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

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## Abstract

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_{S}) ≥70% can eliminate the infection. A vaccine with VE

_{S}<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 VE

_{S}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 VE

_{S}≥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.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model

#### 2.2. Model Parameterization and Fitting

#### 2.3. Product Characteristics of Candidate Vaccines

#### 2.4. Measures of Vaccine Impact

#### 2.5. Vaccination Program Scenarios

#### 2.6. Additional Analyses

#### 2.7. Uncertainty Analysis

## 3. Results

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

_{0}, Text S1D: Probability of a major outbreak. Figure S1: Schematic diagram describing the basic structure of the SARS-CoV-2 vaccine model, 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 scale-up 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 VE

_{S}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, VE

_{S}, 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 R

_{0}, Figure S14: Impact of varying levels of vaccine efficacy in reducing susceptibility, VE

_{S}, on the cumulative number of new SARS-CoV-2 infections when the reproduction number R0 is 3, Table S1: Definitions of population variables and symbols used in the model, Table S2: Model assumptions in terms of parameter values.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Data and Materials Availability

## One Sentence Summary

_{S}) ≥70% at ≥80% coverage could be sufficient to control the pandemic with high cost-effectiveness.

## References

- Yang, Y.; Peng, F.; Wang, R.; Guan, K.; Jiang, T.; Xu, G.; Sun, J.; Chang, C. The deadly coronaviruses: The 2003 SARS pandemic and the 2020 novel coronavirus epidemic in China. J. Autoimmun.
**2020**, 102434. [Google Scholar] [CrossRef] [PubMed] - Lauer, S.A.; Grantz, K.H.; Bi, Q.; Jones, F.K.; Zheng, Q.; Meredith, H.R.; Azman, A.S.; Reich, N.G.; Lessler, J. The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: Estimation and application. Ann. Intern. Med.
**2020**, 172, 577–582. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wu, Z.; McGoogan, J.M. Characteristics of and important lessons from the coronavirus disease 2019 (COVID-19) outbreak in China: Summary of a report of 72314 cases from the Chinese center for disease control and prevention. JAMA
**2020**. [Google Scholar] [CrossRef] [PubMed] - World Health Organization. Report of the WHO-China Joint Mission on Coronavirus Disease 2019 (COVID-19). Available online: https://www.who.int/docs/default-source/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf (accessed on 10 March 2020).
- World Health Organization (WHO). WHO Director-General’s Opening Remarks at the Media Briefing on COVID-19—11 March 2020. Available online: https://www.who.int/dg/speeches/detail/who-director-general-s-opening-remarks-at-the-media-briefing-on-covid-19---11-march-2020 (accessed on 14 March 2020).
- Lu, S. Timely development of vaccines against SARS-CoV-2. Emerg. Microbes Infect.
**2020**, 9, 542–544. [Google Scholar] [CrossRef] [Green Version] - 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] - Baud, D.; Qi, X.; Nielsen-Saines, K.; Musso, D.; Pomar, L.; Favre, G. Real estimates of mortality following COVID-19 infection. Lancet Infect. Dis.
**2020**. [Google Scholar] [CrossRef] [Green Version] - Remuzzi, A.; Remuzzi, G. COVID-19 and Italy: What next? Lancet
**2020**. [Google Scholar] [CrossRef] - McKibbin, W.J.; Fernando, R. The global macroeconomic impacts of COVID-19: Seven scenarios. SSRN Electron. J.
**2020**. [Google Scholar] [CrossRef] [Green Version] - World Health Organization (WHO). Naming the Coronavirus Disease (COVID-19) and the Virus That Causes It. Available online: https://www.who.int/emergencies/diseases/novel-coronavirus-2019/technical-guidance/naming-the-coronavirus-disease-(covid-2019)-and-the-virus-that-causes-it (accessed on 11 March 2020).
- National Institute of Allergy and Infectious Diseases (NIH). NIH Clinical Trial of Investigational Vaccine for COVID-19 Begins. Available online: https://www.nih.gov/news-events/news-releases/nih-clinical-trial-investigational-vaccine-covid-19-begins (accessed on 3 June 2020).
- Ernst, D. Pipeline: Investigational Therapies for COVID-19. Available online: https://www.infectiousdiseaseadvisor.com/home/topics/respiratory/pipeline-investigational-therapies-for-covid-19/ (accessed on 18 March 2020).
- McLean, A.R.; Blower, S.M. Modelling HIV vaccination. Trends Microbiol.
