  Symmetry concepts in parametrized dynamical systems may reduce the number of
external parameters by a suitable normalization prescription. If, under the
action of a symmetry group G, parameter space A becomes a (locally) trivial
principal bundle, A ~ A/G x G, then the normalized dynamics only depends on the
quotient A/G. In this way, the dynamics of fractional variables in homogeneous
epidemic SI(R)S models, with standard incidence, absence of R-susceptibility
and compartment independent birth and death rates, turns out to be isomorphic
to (a marginally extended version of) Hethcote's classic endemic model, first
presented in 1973. The paper studies a 10-parameter master model with constant
and I-linear vaccination rates, vertical transmission and a vaccination rate
for susceptible newborns. As recently shown by the author, all demographic
parameters are redundant. After adjusting time scale, the remaining 5-parameter
model admits a 3-dimensional abelian scaling symmetry. By normalization we end
up with Hethcote's extended 2-parameter model. Thus, in view of symmetry
concepts, reproving theorems on endemic bifurcation and stability in such
models becomes needless.
