On the redundancy of birth and death rates in homogenous epidemic SIR models

Abstract

The dynamics of fractional population sizes y_i=Y_i/N in homogeneous compartment models with time dependent total population N is analyzed. Assuming constant per capita birth and death rates the vector field Y_i'=V_i(Y) naturally projects to a vector field F_i(Y) tangent to the leaves of constant population N. A universal formula for the projected field F_i is given. In this way, in many SIR-type models with standard incidence all demographic parameters become redundant for the dynamical system y_i'=F_i(y). They may be put to zero by shifting remaining parameters appropriately. Normalizing eight examples from the literature this way, they unexpectedly become isomorphic for corresponding parameter ranges. Thus, some recently published results turn out to be already covered by papers 20 years ago.