Unraveling the dynamics of the Omicron and Delta variants of the 2019 coronavirus in the presence of vaccination, mask usage, and antiviral treatment

Abstract

The effectiveness of control interventions against COVID-19 is threatened by the emergence of SARS-CoV-2 variants of concern. We present a mathematical model for studying the transmission dynamics of two of these variants (Delta and Omicron) in the United States, in the presence of vaccination, treatment of individuals with clinical symptoms of the disease and the use of face masks. The model is parameterized and cross-validated using observed daily case data for COVID-19 in the United States for the period from November 2021 (when Omicron first emerged) to March 2022. Rigorous qualitative analysis of the model shows that the disease-free equilibrium of the model is locally-asymptotically stable when the control reproduction number of the model (denoted by $R_c$ ) is less than one. This equilibrium is shown to be globally-asymptotically stable for a special case of the model, where disease-induced mortality is negligible and both vaccine-derived immunity in fully-vaccinated individuals and natural immunity do not wane, when the associated reproduction number is less than one. The epidemiological implication of the latter result is that the combined vaccination-boosting strategy can lead to the elimination of the pandemic if its implementation can bring (and maintain) the associated reproduction number to a value less than one. An analytical expression for the vaccine-derived herd immunity threshold is derived. Using this expression, together with the baseline values of the parameters of the parameterized model, we showed that the vaccine-derived herd immunity can be achieved in the United States (so that the pandemic will be eliminated) if at least $68\%$ of the population is fully-vaccinated with two of the three vaccines approved for use in the United States (Pfizer or Moderna vaccine). Furthermore, this study showed (as of the time of writing in March 2022) that the control reproduction number of the Omicron variant was approximately 3.5 times that of the Delta variant (the reproduction of the latter is computed to be $\approx 0.2782$ ), indicating that Delta had practically died out and that Omicron has competitively-excluded Delta (to become the predominant variant in the United States). Based on our analysis and parameterization at the time of writing of this paper (March 2022), our study suggests that SARS-CoV-2 elimination is feasible by June 2022 if the current baseline level of the coverage of fully-vaccinated individuals is increased by about $20\%$ . The prospect of pandemic elimination is significantly improved if vaccination is combined with a face mask strategy that prioritizes moderately effective and high-quality masks. Having a high percentage of the populace wearing the moderately-effective surgical mask is more beneficial to the community than having low percentage of the populace wearing the highly-effective N95 masks. We showed that waning natural and vaccine-derived immunity (if considered individually) offer marginal impact on disease burden, except for the case when they wane at a much faster rate (e.g., within three months), in comparison to the baseline (estimated to be within 9 months to a year). Treatment of symptomatic individuals has marginal effect in reducing daily cases of SARS-CoV-2, in comparison to the baseline, but it has significant impact in reducing daily hospitalizations. Furthermore, while treatment significantly reduces daily hospitalizations (and, consequently, deaths), the prospects of COVID-19 elimination in the United States are significantly enhanced if investments in control resources are focused on mask usage and vaccination rather than on treatment.

Keywords: COVID-19; Masks; Reproduction number; Vaccination and antiviral treatment; Vaccine-derived herd immunity; Waning immunity.

Fig. 1

Data for COVID-19 daily (a) cases and (b) mortality for the United States for the period of the COVID-19 pandemic from January 2020 to February 2022. The data is obtained from the Johns Hopkins University COVID-19 Dashboard , , . The burden of the Delta variant is in the regions between the green and purple dashed vertical lines (denoted by $\Delta$), while that of the Omicron variant is in the region to the right of the dashed purple vertical line (denoted by $O$).

Fig. 2

Flow diagram of the model (2.8). Although recruitment into the population and natural deaths occur (at the rate $\Lambda$ and $\mu$, respectively), these rates are not illustrated in the flow diagram to make it less crowded and easier to follow. The state variables and parameters are described in Tables S1 and S2 of the Supplementary Information (SI).

