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1.
Clin Respir J ; 16(11): 708-716, 2022 Nov.
Article in English | MEDLINE | ID: covidwho-2052356

ABSTRACT

Coronavirus disease 2019 (COVID-19), the highly contagious viral disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has spread worldwide with millions of cases and more than 5 million deaths to date. SARS-CoV-2 has caused serious damage all over the world with many countries experiencing the third or the fourth wave of the viral disease outbreaks, mainly due to the emergence of mutant variants. Those who unvaccinated remain most vulnerable to COVID-19 and its variants. COVID-19 vaccination, along with prevention strategies, is a critical measure to defense against the disease. COVID-19 vaccination can reduce the spread of virus and help protect susceptible population. Although herd immunity might not be realized solely by vaccination, COVID-19 vaccines have been proved to be effective in reducing the risk of severe disease, hospitalization, and even death. It is recommended that people get vaccinated as soon as they are eligible. This review summarizes the recent SARS-CoV-2 variants that brought challenges for vaccination and herd immunity and discusses promising management strategies.


Subject(s)
COVID-19 , SARS-CoV-2 , Humans , COVID-19/prevention & control , Immunity, Herd , COVID-19 Vaccines , Vaccination
2.
Bull Math Biol ; 84(10): 116, 2022 09 10.
Article in English | MEDLINE | ID: covidwho-2014405

ABSTRACT

COVID-19 is caused by the SARS-CoV-2 virus, which is mainly transmitted directly between humans. However, it is observed that this disease can also be transmitted through an indirect route via environmental fomites. The development of appropriate and effective vaccines has allowed us to target and anticipate herd immunity. Understanding of the transmission dynamics and the persistence of the virus on environmental fomites and their resistive role on indirect transmission of the virus is an important scientific and public health challenge because it is essential to consider all possible transmission routes and route specific transmission strength to accurately quantify the herd immunity threshold. In this paper, we present a mathematical model that considers both direct and indirect transmission modes. Our analysis focuses on establishing the disease invasion threshold, investigating its sensitivity to both transmission routes and isolate route-specific transmission rate. Using the tau-leap algorithm, we perform a stochastic model simulation to address the invasion potential of both transmission routes. Our analysis shows that direct transmission has a higher invasion potential than that of the indirect transmission. As a proof of this concept, we fitted our model with early epidemic data from several countries to uniquely estimate the reproduction numbers associated with direct and indirect transmission upon confirming the identifiability of the parameters. As the indirect transmission possess lower invasion potential than direct transmission, proper estimation and necessary steps toward mitigating it would help reduce vaccination requirement.


Subject(s)
COVID-19 , Immunity, Herd , COVID-19/prevention & control , Humans , Mathematical Concepts , Models, Biological , SARS-CoV-2
3.
Viruses ; 14(8)2022 08 22.
Article in English | MEDLINE | ID: covidwho-1997803

ABSTRACT

It has been very difficult to predict the development of the COVID-19 pandemic based on mathematical models for the spread of infectious diseases, and due to major non-pharmacological interventions (NPIs), it is still unclear to what extent the models would have fit reality in a "do nothing" scenario. To shed light on this question, the case of Sweden during the time frame from autumn 2020 to spring 2021 is particularly interesting, since the NPIs were relatively minor and only marginally updated. We found that state of the art models are significantly overestimating the spread, unless we assume that social interactions significantly decrease continuously throughout the time frame, in a way that does not correlate well with Google-mobility data nor updates to the NPIs or public holidays. This leads to the question of whether modern SEIR-type mathematical models are unsuitable for modeling the spread of SARS-CoV-2 in the human population, or whether some particular feature of SARS-CoV-2 dampened the spread. We show that, by assuming a certain level of pre-immunity to SARS-CoV-2, we obtain an almost perfect data-fit, and discuss what factors could cause pre-immunity in the mathematical models. In this scenario, a form of herd-immunity under the given restrictions was reached twice (first against the Wuhan-strain and then against the alpha-strain), and the ultimate decline in cases was due to depletion of susceptibles rather than the vaccination campaign.


