Wewere interested in the change of the qualitative behaviour of a parameterized system of non-linear di#erential equations as they occur in epidemiology, which are given by rational functions of the parameters. At a certain point of the computation, questions of the following form arose: if a certain rational function of the parameters is equal to one, is it true that another rational function is less than zero for all parameters? We had di#culties to solve this question by the use of commercial algebra systems. However this problem can be formulated as a #rst order formula over the reals and can thus be solved by a quanti#er elimination method. Wehave used the implementation of H. Hong and for three parameters the necessary computation time on a Sparc 2 was about one minute. The method was not only valuable for deciding the formula, but in several cases the returned equivalent quanti#er-free formula gave additional information. 1. Introduction Wewere interested in the change of the q..