On the Redundancy of Birth and Death Rates in Homogeneous Epidemic SIR Models

Abstract

The dynamics of fractional population sizes 𝑦𝑖=𝑌𝑖/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=𝑉𝑖(𝑌) naturally projects to a vector field 𝐹𝑖(𝑌) tangent to the leaves of constant population N. A universal formula for the projected field 𝐹𝑖 is given. In this way, in many SIR-type models with standard incidence, all demographic parameters become redundant for the dynamical system 𝑦˙𝑖=𝐹𝑖(𝑦). They may be put to zero by shifting the 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 have been covered already by papers 20 years ago.