  Identifiability describes the possibility of determining the values of the
unknown parameters that characterize a dynamic system from the knowledge of its
inputs and outputs. This paper finds the general analytical condition that
fully characterizes this property. The condition can be applied to any system,
regardless of its complexity and type of nonlinearity. In the presence of time
varying parameters, it is only required that their time dependence be
analytical. In addition, its implementation requires no inventiveness from the
user as it simply needs to follow the steps of a systematic procedure that only
requires to perform the calculation of derivatives and matrix ranks. Time
varying parameters are treated as unknown inputs and their identifiability is
based on the very recent analytical solution of the unknown input observability
problem. Finally, when a parameter is unidentifiable, the paper also provides
an analytical method to determine infinitely many values for this parameter
that are indistinguishable from its true value. The condition is used to study
the identifiability of two nonlinear models in the field of viral dynamics (HIV
and Covid-19). In particular, regarding the former, a very popular HIV ODE
model is investigated, and the condition allows us to automatically find a new
fundamental result that highlights a serious error in the current state of the
art.