Subject(s)
COVID-19 , COVID-19/epidemiology , Humans , Immunity, Herd , Pandemics/prevention & control , SARS-CoV-2 , Sweden/epidemiology
4.
Philos Trans A Math Phys Eng Sci ; 380(2233): 20210311, 2022 Oct 03.
Article in English | MEDLINE | ID: covidwho-1992466

ABSTRACT

Long-term control of SARS-CoV-2 outbreaks depends on the widespread coverage of effective vaccines. In Australia, two-dose vaccination coverage of above 90% of the adult population was achieved. However, between August 2020 and August 2021, hesitancy fluctuated dramatically. This raised the question of whether settings with low naturally derived immunity, such as Queensland where less than [Formula: see text] of the population is known to have been infected in 2020, could have achieved herd immunity against 2021's variants of concern. To address this question, we used the agent-based model Covasim. We simulated outbreak scenarios (with the Alpha, Delta and Omicron variants) and assumed ongoing interventions (testing, tracing, isolation and quarantine). We modelled vaccination using two approaches with different levels of realism. Hesitancy was modelled using Australian survey data. We found that with a vaccine effectiveness against infection of 80%, it was possible to control outbreaks of Alpha, but not Delta or Omicron. With 90% effectiveness, Delta outbreaks may have been preventable, but not Omicron outbreaks. We also estimated that a decrease in hesitancy from 20% to 14% reduced the number of infections, hospitalizations and deaths by over 30%. Overall, we demonstrate that while herd immunity may not be attainable, modest reductions in hesitancy and increases in vaccine uptake may greatly improve health outcomes. This article is part of the theme issue 'Technical challenges of modelling real-life epidemics and examples of overcoming these'.


Subject(s)
COVID-19 , Immunity, Herd , Australia/epidemiology , COVID-19/epidemiology , COVID-19/prevention & control , Humans , Queensland/epidemiology , SARS-CoV-2 , Vaccination
5.
PLoS Comput Biol ; 18(8): e1010391, 2022 08.
Article in English | MEDLINE | ID: covidwho-1987114

ABSTRACT

The COVID-19 pandemic demonstrated that the process of global vaccination against a novel virus can be a prolonged one. Social distancing measures, that are initially adopted to control the pandemic, are gradually relaxed as vaccination progresses and population immunity increases. The result is a prolonged period of high disease prevalence combined with a fitness advantage for vaccine-resistant variants, which together lead to a considerably increased probability for vaccine escape. A spatial vaccination strategy is proposed that has the potential to dramatically reduce this risk. Rather than dispersing the vaccination effort evenly throughout a country, distinct geographic regions of the country are sequentially vaccinated, quickly bringing each to effective herd immunity. Regions with high vaccination rates will then have low infection rates and vice versa. Since people primarily interact within their own region, spatial vaccination reduces the number of encounters between infected individuals (the source of mutations) and vaccinated individuals (who facilitate the spread of vaccine-resistant strains). Thus, spatial vaccination may help mitigate the global risk of vaccine-resistant variants.


Subject(s)
COVID-19 , Vaccines , COVID-19/epidemiology , COVID-19/prevention & control , Humans , Immunity, Herd , Pandemics/prevention & control , Vaccination
6.
Am J Public Health ; 112(10): 1465-1470, 2022 10.
Article in English | MEDLINE | ID: covidwho-1974455

ABSTRACT

Intermittently, the concept of herd immunity has been a potent, if sometimes ambiguous and controversial, means of framing the global response to the COVID-19 pandemic and envisaging its end. Realizing the full meaning of human herd immunity requires further attention to its connections after World War I with British social theory. Distracted by "obvious" yet unsubstantiated correspondences with veterinary research, historians of the concept have not engaged with the more proximate influence of discussions of social psychology and group dynamics on postwar epidemiology. Understanding the openness of early 20th century epidemiology to social thought deepens our appreciation of the significance of herd or population immunity, as well as suggests new avenues for exchange between public health and contemporary social sciences. (Am J Public Health. 2022;112(10):1465-1470. https://doi.org/10.2105/AJPH.2022.306931).