**1995**, 3, 458–462. [Google Scholar] [CrossRef] - Blower, S.M.; McLean, A.R.; Nüsslein-Volhard, C. Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco. Science
**1994**, 265, 1451–1454. [Google Scholar] [CrossRef] - McLean, A.R.; Blower, S.M. Imperfect vaccines and herd immunity to HIV. Proc. R. Soc. B Boil. Sci.
**1993**, 253, 9–13. [Google Scholar] [CrossRef] - Andersson, K.M.; Paltiel, A.D.; Owens, D.K. The potential impact of an HIV vaccine with rapidly waning protection on the epidemic in Southern Africa: Examining the RV144 trial results. Vaccine
**2011**, 29, 6107–6112. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Andersson, K.M.; Owens, D.K.; Vardas, E.; Gray, G.E.; McIntyre, J.A.; Paltiel, A.D. Predicting the impact of a partially effective HIV vaccine and subsequent risk behavior change on the heterosexual HIV epidemic in low- and middle-income countries: A South African example. J. Acquir. Immune Defic. Syndr.
**2007**, 46, 78–90. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Morrison, D.; Ribeiro, R.M.; Chao, D.L.; Perelson, A.S. Predicting the impact of a nonsterilizing vaccine against human immunodeficiency virus. J Virol.
**2004**, 78, 11340–11351. [Google Scholar] [CrossRef] [Green Version] - Wells, C.R.; Yamin, D.; Ndeffo-Mbah, M.L.; Wenzel, N.; Gaffney, S.G.; Townsend, J.P.; Meyers, L.; Fallah, M.; Nyenswah, T.G.; Altice, F.L.; et al. Harnessing case isolation and ring vaccination to control Ebola. PLoS Negl. Trop. Dis.
**2015**, 9, e0003794. [Google Scholar] [CrossRef] - Boily, M.-C.; Brisson, M.; Mâsse, B.; Anderson, R. The role of mathematical models in vaccine development and public health decision making. In Vaccinology: Principles and Practice; Morrow, W., Sheikh, N., Schmidt, C., Davies, D., Eds.; Wiley-Blackwell: Hoboken, NJ, USA, 2012; pp. 480–508. [Google Scholar]
- Abu-Raddad, L.J.; Boily, M.-C.; Self, S.; Longini, I.M. Analytic insights into the population level impact of imperfect prophylactic HIV vaccines. J. Acquir. Immune Defic. Syndr.
**2007**, 45, 454–467. [Google Scholar] [CrossRef] - Boily, M.-C.; Abu-Raddad, L.J.; Desai, K.; Masse, B.; Self, S.; Anderson, R. Measuring the public-health impact of candidate HIV vaccines as part of the licensing process. Lancet Infect. Dis.
**2008**, 8, 200–207. [Google Scholar] [CrossRef] - Alsallaq, R.A.; Schiffer, J.T.; Longini, I.M.; Wald, A.; Corey, L.; Abu-Raddad, L.J. Population level impact of an imperfect prophylactic vaccine for herpes simplex virus-2. Sex. Transm. Dis.
**2010**, 37, 290–297. [Google Scholar] [CrossRef] [Green Version] - Gay, N.J.; Hesketh, L.M.; Morgan-Capner, P.; Miller, E. Interpretation of serological surveillance data for measles using mathematical models: Implications for vaccine strategy. Epidemiol. Infect.
**1995**, 115, 139–156. [Google Scholar] [CrossRef] [Green Version] - Michael, E.; Malecela-Lazaro, M.N.; Kazura, J.W. Epidemiological modelling for monitoring and evaluation of lymphatic filariasis control. Adv. Parasitol.
**2007**, 65, 191–237. [Google Scholar] [CrossRef] - Basáñez, M.-G.; McCarthy, J.S.; French, M.D.; Yang, G.-J.; Walker, M.; Gambhir, M.; Prichard, R.K.; Churcher, T.S. A research agenda for helminth diseases of humans: Modelling for control and elimination. PLoS Negl. Trop. Dis.
**2012**, 6, e1548. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hill, E.M.; Petrou, S.; De Lusignan, S.; Yonova, I.; Keeling, M.J. Seasonal influenza: Modelling approaches to capture immunity propagation. PLoS Comput. Biol.
**2019**, 15, e1007096. [Google Scholar] [CrossRef] [Green Version] - Sah, P.; Alfaro-Murillo, J.A.; Fitzpatrick, M.C.; Neuzil, K.M.; Meyers, L.A.; Singer, B.H.; Galvani, A.P. Future epidemiological and economic impacts of universal influenza vaccines. Proc. Natl. Acad. Sci. USA
**2019**, 116, 20786–20792. [Google Scholar] [CrossRef] [Green Version] - Gottlieb, S.L.; Jerse, A.E.; Delany-Moretlwe, S.; Deal, C.; Giersing, B.K. Advancing vaccine development for gonorrhoea and the Global STI Vaccine Roadmap. Sex. Health
**2019**, 16, 426–432. [Google Scholar] [CrossRef] - Gottlieb, S.L.; Giersing, B.; Boily, M.-C.; Chesson, H.; Looker, K.J.; Schiffer, J.; Spicknall, I.; Hutubessy, R.; Broutet, N. Modelling efforts needed to advance herpes simplex virus (HSV) vaccine development: Key findings from the World Health Organization Consultation on HSV Vaccine Impact Modelling. Vaccine
**2019**, 37, 7336–7345. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Peasah, S.K.; Meltzer, M.I.; Vu, M.; Moulia, D.L.; Bridges, C.B. Cost-effectiveness of increased influenza vaccination uptake against readmissions of major adverse cardiac events in the US. PLoS ONE
**2019**, 14, e0213499. [Google Scholar] [CrossRef] - Spicknall, I.H.; Looker, K.J.; Gottlieb, S.L.; Chesson, H.W.; Schiffer, J.T.; Elmes, J.; Boily, M.