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Identifiability is a structural property of any ODE model characterized by aset of unknown parameters. It describes the possibility of determining thevalues of these parameters from fusing the observations of the system inputsand outputs. This paper finds the general analytical solution of thisfundamental problem and, based on this, provides a general and automatedanalytical method to determine the identifiability of the unknown parameters.In particular, the method can handle any model, regardless of its complexityand type of non-linearity, and provides the identifiability of the parameterseven when they are time-varying. In addition, it is automatic as it simplyneeds to follow the steps of a systematic procedure that only requires toperform the calculation of derivatives and matrix ranks. Time-varyingparameters are treated as unknown inputs and their identification is based onthe very recent analytical solution of the unknown input observability problem[1, 2]. The method is used to determine the identifiability of the unknowntime-varying parameters that characterize two non-linear models in the field ofviral dynamics (HIV and Covid-19) and a non-linear model that characterizes thegenetic toggle switch. New fundamental properties that characterize thesemodels are determined and discussed in detail through a comparison with thestate-of-the-art results. In particular, regarding the very popular HIV ODEmodel and the genetic toggle switch model, the method automatically finds newimportant results that are in contrast with the results in the currentliterature.

Identifiability of nonlinear ODE Models with Time-Varying Parameters: the General Analytical Solution and Applications in Viral Dynamics