Subject(s)
COVID-19 , Humans , Immunity, Herd , Pandemics/prevention & control , Psychology, Social , Social Conditions , Social Sciences
7.
Animal Model Exp Med ; 5(5): 430-435, 2022 10.
Article in English | MEDLINE | ID: covidwho-1966022

ABSTRACT

The mass inoculation of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) vaccines to induce herd immunity is one of the most effective measures we can deploy in the fight against coronavirus disease 2019 (COVID-19). Pregnant women are prone to a higher risk of COVID-19, and maternal infection is a risk factor for a range of neurological disorders leading to abnormal behavior in adulthood. However, there are limited clinical data to support whether vaccination or infection post-immunization in pregnant women can affect the behavioral cognition of fetuses in adulthood. In this study, human angiotensin-converting enzyme 2 pregnant mice (F0 generation) were immunized with CoronaVac and then infected with SARS-CoV-2. Subsequently, we analyzed the behavioral cognition of their adult offspring (F1 generation) using the open-field test and Morris water maze test. The adult F1 generation did not exhibit any impairments in spontaneous locomotor activity or spatial reference memory.


Subject(s)
COVID-19 , SARS-CoV-2 , Humans , Adult , Female , Mice , Pregnancy , Animals , COVID-19 Vaccines , COVID-19/prevention & control , Immunity, Herd , Vaccination
8.
Med Microbiol Immunol ; 211(4): 195-210, 2022 Aug.
Article in English | MEDLINE | ID: covidwho-1919783

ABSTRACT

In the fight against coronavirus infection, control of the immune response is of decisive importance, an important component of which is the seroprevalence of antibodies to SARS-CoV-2. Immunity to SARS-CoV-2 is formed either naturally or artificially through vaccination. The purpose of this study was to assess the seroprevalence of antibodies to SARS-CoV-2 in the population of Kyrgyzstan. A cross-sectional randomized study of seroprevalence was carried out according to a program developed by Rospotrebnadzor and the St. Petersburg Pasteur Institute, taking into account WHO recommendations. The ethics committees of the Association of Preventive Medicine (Kyrgyzstan) and the St. Petersburg Pasteur Institute (Russia) approved the study. Volunteers (9471) were recruited, representing 0.15% (95% CI 0.14-0.15) of the total population, randomized by age and region. Plasma antibodies (Abs) to the nucleocapsid antigen (Nag) were determined. In vaccinated individuals, Abs to the SARS-CoV-2 receptor-binding domain antigen (RBDag) were determined. Differences were considered statistically significant at p < 0.05. The SARS-CoV-2 Nag Ab seroprevalence was 48.7% (95% CI 47.7-49.7), with a maximum in the 60-69 age group [59.2% (95% CI 56.6-61.7)] and a minimum in group 1-17 years old [32.7% (95 CI: 29.4-36.1)]. The highest proportion of seropositive individuals was in the Naryn region [53.3% (95% CI 49.8-56.8)]. The lowest share was in Osh City [38.1% (95% CI 32.6-43.9)]. The maximum SARS-CoV-2 Nag seropositivity was found in the health-care sector [57.1% (95% CI 55.4-58.8)]; the minimum was seen among artists [38.6% (95% CI 26.0-52.4)]. Asymptomatic SARS-CoV-2 Nag seropositivity was 77.1% (95% CI 75.6-78.5). Vaccination with Sputnik V or Sinopharm produced comparable Ab seroprevalence. SARS-CoV-2 Nag seropositivity in the Kyrgyz population was 48.75% (95% CI 47.7-49.7), with the mass vaccination campaign undoubtedly benefitting the overall situation.


Subject(s)
COVID-19 , Immunity, Herd , SARS-CoV-2 , Adolescent , Antibodies, Viral , COVID-19/epidemiology , Child , Child, Preschool , Cross-Sectional Studies , Humans , Infant , Kyrgyzstan/epidemiology , Seroepidemiologic Studies
9.
PLoS One ; 17(2): e0253638, 2022.
Article in English | MEDLINE | ID: covidwho-1910476

