  In this paper, we focus on the influence of heterogeneity and stochasticity
of the population on the dynamical structure of a basic
susceptible-infected-susceptible (SIS) model. First we prove that, upon a
suitable mathematical reformulation of the basic reproduction number, the
homogeneous system and the heterogeneous system exhibit a completely analogous
global behaviour. Then we consider noise terms to incorporate the fluctuation
effects and the random import of the disease into the population and analyse
the influence of heterogeneity on warning signs for critical transitions (or
tipping points). This theory shows that one may be able to anticipate whether a
bifurcation point is close before it happens. We use numerical simulations of a
stochastic fast-slow heterogeneous population SIS model and show various
aspects of heterogeneity have crucial influences on the scaling laws that are
used as early-warning signs for the homogeneous system. Thus, although the
basic structural qualitative dynamical properties are the same for both
systems, the quantitative features for epidemic prediction are expected to
change and care has to be taken to interpret potential warning signs for
disease outbreaks correctly.
