Local identifiability of nonlinear ODE models with time-varying parameters: The general algebraic condition

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 algebraic condition that characterizes this property locally. It is the first-ever necessary and sufficient condition that can be successfully applied in the presence of time-varying parameters. The only requirement is that the time dependence of the parameters is analytical. The use of this condition requires no inventiveness from the user as it simply needs to follow the steps of a systematic procedure that only requires performing the calculation of derivatives and matrix ranks. Time-varying parameters are treated as unknown inputs and the condition is constructed from the recent analytical solution of the unknown input observability problem [1], [2]. The condition is used to study two well-known nonlinear biological models. The former characterizes the dynamics of the HIV, the latter is commonly used to characterize the genetic toggle switch. In both cases, the condition allows us to obtain new results that contradict the current state of the art. The paper also analyzes these discrepancies, showing that they arise from intrinsic difficulties faced by existing methods when applied to identifiability problems involving time-varying parameters.