-C. Review of mathematical models of HSV-2 vaccination: Implications for vaccine development. Vaccine
**2019**, 37, 7396–7407. [Google Scholar] [CrossRef] - Ayoub, H.H.; Chemaitelly, H.; Mumtaz, G.R.; Seedat, S.; Awad, S.F.; Makhoul, M.; Abu-Raddad, L.J. Characterizing key attributes of the epidemiology of COVID-19 in China: Model-based estimations. medRxiv
**2020**. [Google Scholar] [CrossRef] [Green Version] - Halloran, M.E.; Haber, M.; Longini, I.M. Interpretation and estimation of vaccine efficacy under heterogeneity. Am. J. Epidemiol.
**1992**, 136, 328–343. [Google Scholar] [CrossRef] - Halloran, M.; Haber, M.; Longini, I.M.; Struchiner, C.J. Direct and indirect effects in vaccine efficacy and effectiveness. Am. J. Epidemiol.
**1991**, 133, 323–331. [Google Scholar] [CrossRef] - Halloran, M.E.; Struchiner, C.J.; Longini, I.M. Study designs for evaluating different efficacy and effectiveness aspects of vaccines. Am. J. Epidemiol.
**1997**, 146, 789–803. [Google Scholar] [CrossRef] - Halloran, M.; Watelet, L.; Struchiner, C.J. Epidemiologic effects of vaccines with complex direct effects in an age-structured population. Math. Biosci.
**1994**, 121, 193–225. [Google Scholar] [CrossRef] - Matrajt, L.; Longini, I.M. Critical immune and vaccination thresholds for determining multiple influenza epidemic waves. Epidemics
**2012**, 4, 22–32. [Google Scholar] [CrossRef] [Green Version] - Hill, A.N.; Longini, I.M. The critical vaccination fraction for heterogeneous epidemic models. Math. Biosci.
**2003**, 181, 85–106. [Google Scholar] [CrossRef] - MATLAB. The Language of Technical Computing; The MathWorks, Inc.: Natick, MA, USA, 2019. [Google Scholar]
- Guan, W.-J.; Ni, Z.-Y.; Hu, Y.; Liang, W.-H.; Ou, C.-Q.; He, J.-X.; Liu, L.; Shan, H.; Lei, C.-L.; Hui, D.S.C.; et al. Clinical characteristics of coronavirus disease 2019 in China. N. Engl. J. Med.
**2020**. [Google Scholar] [CrossRef] - Huang, C.; Wang, Y.; Li, X.; Ren, L.; Zhao, J.; Hu, Y.; Zhang, L.; Fan, G.; Xu, J.; Gu, X.; et al. Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China. Lancet
**2020**, 395, 497–506. [Google Scholar] [CrossRef] [Green Version] - Novel Coronavirus Pneumonia Emergency Response Epidemiology Team. The epidemiological characteristics of an outbreak of 2019 novel coronavirus diseases (COVID-19) in China. Zhonghua Liu Xing Bing Xue Za Zhi
**2020**, 41, 145–151. [Google Scholar] [CrossRef] - United Nations Department of Economic and Social Affairs Population Dynamics. The 2019 Revision of World Population Prospects. Available online: https://population.un.org/wpp/ (accessed on 1 March 2020).
- COVID-19 Outbreak Live Update. Available online: https://www.worldometers.info/coronavirus/ (accessed on 14 March 2020).
- Makhoul, M.; Ayoub, H.H.; Chemaitelly, H.; Seedat, S.; Mumtaz, G.R.; Abu-Raddad, L.J. Epidemiological impact of SARS-CoV-2 vaccination: Mathematical modeling analyses. medRxiv
**2020**. submitted for publication. [Google Scholar] - McKay, M.D.; Beckman, R.J.; Conover, W.J. A Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics
**1979**, 21, 239–245. [Google Scholar] [CrossRef] - Sanchez, M.A.; Blower, S.M. Uncertainty and sensitivity analysis of the basic reproductive rate - Tuberculosis as an example. Am. J. Epidemiol.
**1997**, 145, 1127–1137. [Google Scholar] [CrossRef] [Green Version] - Davies, N.G.; Klepac, P.; Liu, Y.; Prem, K.; Jit, M.; Eggo, R.M. Age-dependent effects in the transmission and control of COVID-19 epidemics. Nat. Med.
**2020**. [Google Scholar] [CrossRef] - Zhu, Y.; Bloxham, C.J.; Hulme, K.D.; Sinclair, J.E.; Tong, Z.W.M.; Steele, L.E.; Noye, E.C.; Lu, J.; Chew, K.Y.; Pickering, J.; et al. Children are unlikely to have been the primary source of household SARS-CoV-2 infections. SSRN Electron. J.
**2020**, 2020. [Google Scholar] [CrossRef] - Li, R.; Pei, S.; Chen, B.; Song, Y.; Zhang, T.; Yang, W.; Shaman, J. Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV2). Science
**2020**, 368, 489–493. [Google Scholar] [CrossRef] [Green Version] - Verity, R.; Okell, L.C.; Dorigatti, I.; Winskill, P.; Whittaker, C.; Imai, N.; Cuomo-Dannenburg, G.; Thompson, H.; Walker, P.G.T.; Fu, H.; et al. Estimates of the severity of coronavirus disease 2019: A model-based analysis. Lancet
**2020**, 20, 669–677. [Google Scholar] [CrossRef] - He, W.; Yi, G.Y.; Zhu, Y. Estimation of the basic reproduction number, average incubation time, asymptomatic infection rate, and case fatality rate for COVID-19: Meta-analysis and sensitivity analysis. J. Med. Virol.
**2020**. [Google Scholar] [CrossRef] - MIDAS Online COVID-19 Portal. COVID-19 Parameter Estimates: Basic Reproduction Number. Available online: https://github.com/midas-network/COVID-19/tree/master/parameter_estimates/2019_novel_coronavirus (accessed on 19 May 2020).

