  Recently, [8] has proposed that heterogeneity of infectiousness (and
susceptibility) across individuals in infectious diseases, plays a major role
in affecting the Herd Immunity Threshold (HIT). Such heterogeneity has been
observed in COVID-19 and is recognized as overdispersion (or
"super-spreading"). The model of [8] suggests that super-spreaders contribute
significantly to the effective reproduction factor, R, and that they are likely
to get infected and immune early in the process. Consequently, under R_0 = 3
(attributed to COVID-19), the Herd Immunity Threshold (HIT) is as low as 5%, in
contrast to 67% according to the traditional models [1, 2, 4, 10]. This work
follows up on [8] and proposes that heterogeneity of infectiousness
(susceptibility) has two "faces" whose mix affects dramatically the HIT: (1)
Personal-Trait-, and (2) Event-Based- Infectiousness (Susceptibility). The
former is a personal trait of specific individuals (super-spreaders) and is
nullified once those individuals are immune (as in [8]). The latter is
event-based (e.g cultural super-spreading events) and remains effective
throughout the process, even after the super-spreaders immune. We extend [8]'s
model to account for these two factors, analyze it and conclude that the HIT is
very sensitive to the mix between (1) and (2), and under R_0 = 3 it can vary
between 5% and 67%. Preliminary data from COVID-19 suggests that herd immunity
is not reached at 5%. We address operational aspects and analyze the effects of
lockdown strategies on the spread of a disease. We find that herd immunity (and
HIT) is very sensitive to the lockdown type. While some lockdowns affect
positively the disease blocking and increase herd immunity, others have adverse
effects and reduce the herd immunity.
