We model and calculate the fraction of infected population necessary to reach herd immunity, taking into account the heterogeneity in infectiousness and susceptibility, as well as the correlation between those two parameters. We show that these cause the effective reproduction number to decrease more rapidly, and consequently have a drastic effect on the estimate of the necessary percentage of the population that has to contract the disease for herd immunity to be reached. We quantify the difference between the size of the infected population when the effective reproduction number decreases below 1 vs the ultimate fraction of population that had contracted the disease. This sheds light on an important distinction between herd immunity and the end of the disease and highlights the importance of limiting the spread of the disease even if we plan to naturally reach herd immunity. We analyze the effect of various lock-down scenarios on the resulting final fraction of infected population. We discuss implications to COVID-19 and other pandemics and compare our theoretical results to population-based simulations. We consider the dependence of the disease spread on the architecture of the infectiousness graph and analyze different graph architectures and the limitations of the graph models.

考虑到传染性和易感性的异质性，以及这两个参数之间的相关性，我们对获得群体免疫力所必需的感染人群比例进行建模和计算。我们表明，这些因素导致有效繁殖数量下降得更快，因此对估计必须感染该疾病以达到群免疫的人口的必要百分比的估计产生了巨大影响。当有效繁殖数量减少到1以下时，我们量化了感染人群的大小与感染该疾病的最终人群之间的差异。这阐明了畜群免疫力与疾病终结之间的重要区别，并强调了即使我们计划自然实现畜群免疫力，也必须限制疾病传播。我们分析了各种锁定方案对最终感染人群的影响。我们讨论了对COVID-19和其他流行病的影响，并将我们的理论结果与基于人口的模拟进行了比较。我们考虑疾病传播对传染性图结构的依赖性，并分析不同的图结构和图模型的局限性。我们讨论了对COVID-19和其他流行病的影响，并将我们的理论结果与基于人口的模拟进行了比较。我们考虑疾病传播对传染性图结构的依赖性，并分析不同的图结构和图模型的局限性。我们讨论了对COVID-19和其他流行病的影响，并将我们的理论结果与基于人口的模拟进行了比较。我们考虑疾病传播对传染性图结构的依赖性，并分析不同的图结构和图模型的局限性。