Simple 2-Patch Vaccine Model

Impact of Population Mixing Between a Vaccinated Majority and Unvaccinated Minority on Disease Dynamics: Implications for SARS-CoV-2AbstractBackground: The speed of vaccine development has been a singular achievement during the SARS-CoV-2 pandemic. However, anti-vaccination movements and disinformation efforts have resulted in suboptimal uptake of available vaccines. Vaccine opponents often frame their opposition in terms of the rights of the unvaccinated. Our objective was to explore the impact of mixing of vaccinated and unvaccinated populations on risk among vaccinated individuals.Methods: We constructed a simple Susceptible-Infectious-Recovered (SIR) compartmental model of a respiratory infectious disease with two connected sub-populations: vaccinated individuals and unvaccinated individuals (Figure 1). We modeled the non-random mixing of these two groups using a matrix approach with a mixing constant varied to simulate a spectrum of patterns ranging from random mixing to complete assortativity. We evaluated the dynamics of an epidemic within each subgroup, and in the population as a whole, and also evaluated the contact-frequency-adjusted contribution of unvaccinated individuals to risk among the vaccinated.Results: As expected, the relative risk of infection was markedly higher among unvaccinated individuals than among vaccinated individuals. However, the contact-adjusted contribution of unvaccinated individuals to infection risk during the epidemic was disproportionate with unvaccinated individuals contributing to infection risk among the vaccinated at a rate up to 6.4 times higher than would have been expected based on contact numbers alone in the base case. As assortativity increased the final attack rate decreased among vaccinated individuals, but the contact-adjusted contribution to risk among vaccinated individuals derived from contact with unvaccinated individuals increased.Interpretation: While risk associated with avoiding vaccination during a virulent pandemic accrues chiefly to the unvaccinated, the choices of these individuals are likely to impact the health and safety of vaccinated individuals in a manner disproportionate to the fraction of unvaccinated individuals in the population.

《疫苗接种多数群体与未接种少数群体间的人群混合对疾病传播动力学的影响——对新型冠状病毒（SARS-CoV-2）的启示》

摘要
背景：新型冠状病毒（SARS-CoV-2）大流行期间，疫苗研发速度堪称独一无二的非凡成就。然而，反疫苗运动与虚假信息传播导致现有疫苗的接种覆盖率未达理想水平。反疫苗人士常以未接种群体的权利为说辞，为其反对疫苗接种的立场辩解。本研究旨在探究接种与未接种人群的混合模式对接种个体感染风险的影响。

方法：我们构建了一个针对呼吸道传染病的简易易感-感染-恢复（Susceptible-Infectious-Recovered, SIR）分室模型，将人群划分为两个连通的子群体：接种个体与未接种个体（图1）。我们采用矩阵方法模拟两组间的非随机混合模式，并通过调整混合常数，模拟从随机混合到完全同配性（assortativity）的全谱系混合场景。我们评估了各亚群及整体人群的流行病传播动力学，并量化了未接种个体经接触频率校正后，对接种个体感染风险的贡献程度。

结果：正如预期，未接种个体的相对感染风险显著高于接种个体。但在疫情流行期间，未接种个体经接触校正后的感染风险贡献比例却不成比例：在基准场景中，未接种个体对接种个体的感染风险贡献速率最高可达仅按接触数估算值的6.4倍。随着同配性程度提升，接种个体的最终罹患率有所下降，但未接种个体通过接触对接种个体造成的风险贡献，经接触频率校正后反而有所升高。

阐释：尽管在强致病性大流行期间，规避疫苗接种所带来的健康风险主要由未接种个体自身承担，但这类个体的选择仍可能以远超其在人群中所占比例的方式，对接种个体的健康与安全造成影响。