We treated immunity after vaccination as an all-or-none phenomenon, with a fraction of vaccinated people (as defined by vaccine effectiveness) entering the model in the immune state and the remainder being left in the susceptible state. We also assumed that some fraction of the unvaccinated population had immunity at baseline owing to previous infection and that a fraction of the population was vaccinated. After an infectious period, infectious people with SARS-CoV-2 recover with immunity. In the context of an airborne virus like SARS-CoV-2, 14 – 20 effective contact may be conceptualized as “sharing air” with an infective case. Susceptible people move into the infectious compartment after effective contacts (i.e., contacts of a nature and duration sufficient to permit transmission) with people who are infected. We divided the compartments to reflect 2 connected subpopulations: vaccinated and unvaccinated people. 26 The model is described in Appendix 1 (available at People are represented as residing in 3 possible “compartments:” susceptible to infection (S), infected and infectious (I), and recovered from infection with immunity (R). We constructed a simple compartmental model of a respiratory viral disease. We sought to contrast contribution to epidemic size and risk estimates by subpopulation, and to understand the impact of mixing between vaccinated and unvaccinated groups on expected disease dynamics. 13, 24, 25 To better understand the implications of the interplay between vaccinated and unvaccinated populations under different assumptions about population mixing, we constructed a simple susceptible–infectious–recovered model to reproduce the dynamics of interactions between vaccinated and unvaccinated subpopulations in a predominantly vaccinated population. Simple mathematical models can often provide important insights into the behaviour of complex communicable diseases systems. This is true of communicable infectious diseases but also applies to public health statutes that limit indoor cigarette smoking 21 and legal restrictions on driving under the influence of alcohol and other intoxicants. Historically, behaviours that create health risks for the community as well as individuals have been the subject of public health regulation. Furthermore, the airborne spread of SARS-CoV-2 14 – 20 means that close-range physical mixing of people from vaccinated and unvaccinated groups is not necessary for between-group disease transmission. Although assortative (like-with-like) mixing 13 is characteristic of many communicable disease systems and may be expected to limit interaction between vaccinated and unvaccinated subpopulations to some degree, the normal functioning of society means that complete like-with-like mixing is not observed in reality. Nonvaccination is expected to result in amplification of disease transmission in unvaccinated subpopulations, but the communicable nature of infectious diseases means that this also heightens risk for vaccinated populations, when vaccines confer imperfect immunity. 8 – 10 Although the decision not to receive vaccination is often framed in terms of the rights of individuals to opt out, 11, 12 such arguments neglect the potential harms to the wider community that derive from poor vaccine uptake. However, antivaccine sentiment, fuelled in part by organized disinformation efforts, has resulted in suboptimal uptake of readily available vaccines in many countries, with adverse health and economic consequences. The emergence of the immune-evasive Omicron variant may undermine some of these gains, although provision of booster vaccine doses may restore vaccination to a high level of potency, and vaccines developed specifically to enhance immunity to the Omicron variant may emerge in 2022. 1 While the ability to vaccinate to herd immunity has been held back by the increasing transmissibility of novel SARS-CoV-2 variants of concern (e.g., Delta and Omicron variants), 2, 3 and global distribution of vaccines is inequitable, 4 the effectiveness of SARS-CoV-2 vaccines in reducing severity of disease and disrupting onward transmission even when breakthrough infections occur is likely to have saved many lives. The remarkable speed of vaccine development, production and administration during the COVID-19 pandemic is a singular human achievement.
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