  The purpose of this paper is to analyze the interplay of deterministic and
stochastic models for epidemic diseases. Deterministic models for epidemic
diseases are prone to predict global stability. If the natural birth and death
rates are assumed small in comparison to disease parameters like the contact
rate and the recovery rate, then the globally stable endemic equilibrium
corresponds to a tiny proportion of infected individuals. Asymptotic
equilibrium levels corresponding to low numbers of individuals invalidate the
deterministic results.
  Diffusion effects force frequency functions of the stochastic model to
possess similar stability properties as the deterministic model. Particular
simulations of the stochastic model are, however, oscillatory and predict
oscillatory patterns. Smaller or isolated populations show longer periods, more
violent oscillations, and larger probabilities of extinction.
  We prove that evolution maximizes the infectiousness of the disease as
measured by the ability to increase the proportion of infected individuals.
This hold provided the stochastic oscillations are moderate enough to remain in
the vicinity of the deterministic equilibrium.
  We close our paper with a discussion of the herd-immunity concept and stress
its close relation to vaccination-campaigns.