**Figure 1.**Impact of SARS-CoV-2 vaccination on the 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. The duration of vaccine protection is 10 years. Impact was assessed at $V{E}_{S}=50\%$, $V{E}_{I}=50\%$, $V{E}_{{P}_{1}}=50\%$, $V{E}_{{P}_{2}}=50\%$, $V{E}_{S}=V{E}_{I}=V{E}_{{P}_{1}}=50\%$.

**Figure 2.**Impact of SARS-CoV-2 vaccination on the 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 $V{E}_{S}=50\%$, $V{E}_{I}=50\%$, $V{E}_{{P}_{1}}=50\%$, $V{E}_{{P}_{2}}=50\%$, $V{E}_{S}=V{E}_{I}=V{E}_{{P}_{1}}=50\%$.

**Figure 3.**SARS-CoV-2 vaccine effectiveness. Number of vaccinated persons needed to avert (

**A**) one infection, (

**B**) one severe disease case, (

**C**) one critical disease case, and (

**D**) one death, by the end of the epidemic cycle, that is, after the epidemic has reached its peak and declined to a negligible level. The scenario assumes vaccine scale-up to 80% coverage before epidemic onset. Duration of vaccine protection is 10 years. Impact was assessed at $V{E}_{S}=50\%$, $V{E}_{I}=50\%$, $V{E}_{{P}_{1}}=50\%$, $V{E}_{{P}_{2}}=50\%$, $V{E}_{S}=V{E}_{I}=V{E}_{{P}_{1}}=50\%$. Panel A does not include the result for $V{E}_{{P}_{2}}=50\%$, as this efficacy has no impact on the number of infections—it affects only severe and critical disease and death.

