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.

疫苗接种人群中与未接种人群混合对疾病动态的影响：对SARS-CoV-2的启示
摘要
背景：在SARS-CoV-2大流行期间，疫苗的研发速度是一项非凡的成就。然而，反疫苗运动和虚假信息的传播导致了疫苗接种率的低下。疫苗的反对者常常以未接种人群的权利为由来表达他们的反对立场。我们的目标是探讨疫苗接种人群与未接种人群混合对接种人群风险的影响。
方法：我们构建了一个简单的易感-感染-康复（SIR）模型，该模型描述了一种呼吸系统传染病的两个相互连接的亚人群：已接种疫苗的人群和未接种疫苗的人群（见图1）。我们采用矩阵方法模拟这两组人群的非随机混合，并通过改变混合常数来模拟从随机混合到完全同质性的一系列模式。我们评估了每个亚组和整个群体内的流行病动力学，以及未接种疫苗人群在疫情期间对接种人群感染风险的调整接触频率的贡献。
结果：正如预期的那样，未接种疫苗人群的感染相对风险明显高于已接种疫苗人群。然而，在疫情期间，未接种疫苗人群对感染风险的调整接触频率的贡献不成比例，未接种疫苗人群对接种人群感染风险的贡献率高达基于接触数量预期的6.4倍。随着同质性的增加，接种人群的最终感染率降低，但来自与未接种疫苗人群接触的接种人群风险贡献率增加。
解释：尽管在恶性大流行期间避免接种疫苗的风险主要影响未接种人群，但这些个体的选择可能会以与未接种人群在人口中所占比例不成比例的方式影响接种人群的健康和安全。