  The purpose of this paper is to analyze the mechanism for the interplay of
deterministic and stochastic models for contagious diseases. Deterministic
models for contagious diseases are prone to predict global stability. Small
natural birth and death rates in comparison to disease parameters like the
contact rate and the removal rate ensures that the globally stable endemic
equilibrium corresponds to a tiny average 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 predict, however, oscillatory patterns.
Small and 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 holds provided the stochastic oscillations are moderate enough to keep the
proportion of susceptible individuals near a deterministic equilibrium.
  We close our paper with a discussion of the herd-immunity concept and stress
its close relation to vaccination-programs.
