In this paper, an approach for analysing the structural indistinguishability between two uncontrolled (or autonomous) analytic systems is presented. The approach involves constructing, if possible, a smooth mapping between the trajectories of two candidate models. If either of the models satisfies an observability criterion, then such a transformation always exists when the models are indistinguishable from their outputs. The approach is illustrated by examples from epidemiology and chemical reaction kinetics. One important outcome is that the susceptible, infectious, recovered (SIR) and SIR with temporary immunity (SIRS) models are shown to be indistinguishable when a proportion of the number of infectives is measured. (C) 2004 Elsevier Ltd. All rights reserved.