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
. 2022 Jan 16;14(1):158.
doi: 10.3390/v14010158.

Modeling the Transmission of the SARS-CoV-2 Delta Variant in a Partially Vaccinated Population

Ugo Avila-Ponce de León  1 , Eric Avila-Vales  2 , Kuanlin Huang  3
Affiliations

Modeling the Transmission of the SARS-CoV-2 Delta Variant in a Partially Vaccinated Population

Ugo Avila-Ponce de León et al. Viruses. .

Abstract

In a population with ongoing vaccination, the trajectory of a pandemic is determined by how the virus spreads in unvaccinated and vaccinated individuals that exhibit distinct transmission dynamics based on different levels of natural and vaccine-induced immunity. We developed a mathematical model that considers both subpopulations and immunity parameters, including vaccination rates, vaccine effectiveness, and a gradual loss of protection. The model forecasted the spread of the SARS-CoV-2 delta variant in the US under varied transmission and vaccination rates. We further obtained the control reproduction number and conducted sensitivity analyses to determine how each parameter may affect virus transmission. Although our model has several limitations, the number of infected individuals was shown to be a magnitude greater (~10×) in the unvaccinated subpopulation compared to the vaccinated subpopulation. Our results show that a combination of strengthening vaccine-induced immunity and preventative behavioral measures like face mask-wearing and contact tracing will likely be required to deaccelerate the spread of infectious SARS-CoV-2 variants.

Keywords: SARS-CoV-2 variants; breakthrough cases; mathematical model; sensitivity analysis: control reproduction number; vaccines dynamics.

PubMed Disclaimer

Conflict of interest statement

The authors declare no conflict of interest.

Figures

Figure 1
Figure 1
Schematic of the model considering vaccinated vs. unvaccinated individuals and the projection of newly infected symptomatic and asymptomatic cases. Flow dynamics of the mathematical model. S: susceptible, P: partial immunity (one vaccine dose), and V: full immunity (two vaccine doses). E: unvaccinated, exposed. I: unvaccinated, infected, and symptomatic, A: unvaccinated, infected, and asymptomatic. EB: vaccinated, exposed. IB: vaccinated, infected, and symptomatic (“breakthrough cases”). AB: vaccinated, infected, and asymptomatic (“breakthrough cases”). R: recovered. D: deceased.
Figure 2
Figure 2
Modeled dynamics of the new infections that are symptomatic (top row) and asymptomatic (bottom row) from the SARS-CoV-2 delta variant considering different vaccination rates. The red line presents zero vaccination, the green line represents a 50% decrease in VR, the blue line means baseline VR, and the black dotted line denotes 200% VR.
Figure 3
Figure 3
Modeled projections of symptomatic infections in unvaccinated and vaccinated subpopulations under different vaccination rates and vaccine effectiveness. The case counts of the unvaccinated individuals are depicted by the red line and the unit labels on the left-side y-axis, whereas the infected, vaccinated (breakthrough) cases are depicted by the blue line and the unit labels on the right-side y-axis.
Figure 4
Figure 4
Control reproduction number for symptomatic transmission considering full immunity and recovered individuals.
Figure 5
Figure 5
Control reproduction number for symptomatic transmission considering partial immunity and recovered individuals. The assumed vaccine effectiveness (εL1) parameter under each scenario is denoted on top of each panel and is inferred based on real-world data of individuals acquiring partial immunity as those induced by one dose of the BNT162b2 vaccine [27]. The heatmaps in the left row consider a low transmission rate, the heatmaps in the center row panels consider a baseline transmission rate, and the heatmaps in the right row panels consider a high transmission rate.
Figure 6
Figure 6
Elasticity index showing each modeled parameter’s effect on the control reproduction number, as determined by the local sensitivity analysis. For each column, a positive value represents an increase of the parameter is associated with an increment in the control reproduction number, whereas a negative value means a decrease.
Figure 7
Figure 7
Global sensitivity analyses of the parameters correlated with the symptomatic infections in (A) unvaccinated and (B) vaccinated individuals. The relationship between each parameter and the infected subpopulation is measured by the partial rank correlation coefficient (PRCC); a positive PRCC indicates an increase in the parameters is correlated with an increase of the said sub-population, and a negative PRCC vice versa. The significance level of each PRCC is indicated in Figure S6.
See this image and copyright information in PMC

Similar articles

See all similar articles

Cited by

References

    1. Dong E., Du H., Gardner L. An Interactive Web-Based Dashboard to Track COVID-19 in Real Time. Lancet Infect. Dis. 2020;20:533–534. doi: 10.1016/S1473-3099(20)30120-1. - DOI - PMC - PubMed
    1. Haug N., Geyrhofer L., Londei A., Dervic E., Desvars-Larrive A., Loreto V., Pinior B., Thurner S., Klimek P. Ranking the Effectiveness of Worldwide COVID-19 Government Interventions. Nat. Hum. Behav. 2020;4:1303–1312. doi: 10.1038/s41562-020-01009-0. - DOI - PubMed
    1. Perra N. Non-Pharmaceutical Interventions during the COVID-19 Pandemic: A Review. Phys. Rep. 2021;913:1–52. doi: 10.1016/j.physrep.2021.02.001. - DOI - PMC - PubMed
    1. IHME COVID-19 Forecasting Team Modeling COVID-19 Scenarios for the United States. Nat. Med. 2021;27:94–105. doi: 10.1038/s41591-020-1132-9. - DOI - PMC - PubMed
    1. Gondim J.A.M. Preventing Epidemics by Wearing Masks: An Application to COVID-19. Chaos Solitons Fractals. 2021;143:110599. doi: 10.1016/j.chaos.2020.110599. - DOI - PMC - PubMed

Publication types

Substances

Supplementary concepts