ABSTRACT

Population immunity (herd immunity) to SARS-CoV-2 derives from two sources: vaccinations or cases of infection with the virus. Infections can be diagnosed as COVID-19 and registered, or they can be asymptomatic, oligosymptomatic, or even full-blown but undiagnosed and unregistered when patients recovered at home. Estimation of population immunity to SARS-CoV-2 is difficult and remains a subject of speculations. Here we present a population screening for SARS-CoV-2 specific IgG and IgA antibodies in Polish citizens (N = 501) who had never been positively diagnosed with or vaccinated against SARS-CoV-2. Serum samples were collected in Wroclaw (Lower Silesia) on 15th and 22nd May 2021. Sera from hospitalized COVID-19 patients (N = 22) or from vaccinated citizens (N = 14) served as positive controls. Sera were tested with Microblot-Array COVID-19 IgG and IgA (quantitative) that contain specific SARS-CoV-2 antigens: NCP, RBD, Spike S2, E, ACE2, PLPro protein, and antigens for exclusion cross-reactivity with other coronaviruses: MERS-CoV, SARS-CoV, HCoV 229E Np, HCoV NL63 Np. Within the investigated population of healthy individuals who had never been positively diagnosed with or vaccinated against SARS-CoV-2, we found that 35.5% (178 out of 501) were positive for SARS-CoV-2-specific IgG and 52.3% (262 out of 501) were positive for SARS-CoV-2-specific IgA; 21.2% of the investigated population developed virus-specific IgG or IgA while being asymptomatic. Anti-RBD IgG, which represents virus-neutralizing potential, was found in 25.6% of individuals (128 out of 501). These patients, though positive for anti-SARS-CoV-2 antibodies, cannot be identified in the public health system as convalescents due to undiagnosed infections, and they are considered unaffected by SARS-CoV-2. Their contribution to population immunity against COVID-19 should however be considered in predictions and modeling of the COVID-19 pandemic. Of note, the majority of the investigated population still lacked anti-RBD IgG protection (74.4%); thus vaccination against COVID-19 is still of the most importance for controlling the pandemic.


Subject(s)
Asymptomatic Infections/epidemiology , COVID-19 Vaccines/therapeutic use , COVID-19/epidemiology , COVID-19/immunology , Immunity, Herd , Pandemics/prevention & control , SARS-CoV-2/immunology , Vaccination/methods , Adolescent , Adult , Aged , Antibodies, Viral/blood , Antibodies, Viral/immunology , Antigens, Viral/immunology , COVID-19/blood , COVID-19/prevention & control , Cross Reactions , Female , Humans , Immunoglobulin A/blood , Immunoglobulin A/immunology , Immunoglobulin G/blood , Immunoglobulin G/immunology , Male , Middle Aged , Poland/epidemiology , Treatment Outcome , Young Adult
10.
J Math Biol ; 85(1): 2, 2022 06 30.
Article in English | MEDLINE | ID: covidwho-1906031

ABSTRACT

We study a susceptible-exposed-infected-recovered (SEIR) model considered by Aguas et al. (In: Herd immunity thresholds for SARS-CoV-2 estimated from unfolding epidemics, 2021), Gomes et al. (In: J Theor Biol. 540:111063, 2022) where individuals are assumed to differ in their susceptibility or exposure to infection. Under this heterogeneity assumption, epidemic growth is effectively suppressed when the percentage of the population having acquired immunity surpasses a critical level - the herd immunity threshold - that is lower than in homogeneous populations. We derive explicit formulas to calculate herd immunity thresholds and stable configurations, especially when susceptibility or exposure are gamma distributed, and explore extensions of the model.


Subject(s)
COVID-19 , Epidemics , COVID-19/epidemiology , Humans , Immunity, Herd , Reinfection/epidemiology , SARS-CoV-2
11.
Sci Rep ; 12(1): 598, 2022 01 12.
Article in English | MEDLINE | ID: covidwho-1900532

ABSTRACT

After a year of living with the COVID-19 pandemic and its associated consequences, hope looms on the horizon thanks to vaccines. The question is what percentage of the population needs to be immune to reach herd immunity, that is to avoid future outbreaks. The answer depends on the basic reproductive number, R0, a key epidemiological parameter measuring the transmission capacity of a disease. In addition to the virus itself, R0 also depends on the characteristics of the population and their environment. Additionally, the estimate of R0 depends on the methodology used, the accuracy of data and the generation time distribution. This study aims to reflect on the difficulties surrounding R0 estimation, and provides Spain with a threshold for herd immunity, for which we considered the different combinations of all the factors that affect the R0 of the Spanish population. Estimates of R0 range from 1.39 to 3.10 for the ancestral SARS-CoV-2 variant, with the largest differences produced by the method chosen to estimate R0. With these values, the herd immunity threshold (HIT) ranges from 28.1 to 67.7%, which would have made 70% a realistic upper bound for Spain. However, the imposition of the delta variant (B.1.617.2 lineage) in late summer 2021 may have expanded the range of R0 to 4.02-8.96 and pushed the upper bound of the HIT to 90%.