**Figure 4.**Effectiveness of age-group prioritization using a SARS-CoV-2 vaccine with VE

_{s}of 50%. Number of vaccinated persons needed to avert (

**A**) one infection, (

**B**) one severe disease case, (

**C**) one critical disease case, and (

**D**) one death by prioritizing different age groups for vaccination. Scenario assumes vaccine scale-up to 80% coverage before epidemic onset and duration of vaccine protection of 10 years. Effectiveness is assessed at the end of the epidemic cycle, that is, after the epidemic has reached its peak and declined to a negligible level.

**Figure 5.**Impact of varying levels of vaccine efficacy in reducing susceptibility (VE

_{s}) on (

**A**) cumulative number of new SARS-CoV-2 infections (final epidemic size) and (

**B**) number of vaccinated persons needed to avert one SARS-CoV-2 infection. Scenario assumes vaccine scale-up to 80% coverage before epidemic onset. Duration of vaccine protection is 10 years. Measures are assessed at the end of the epidemic cycle, that is, after the epidemic has reached its peak and declined to a negligible level.

**Figure 6.**Impact of vaccination with reduced adherence to social distancing for those vaccinated. Figure shows the impact of varying levels of behavior compensation post-vaccination on the vaccine-induced reduction in the cumulative number of new SARS-CoV-2 infections by the end of the epidemic cycle. Scenario assumes vaccine scale-up to 80% coverage before epidemic onset, VE

_{s}is 50%, and duration of vaccine protection is 10 years.

**Figure 7.**Probability of occurrence of a major outbreak following vaccination. Probability of occurrence of a major outbreak upon virus introduction at varying levels of (

**A**) $V{E}_{S}$, (

**B**) $V{E}_{I}$, (

**C**) $V{E}_{{P}_{1}}$, and (

**D**) $V{E}_{S}=V{E}_{I}=V{E}_{{P}_{1}}$. Scenario assumes vaccine scale-up to 80% coverage before epidemic onset. Duration of vaccine protection is 10 years. The figure does not include the result for $V{E}_{{P}_{2}}$, as this efficacy has no impact on the probability of occurrence of a major outbreak. The analysis and derivation for the probability of occurrence of a major outbreak can be found in Text S1D of the Supplementary Material.

Vaccine Characteristic | Definition | Description |
---|---|---|

$V{E}_{S}$ | Vaccine efficacy in reducing susceptibility | Proportional reduction in the susceptibility to infection acquisition among those vaccinated compared to those unvaccinated |

$V{E}_{I}$ | Vaccine efficacy in reducing infectiousness | Proportional reduction in infectiousness (lower viral load due to vaccine-primed immune response) among those who are vaccinated but acquire the infection compared to those unvaccinated |

$V{E}_{{P}_{1}}$ | Vaccine efficacy in reducing the duration of infection | Proportional reduction in the duration of mild infection (faster infection clearance due to vaccine-primed immune response) among those who are vaccinated but still acquire the infection compared to those unvaccinated |

$V{E}_{{P}_{2}}$ | Vaccine efficacy in reducing the fraction of individuals with severe or critical infection | Proportional reduction in the fraction of individuals with severe or critical infection (lower probability of developing severe or critical infection due to vaccine-primed immune response) among those who are vaccinated but still acquire the infection compared to those unvaccinated |

$D$ | Duration of vaccine protection | Duration of protection that the vaccine will elicit |

$r$ | Behavior compensation post-vaccination | Proportional increase in social contact rate (reduced social distancing) among those who are vaccinated compared to those unvaccinated |

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**MDPI and ACS Style**

Makhoul, M.; Ayoub, H.H.; Chemaitelly, H.; Seedat, S.; Mumtaz, G.R.; Al-Omari, S.; Abu-Raddad, L.J.
Epidemiological Impact of SARS-CoV-2 Vaccination: Mathematical Modeling Analyses. *Vaccines* **2020**, *8*, 668.
https://doi.org/10.3390/vaccines8040668

**AMA Style**

Makhoul M, Ayoub HH, Chemaitelly H, Seedat S, Mumtaz GR, Al-Omari S, Abu-Raddad LJ.
Epidemiological Impact of SARS-CoV-2 Vaccination: Mathematical Modeling Analyses. *Vaccines*. 2020; 8(4):668.
https://doi.org/10.3390/vaccines8040668

**Chicago/Turabian Style**

Makhoul, Monia, Houssein H. Ayoub, Hiam Chemaitelly, Shaheen Seedat, Ghina R. Mumtaz, Sarah Al-Omari, and Laith J. Abu-Raddad.
2020. "Epidemiological Impact of SARS-CoV-2 Vaccination: Mathematical Modeling Analyses" *Vaccines* 8, no. 4: 668.
https://doi.org/10.3390/vaccines8040668