  We generalize the Susceptible-Infected-Removed model for epidemics to take
into account generic effects of heterogeneity in the degree of susceptibility
to infection in the population. We introduce a single new parameter
corresponding to a power-law exponent of the susceptibility distribution that
characterizes the population heterogeneity. We show that our generalized model
is as simple as the original model which is contained as a limiting case.
Because of this simplicity, numerical solutions can be generated easily and key
properties of the epidemic wave can still be obtained exactly. In particular,
we present exact expressions for the herd immunity level, the final size of the
epidemic, as well as for the shape of the wave and for observables that can be
quantified during an epidemic. We find that in strongly heterogeneous
populations the epidemic reaches only a small fraction of the population. This
implies that the herd immunity level can be much lower than in commonly used
models with homogeneous populations. Using our model to analyze data for the
SARS-CoV-2 epidemic in Germany shows that the reported time course is
consistent with several scenarios characterized by different levels of
immunity. These scenarios differ in population heterogeneity and in the time
course of the infection rate, for example due to mitigation efforts or
seasonality. Our analysis reveals that quantifying the effects of mitigation
requires knowledge on the degree of heterogeneity in the population. Our work
shows that key effects of population heterogeneity can be captured without
increasing the complexity of the model. We show that information about
population heterogeneity will be key to understand how far an epidemic has
progressed and what can be expected for its future course.