Fig. 3

(a) Time series illustration of the least squares fit of the model (2.8), showing the model’s output for the daily cases (blue curve) compared to the observed daily confirmed cases for the United States (red dots) from November 28, 2021 to January 31, 2022 (segment to the left of the dashed vertical black line). (b) Simulation result of the model (2.8), showing cumulative COVID-19 cases for the United States as a function of time, using the fixed and estimated baseline parameter values given in Tables S3 and S4. The segment from February 1, 2022 to March 18, 2022 (i.e., solid green and magenta curves or the entire segment to the right of the dashed black vertical line) illustrates the performance of the model (2.8) in predicting the daily and cumulative cases in the United States.

Fig. 4

Contour plots of the control reproduction number of the model (2.8), ${\mathbb{R}}_c$, as a function vaccine coverage (i.e., proportion of fully-vaccinated individuals, $f_v$) and cross-protective vaccine efficacy against the variants (${\epsilon }_v=\min \left({\epsilon }_{vf},{\epsilon }_{vb}\right)$) for the case when (a) mask coverage is maintained at its baseline value, (b) surgical mask is prioritized and the coverage in its usage is increased by $10\%$ from its baseline value, (c) N95 mask is prioritized and the coverage in its usage is increased by $10\%$ from its baseline value. The values of all other parameters used in the simulations are as given by the baseline values in Tables S3 and S4.

Fig. 5

Simulations of the model (2.8) showing the effect of increases or decreases in fully-vaccinated vaccination coverage rate (${\xi }_{vf}$) on the COVID-19 pandemic in the United States. (a) Daily cases, as a function of time, for various values of the fully-vaccinated vaccination coverage rate. (b) Cumulative cases, as a function of time, for various values of the fully-vaccinated vaccination coverage rate. The values of all other parameters used in these simulations are given by the baseline values in Tables S3 and S4.

Fig. 6

Simulations of the model (2.8) showing the incremental impact of mask coverage ($c_m$) and mask type (cloth masks, with ${\epsilon }_m=0.3$; surgical masks, with ${\epsilon }_m=0.7$; and N95 respirators, with ${\epsilon }_m=0.95$) on the daily ((a) and (c)) and cumulative ((b) and (d)) COVID-19 cases in the United States, as a function of time. The values of the other parameters used in these simulations are as given in Tables  S3 and S4.

Fig. 7

Simulations of the model (2.8), showing the combined incremental impact of mask coverage ($c_m$), mask type (cloth masks, with ${\epsilon }_m=0.3$; surgical masks, with ${\epsilon }_m=0.7$; and N95 respirators, with ${\epsilon }_m=0.95$) and fully-vaccinated vaccination coverage rate (${\xi }_{vf}$) on the daily ((a)) and cumulative ((b)) COVID-19 cases in the United States, as a function of time. In these simulations, the mask and fully-vaccinated vaccination coverage rates are increased by $20\%$ from their respective baseline values. The values of the other parameters used in the simulations are given in Tables  S3 and S4.

Fig. 8

Simulations of the model (2.8), showing (a)-(c) new daily and (d)-(f) cumulative COVID-19 cases in the United States, as a function of time, for various values of the waning rate of (a) and (d) vaccine-derived immunity in fully-vaccinated and boosted individuals (${\omega }_v={\omega }_{vf}={\omega }_{vb}$), (b) and (e) natural immunity for the Delta and Omicron variants (${\omega }_r={\omega }_{dr}={\omega }_{or}$), and (c) and (f) both vaccine-derived and natural immunity ($\omega ={\omega }_{vr}={\omega }_{vf}={\omega }_{vb}={\omega }_{dr}={\omega }_{or}$). The durations for the waning of immunity were taken to be 3 months (i.e., $\omega =0.0110$) per day), 6 months ($\omega =0.0055$per day), 9 months ($\omega =0.0037$per day) and 48 months ($\omega =0.0007$per day), respectively. The values of the other parameters used in these simulations are as given in Tables S3-S4.

Fig. 9

Simulations of the model (2.8) depicting the impact of treatment of symptomatic infectious and hospitalized individuals infectious on the (a) confirmed daily COVID-19 cases, and (b) daily COVID-19 hospitalizations in the United States. The treatment rate ($\tau$) is given by $\tau ={\tau }_{j1}={\tau }_{j2}={\tau }_{jh},j\in \{d,o\}$. The other parameter values used for the simulations are presented in Tables  S3 and S4.