Skip to main page content
U.S. flag

An official website of the United States government

Dot gov

The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

Https

The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation
. 2023 May 11;18(5):e0285466.
doi: 10.1371/journal.pone.0285466. eCollection 2023.

Time-dependent vaccine efficacy estimation quantified by a mathematical model

Jennifer Loria  1   2 , Vinicius V L Albani  3 , Francisco A B Coutinho  4 , Dimas T Covas  5 , Claudio J Struchiner  6 , Jorge P Zubelli  7 , Eduardo Massad  6   8
Affiliations

Time-dependent vaccine efficacy estimation quantified by a mathematical model

Jennifer Loria et al. PLoS One. .

Abstract

In this paper we calculate the variation of the estimated vaccine efficacy (VE) due to the time-dependent force of infection resulting from the difference between the moment the Clinical Trial (CT) begins and the peak in the outbreak intensity. Using a simple mathematical model we tested the hypothesis that the time difference between the moment the CT begins and the peak in the outbreak intensity determines substantially different values for VE. We exemplify the method with the case of the VE efficacy estimation for one of the vaccines against the new coronavirus SARS-CoV-2.

PubMed Disclaimer

Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Fig 1
Fig 1. Schematic representation of the epidemiological model representing its different compartments.
Fig 2
Fig 2. Force of infection in the city of Santos, Southeast Region of Brazil, modeled as a Gaussian function.
Fig 3
Fig 3. The cohort’s burden of infection during the clinical trial that begins at t = 0.
Fig 4
Fig 4. Evolution of the vaccine efficacy as a function of the disease outbreak peak when the vaccine and the placebo inoculation start at tv = tp = 0.
Fig 5
Fig 5. Evolution of the vaccine efficacy as a function of the disease outbreak peak (τ) when the vaccine inoculation starts at tv = 0 and the placebo inoculation starts at tp = 0, 5, 10, and 15 days.
The quantity τ is the time delay in days from the clinical trial launch (which is set to t = 0) to the outbreak peak.
Fig 6
Fig 6. Evolution of the vaccine efficacy as a function of the disease outbreak peak (τ) when the vaccine inoculation starts at tv = 10 and the placebo inoculation starts at tp = 0, 5, 10, and 15 days.
The quantity τ is the time delay in days from the clinical trial launch (which is set to t = 0) to the outbreak peak.
Fig 7
Fig 7. Evolution of the vaccine efficacy as a function of the disease outbreak peak (τ) when the vaccine inoculation starts at tv = 15 and the placebo inoculation starts at tp = 10, 15, 20, and 25 days.
The quantity τ is the time delay in days from the clinical trial launch (which is set to t = 0) to the outbreak peak.
Fig 8
Fig 8. The cohort’s infection burden during the clinical trial.
Fig 9
Fig 9. Evolution of the vaccine efficacy as a function of the disease outbreak peak when the vaccine and the placebo inoculation start at tv = tp = 0.
The clinical trial ends at the first day when the total number of infected individuals inoculated with the vaccine and the placebo reached at least 200.
Fig 10
Fig 10. Evolution of the vaccine efficacy as a function of the disease outbreak peak when the peak in the vaccine and the placebo inoculation are at Tv = Tp = 0 and the clinical trial ends at the first day when the total number of infected individuals inoculated with the vaccine and the placebo reached at least 200.
The placebo and vaccine daily inoculation follows the Gaussian function in Eq (19).
Fig 11
Fig 11. Evolution of the vaccine efficacy as a function of the disease outbreak peak when the peak of the vaccine and the placebo inoculation are at Tv = 0 and Tp = 10, respectively, and the clinical trial ends at the first day when the total number of infected individuals inoculated with the vaccine and the placebo reached at least 200.
The placebo and vaccine daily inoculation follows the Gaussian function in Eq (19).
Fig 12
Fig 12. Evolution of the vaccine efficacy as a function of the disease outbreak peak when the peak of the vaccine and the placebo inoculation are at Tv = 10 and Tp = 0, respectively, and the clinical trial ends at the first day when the total number of infected individuals inoculated with the vaccine and the placebo reached at least 200.
The placebo and vaccine daily inoculation follows the Gaussian function in Eq (19).
Fig 13
Fig 13. Daily numbers of individuals inoculated with the vaccine and the placebo during the CORONAVAC® trial in Brazil.
The solid line represents a continuous function fitted to the data.
See this image and copyright information in PMC

References

    1. Scheppler L, De Clercq N, McGoldrick M, Dias J. Regulatory Harmonization and Streamlining of Clinical Trial Applications globally should lead to faster clinical development and earlier access to life-saving vaccines. Vaccine. 2021;. - PubMed
    1. Bellomo N, Bingham R, Chaplain MA, Dosi G, Forni G, Knopoff DA, et al.. A multiscale model of virus pandemic: Heterogeneous interactive entities in a globally connected world. Mathematical Models and Methods in Applied Sciences. 2020;30(08):1591–1651. doi: 10.1142/s0218202520500323 - DOI - PMC - PubMed
    1. Bellomo N, Burini D, Dosi G, Gibelli L, Knopoff D, Outada N, et al.. What is life? A perspective of the mathematical kinetic theory of active particles. Mathematical Models and Methods in Applied Sciences. 2021;31(09):1821–1866. doi: 10.1142/S0218202521500408 - DOI
    1. Delany I, Rappuoli R, De Gregorio E. Vaccines for the 21st century. EMBO molecular medicine. 2014;6(6):708–720. doi: 10.1002/emmm.201403876 - DOI - PMC - PubMed
    1. World Health Organization, Global Immunization Vision and Strategy 2006–2015; 2016.

Publication types