return delay | 1 day | |

${\tau}_{1}^{j}$ ($j=\{a,s\}$) | Age of onset of virus detectability in non-vaccinated | 0.554 days | |

${\tau}_{1}^{j}$ ($j=\{va,vs\}$) | Age of onset of virus detectability in vaccinated | 0.974 days | |

${\tau}_{2}^{j}$ ($j=\{a,s\}$) | Age of onset of infectiousness in non-vaccinated | $2.5$ days | [31] |

${\tau}_{2}^{j}$ ($j=\{va,vs\}$) | Age of onset of infectiousness in vaccinated | $2.8$ days | [27] |

${\tau}_{3}^{j}$ ($j=\{a,s\}$) | Age of end of infectiousness in non-vaccinated | $10.5$ days | [30,32] |

${\tau}_{3}^{j}$ ($j=\{va,vs\}$) | Age of end of infectiousness in vaccinated | $9.8$ days | [27] |

${\tau}_{4}^{j}$ ($j=\{a,s\}$) | Age of loss of virus detectability in non-vaccinated | 10.95 days | |

${\tau}_{4}^{j}$ ($j=\{va,vs\}$) | Age of loss of virus detectability in vaccinated | 10.22 days | |

$\nu $ | Vaccination rate | varied | |

${t}_{1}$ | Time when additional vaccination is initiated | 20 days | |

C | testing capacity | varied | |

Initial Conditions | Description | Value | Source |

$S\left(0\right)$ | Susceptible population | $0.95-V\left(0\right)$ | |

$V(\alpha ,0)=V\left(0\right)={V}_{0}$ | Vaccination level | varied | |

${i}_{s}(\tau ,0)$ | Infected symptomatic population | $0.05f\delta \left(\tau \right)$ | |

${i}_{a}(\tau ,0)$ | Infected asymptomatic population | $0.05(1-f)\delta \left(\tau \right)$ | |

${i}_{vs}(\tau ,0)$ | Infected vaccinated symptomatic population | 0 | |

${i}_{va}(\tau ,0)$ | Infected vaccinated asymptomatic population | 0 | |

$R\left(0\right)$ | Recovered population | 0 | |

${R}_{v1}\left(0\right)$ | Vaccinated after natural infection | 0 | |

${R}_{v2}\left(0\right)$ | Vaccinated, infected, and recovered | 0 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Forde, J.E.; Ciupe, S.M.
Modeling the Influence of Vaccine Administration on COVID-19 Testing Strategies. *Viruses* **2021**, *13*, 2546.
https://doi.org/10.3390/v13122546

**AMA Style**

Forde JE, Ciupe SM.
Modeling the Influence of Vaccine Administration on COVID-19 Testing Strategies. *Viruses*. 2021; 13(12):2546.
https://doi.org/10.3390/v13122546

**Chicago/Turabian Style**

Forde, Jonathan E., and Stanca M. Ciupe.
2021. "Modeling the Influence of Vaccine Administration on COVID-19 Testing Strategies" *Viruses* 13, no. 12: 2546.
https://doi.org/10.3390/v13122546