Epidemiological Impact of SARS-CoV-2 Vaccination: Mathematical Modeling Analyses

This study aims to inform SARS-CoV-2 vaccine development/licensure/decision-making/implementation, using mathematical modeling, by determining key preferred vaccine product characteristics and associated population-level impacts of a vaccine eliciting long-term protection. A prophylactic vaccine with efficacy against acquisition (VES) ≥70% can eliminate the infection. A vaccine with VES <70% may still control the infection if it reduces infectiousness or infection duration among those vaccinated who acquire the infection, if it is supplemented with <20% reduction in contact rate, or if it is complemented with herd-immunity. At VES of 50%, the number of vaccinated persons needed to avert one infection is 2.4, and the number is 25.5 to avert one severe disease case, 33.2 to avert one critical disease case, and 65.1 to avert one death. The probability of a major outbreak is zero at VES ≥70% regardless of the number of virus introductions. However, an increase in social contact rate among those vaccinated (behavior compensation) can undermine vaccine impact. In addition to the reduction in infection acquisition, developers should assess the natural history and disease progression outcomes when evaluating vaccine impact.


Table of Content
Text S1A. Model structure. Figure S1. Schematic diagram describing the basic structure of the SARS-CoV-2 vaccine model. Table S1. Definitions of population variables and symbols used in the model. Text S1B. Parameter values. Text S1C. The basic reproduction number R0. Text S1D. Probability of a major outbreak. Figure S2. Impact of SARS-CoV-2 vaccination on the cumulative number of A) new infections, B) new severe disease cases, C) new critical disease cases, and D) new deaths in the scenario assuming vaccine scale-up to 80% coverage before epidemic onset. Figure S3. Role of SARS-CoV-2 vaccination in reducing the cumulative number of A) new infections, B) new severe disease cases, C) new critical disease cases, and D) new deaths in the scenario assuming vaccine scale-up to 80% coverage before epidemic onset. Figure S4. Impact of SARS-CoV-2 vaccination on the cumulative number of A) new infections, B) new severe disease cases, C) new critical disease cases, and D) new deaths in the scenario assuming vaccine introduction during the exponential growth phase of the epidemic, with scaleup to 80% coverage within one month. Figure S5. Role of SARS-CoV-2 vaccination in reducing the cumulative number of A) new infections, B) new severe disease cases, C) new critical disease cases, and D) new deaths in the scenario assuming vaccine introduction during the exponential growth phase of the epidemic, with scale-up to 80% coverage within one month. Figure S6. Temporal evolution of SARS-CoV-2 vaccine effectiveness in the scenario assuming vaccine scale-up to 80% coverage before epidemic onset. Figure S7. Temporal evolution of SARS-CoV-2 vaccine effectiveness in the scenario assuming vaccine introduction during the exponential growth phase of the epidemic, with scale-up to 80% coverage within one month. Figure S8. Temporal evolution of effectiveness of age-group prioritization using a SARS-CoV-2 vaccine with VES of 50%. Figure S9. Impact of a social-distancing intervention reducing the contact rate in the population on the cumulative number of new SARS-CoV-2 infections, when introduced to supplement the impact of a vaccine that has 50% efficacy in reducing susceptibility, VES. Figure S10. Probability of occurrence of a major outbreak following vaccination. Figure S11. Sensitivity analyses assessing vaccine effectiveness (number of vaccinated persons needed to avert one infection) at A) varying levels of vaccine coverage and B) high levels of assortativeness in age group mixing. Figure S12. Uncertainty analysis. Figure S13. Vaccine effectiveness of age-group prioritization and the reproduction number R0. Figure S14. Impact of varying levels of vaccine efficacy in reducing susceptibility, VES, on the cumulative number of new SARS-CoV-2 infections when the reproduction number R0 is 3.

A. Model structure
We extended a recently-developed age-structured deterministic compartmental model to describe the impact of vaccination on severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) transmission dynamics and progression of the resulting disease, Coronavirus Disease 2019 (COVID-2019), in a given population. The model stratifies the unvaccinated and vaccinated populations into compartments according to age group (0-9, 10-19, 20-29,…, ≥80 years), infection status (uninfected, infected), infection stage (mild, severe, critical), and disease stage (severe, critical). Transmission and disease progression dynamics in vaccinated and unvaccinated cohorts are described using age-specific sets of nonlinear ordinary differential equations, where each age group a ( 1, 2,...9 a  ) refers to a 10-year age band (0-9,10-19,..70-79) apart from the last group including all those aged  80 years. The model is illustrated in Figure S1. The following set of equations describes the transmission dynamics among unvaccinated and vaccinated populations in the first age group: Unvaccinated population aged 0-9 years: Vaccinated population aged 0-9 years: For subsequent age groups, the following set of equations was used: Unvaccinated populations aged 10+ years: Vaccinated populations aged 10+ years: The definitions of population variables and symbols used in the equations are in Table S1.
while that of vaccinated susceptible populations ( ) V a is given by where  is the overall infectious contact rate. The mixing among the different age groups is dictated by the mixing matrix , a a H . This matrix provides the probability that an individual in the a age group will mix with an individual in the a age group (regardless of vaccination status).
The mixing matrix is given by  Age e  , the mixing is fully assortative, that is individuals mix only with members in their own age group.

