  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.