Subject(s)
COVID-19/immunology , Immunity, Herd , Data Interpretation, Statistical , Differential Threshold , Humans , Models, Biological , Spain
12.
Arch Virol ; 167(9): 1773-1783, 2022 Sep.
Article in English | MEDLINE | ID: covidwho-1899184

ABSTRACT

Currently, health authorities around the world are struggling to limit the spread of COVID-19. Since the beginning of the pandemic, social distancing has been the most important strategy used by most countries to control disease spread by flattening and elongating the epidemic curve. Another strategy, herd immunity, was also applied by some countries through relaxed control measures that allow the free spread of natural infection to build up solid immunity within the population. In 2021, COVID-19 vaccination was introduced with tremendous effort as a promising strategy for limiting the spread of disease. Therefore, in this review, we present the current knowledge about social distancing, herd immunity strategies, and aspects of their implementation to control the COVID-19 pandemic in the presence of the newly developed vaccines. Finally, we suggest a short-term option for controlling the pandemic during vaccine application.


Subject(s)
COVID-19 , COVID-19/prevention & control , COVID-19 Vaccines , Humans , Immunity, Herd , Pandemics/prevention & control , Physical Distancing , SARS-CoV-2
13.
Infect Disord Drug Targets ; 22(4): e170122200309, 2022.
Article in English | MEDLINE | ID: covidwho-1892489

ABSTRACT

Herd immunity can only be achieved when it is extended to the population level. When a sufficiently significant fraction of immune individuals exists in a group, it confers indirect protection from infection to vulnerable individuals. This population-level effect is frequently considered in the context of vaccination programs, which attempt to build herd immunity so that people who cannot be vaccinated, such as the very young or those with impaired immune systems, are nonetheless protected from disease. Clinical signs are at times poor predictor of transmissibility for some infections, such as COVID-19, because asymptomatic hosts can be extremely infectious and contribute to the spread of the virus. COVID-19 is a quickly evolving issue that has been widely spread throughout the world. This article elaborates the idea and goal of herd immunity, the necessary conditions for realizing herd immunity, the restrictive requirements for applying herd immunity, and the obstacles experienced in achieving herd immunity in the context of COVID-19. This mini-article explains the concept and purpose of herd immunization in the context of COVID-19.


Subject(s)
COVID-19 , Immunity, Herd , COVID-19/prevention & control , Humans , Immunization Programs , SARS-CoV-2 , Vaccination
15.
J R Soc Med ; 115(6): 239-240, 2022 06.
Article in English | MEDLINE | ID: covidwho-1865224
16.
Epidemics ; 39: 100581, 2022 06.
Article in English | MEDLINE | ID: covidwho-1851044

ABSTRACT

We present a country specific method to calculate the COVID-19 vaccination coverage needed for herd immunity by considering age structure, age group-specific contact patterns, relative infectivity and susceptibility of children to adults, vaccination effectiveness and seroprevalence prior to vaccination. We find that across all six countries, vaccination of adults age 60 and above has little impact on Reff and this is could be due to the smaller number of contacts between this age group and the rest of the population according to the contact matrices used. If R0 is above 6, herd immunity by vaccine alone is unattainable for most countries either if vaccination is only available for adults or that vaccine effectiveness is lower at 65% against symptomatic infections. In this situation, additional control measures, booster shots, if they improve protection against infection, or the extension of vaccination to children, are required. For a highly transmissible variant with R0 up to 8, herd immunity is possible for all countries and for all four scenarios of varying relative infectivity and susceptibility of children compared to adults, if vaccine effectiveness is very high at 95% against symptomatic infections and that high vaccination coverage is achieved for both adults and children. In addition, we show that the effective reproduction number will vary between countries even if the same proportion of the population is vaccinated, depending on the demographics, the contact rates and the previous pre-vaccination seroprevalence in the country. This therefore means that care must be taken in extrapolating population level impacts of certain vaccine coverages from one country to another.


