Bias Due to Averaging the Logistic and SI Models

Abstract

Modelers have choices in how they approach a problem, with different approaches potentially leading to different outcomes. Sometimes one approach gives a consistently lower (or higher) result than another. The theorem and corollaries in this study show that if the logistic equation or, equivalently, the SI model, are perturbed at time zero by a range of values with mean zero, the resulting trajectories must average to a value below (for logistic and I) or above (for S) the solution with average initial condition. The proof of the theorem shows that this phenomenon is the result of algebraic properties of the nonlinear quadratic term, although we note it can be extended to a larger class of systems. More importantly it shows that the only necessary criterion is that the perturbations average to zero. The source of them and the properties of their distribution does not matter to the result of the theorem but does affect the magnitude of the proven difference.