  As shown recently by the author, constant population SI(R)S models map to
Hethcote's classic endemic model originally proposed in 1973. This unifies a
whole class of models with up to 10 parameters, all being isomorphic to a
simple 2-parameter master model for endemic bifurcation. In this work this
procedure is extended to a 14-parameter SSISS Model, including social behavior
parameters, a (diminished) susceptibility of the R-compartment and unbalanced
constant per capita birth and death rates, thus covering many prominent models
in the literature. Under mild conditions, in the dynamics for fractional
variables in this model all vital parameters become redundant at the cost of
possibly negative incidence rates. There is a symmetry group G acting on
parameter space A, such that systems with G-equivalent parameters are
isomorphic and map to the same normalized system. Using (Xrep,I) as canonical
coordinates, Xrep the replacement number, normalization reduces to parameter
space A/G with 5 parameters only. This approach reveals unexpected relations
between various models in the literature. Part two of this work will analyze
equilibria, stability and backward bifurcation and part three will further
reduce the number of essential parameters from 5 to 3.