B. Parameter values
The input parameters of the model were chosen based on current empirical data for SARS-CoV-2 natural history and epidemiology. The parameter values are listed in Table S2.

C. The basic reproduction number R0
Using the second generation matrix method described by Heffernan et al. [12], the basic reproduction number in absence of vaccination is given by where ( ) w a is the proportion of the population in each age group.

D. Probability of a major outbreak
Based on Whittle's method [13], and by constructing Bailey's ratios [13], the probability of a major outbreak was derived, that is the probability that the fraction of susceptible individuals that become infected is ≥  , where  is a specific chosen level of the final attack rate.

Case 2
For   Meanwhile, in the case of vaccinating a fraction of the population, the probability of a major outbreak is given by the following three cases: , the probability of a major outbreak lies between 0 and

Case 3
For , the probability of a major outbreak is 0.

Figure S2. Impact of SARS-CoV-2 vaccination on the cumulative number of A) new infections, B) new severe disease cases, C) new critical disease cases, and D) new deaths in the scenario assuming vaccine scale-up to
80% coverage before epidemic onset. Duration of vaccine protection is 10 years. Impact was assessed at 50%

Figure S3. Role of SARS-CoV-2 vaccination in reducing the cumulative number of A) new infections, B) new severe disease cases, C) new critical disease cases, and D) new deaths in the scenario assuming vaccine scale-
up to 80% coverage before epidemic onset. Duration of vaccine protection is 10 years. Impact was assessed at 50%

Figure S4. Impact of SARS-CoV-2 vaccination on the cumulative number of A) new infections, B) new severe disease cases, C) new critical disease cases, and D) new deaths in the scenario assuming vaccine introduction during the exponential growth phase of the epidemic, with scale-up to 80% coverage within one month.
Duration of vaccine protection is 10 years. Impact was assessed at 50%

Figure S11. Sensitivity analyses assessing vaccine effectiveness (number of vaccinated persons needed to avert one infection) at A) varying levels of vaccine coverage and B) high levels of assortativeness in age group mixing.
Effectiveness is assessed at end of epidemic cycle, that is after the epidemic has reached its peak and declined to a negligible level. Calculations assume vaccine scale-up to the targeted coverage before epidemic onset. Impact was assessed at 50% S VE  . Duration of vaccine protection is 10 years. Figure S12. Uncertainty analysis. Model predictions for the mean cumulative number of new infections and associated 95% uncertainty interval (UI) at various levels of vaccine efficacy in reducing susceptibility (VES ) generated through 500 simulation runs. Scenario assumes vaccine scale-up to 80% coverage before epidemic onset. Duration of vaccine protection is 10 years. The solid black line, dashed lines, and shades show respectively, the mean, 95% uncertainty interval, and individual estimates for the cumulative number of new infections across all 500 uncertainty runs.
Figure S13. Vaccine effectiveness of age-group prioritization and the reproduction number R0. Model predictions for SARS CoV-2 attack rate at various levels of R0, indicating also the impact on R0 of prioritizing those 60-69 years of age for vaccination or vaccinating all age groups. The blue dashed-dotted line shows the modelpredicted attack rate at various levels of R0. The blue, red, and green stars show, respectively, the model-predicted attack rates in absence of vaccination, by prioritizing vaccination at 80% coverage for those 60-69 years of age, and by extending vaccination at 80% coverage to all age groups. The figure highlights how effectiveness of the vaccine (number of vaccinations needed to avert one infection) depends on the position on the R0 curve-prioritizing vaccination for any single age group, regardless of that age group, has overall lower effectiveness than extending vaccination to all age groups. The reason is that vaccinating one age group reduces R0 only marginally, whereas vaccinating all age groups reduces R0 to an epidemic domain where small reductions in R0 can have more substantial impact on epidemic size. Figure S14. Impact of varying levels of vaccine efficacy in reducing susceptibility, VES, on the cumulative number of new SARS-CoV-2 infections when the reproduction number R0 is 3. Scenario assumes vaccine scaleup to 80% coverage before epidemic onset. Duration of vaccine protection is 10 years.