Subject(s)
COVID-19 , Immunity, Herd , Adult , COVID-19/epidemiology , COVID-19/prevention & control , COVID-19 Vaccines , Child , Humans , Middle Aged , Seroepidemiologic Studies , Vaccination/methods , Vaccination Coverage
17.
PLoS One ; 17(5): e0267840, 2022.
Article in English | MEDLINE | ID: covidwho-1843583

ABSTRACT

We introduce a novel compartmental model accounting for the effects of vaccine efficacy, deployment rates and timing of initiation of deployment. We simulate different scenarios and initial conditions, and we find that higher abundancy and rate of deployment of low efficacy vaccines lowers the cumulative number of deaths in comparison to slower deployment of high efficacy vaccines. We also forecast that, at the same daily deployment rate, the earlier introduction of vaccination schemes with lower efficacy would also lower the number of deaths with respect to a delayed introduction of high efficacy vaccines, which can however, still achieve lower numbers of infections and better herd immunity.


Subject(s)
Vaccination , Vaccines , Immunity, Herd
18.
Appl Health Econ Health Policy ; 20(3): 395-404, 2022 05.
Article in English | MEDLINE | ID: covidwho-1803217

ABSTRACT

BACKGROUND: Herd immunity (HI) is a key benefit of vaccination programs, but the effects are not routinely included in cost-effectiveness analyses (CEAs). OBJECTIVE: This study investigated how the inclusion of HI in CEAs may influence the reported value of immunizations in low- and middle-income countries (LMICs) and illustrated the implications for COVID-19 immunization. METHODS: We reviewed immunization CEAs published from 2000 to 2018 focusing on LMICs using data from the Tufts Medical Center CEA Registries. We investigated the proportion of studies that included HI, the methods used, and the incremental cost-effectiveness ratios (ICERs) reported. When possible, we evaluated how ICERs would change with and without HI. RESULTS: Among the 243 immunization CEAs meeting inclusion criteria, 44 studies (18%) included HI. Of those studies, 11 (25%) used dynamic transmission models, whereas the remainder used static models. Sixteen studies allowed for ICER calculations with and without HI (n = 48 ratios). The inclusion of HI always resulted in more favorable ratios. In 20 cases (42%), adding HI decreased the ICERs enough to cross at least one or more common cost-effectiveness benchmarks for LMICs. Among pneumococcal vaccination studies, including HI in the analyses decreased seven of 24 ICERs enough to cross at least one cost-effectiveness benchmark. CONCLUSION: The full value of immunization may be underestimated without considering a scenario in which HI is achieved. Given the evidence in pneumococcal CEAs, COVID-19 vaccine value assessments should aim to show ICERs with and without HI to inform decision-making in LMICs.


Subject(s)
COVID-19 , Developing Countries , COVID-19/prevention & control , COVID-19 Vaccines , Cost-Benefit Analysis , Humans , Immunity, Herd
19.
Nat Rev Immunol ; 22(6): 333-334, 2022 06.
Article in English | MEDLINE | ID: covidwho-1799586
20.
PLoS One ; 17(2): e0263155, 2022.
Article in English | MEDLINE | ID: covidwho-1793533

ABSTRACT

With limited availability of vaccines, an efficient use of the limited supply of vaccines in order to achieve herd immunity will be an important tool to combat the wide-spread prevalence of COVID-19. Here, we compare a selection of strategies for vaccine distribution, including a novel targeted vaccination approach (EHR) that provides a noticeable increase in vaccine impact on disease spread compared to age-prioritized and random selection vaccination schemes. Using high-fidelity individual-based computer simulations with Oslo, Norway as an example, we find that for a community reproductive number in a setting where the base pre-vaccination reproduction number R = 2.1 without population immunity, the EHR method reaches herd immunity at 48% of the population vaccinated with 90% efficiency, whereas the common age-prioritized approach needs 89%, and a population-wide random selection approach requires 61%. We find that age-based strategies have a substantially weaker impact on epidemic spread and struggle to achieve herd immunity under the majority of conditions. Furthermore, the vaccination of minors is essential to achieving herd immunity, even for ideal vaccines providing 100% protection.


Subject(s)
COVID-19 Vaccines/supply & distribution , COVID-19/prevention & control , COVID-19/genetics , COVID-19/immunology , COVID-19 Vaccines/administration & dosage , COVID-19 Vaccines/pharmacology , Epidemics , Humans , Immunity, Herd/immunology , Models, Theoretical , SARS-CoV-2/immunology , SARS-CoV-2/pathogenicity , Vaccination , Vaccines
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