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

TL;DR

This work delivers a general analytical framework and automated tools for both observability of the state and identifiability of all unknown parameters in nonlinear ODE models with time-varying parameters (treated as unknown inputs). It introduces a systematic procedure (Algorithm AlgoFull) to obtain the observability codistribution and a companion procedure (Algorithm AlgoFullIDE) to assess identifiability, including continuous transformations (Lie-group symmetries) that describe indistinguishable states and inputs. The authors demonstrate the method on three viral-dynamics models (HIV, Covid SEIAR) and a genetic toggle switch, uncovering new identifiability results that contradict prior literature and revealing when external information is required to uniquely determine parameters. The framework is general, automatic, and applicable across domains where ODEs with unknown inputs arise, enabling robust model specification, sensor design, and parameter inference.

Abstract

Identifiability is a structural property of any ODE model characterized by a set of unknown parameters. It describes the possibility of determining the values of these parameters from fusing the observations of the system inputs and outputs. This paper finds the general analytical solution of this fundamental problem and, based on this, provides a general and automated analytical method to determine the identifiability of the unknown parameters. In particular, the method can handle any model, regardless of its complexity and type of non-linearity, and provides the identifiability of the parameters even when they are time-varying. In addition, it is automatic 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 identification is based on the very recent analytical solution of the unknown input observability problem [1, 2]. The method is used to determine the identifiability of the unknown time-varying parameters that characterize two non-linear models in the field of viral dynamics (HIV and Covid-19) and a non-linear model that characterizes the genetic toggle switch. New fundamental properties that characterize these models are determined and discussed in detail through a comparison with the state-of-the-art results. In particular, regarding the very popular HIV ODE model and the genetic toggle switch model, the method automatically finds new important results that are in contrast with the results in the current literature.

Figures (13)

• Figure 1: The vehicle polar coordinates ($\rho,~\phi$), the orientation ($\theta$), and the angle measured by the camera ($\beta$).
• Figure 2: Several indistinguishable profiles for the time-varying parameter $\eta'(t,~\tau)$ for several values of the parameter $\tau$ ranging from $-3$ up to $\tau_*\cong2.2532$. The dashed purple line is the profile given by (\ref{['EquationHIVEtaDataSet']}), i.e., $\eta'(t,~\tau)$ for $\tau=0$.
• Figure 3: The profiles $T_U'(t,~\tau)$ for several values of the parameter $\tau$ ranging from $-3$ up to $\tau_*\cong2.2532$. The dashed purple line is the profile for $\tau=0$.
• Figure 4: The profiles $T_I'(t,~\tau)$ for several values of the parameter $\tau$ ranging from $-3$ up to $\tau_*\cong2.2532$. The dashed purple line is the profile for $\tau=0$.
• Figure 5: The two outputs $y_1(t)=V$ (solid blue line) and $y_2(t)=T_U+T_I$ (dashed red line). The profiles are independent of the parameter $\tau$, as expected.

Theorems & Definitions (9)

• Definition 2.1: Unknown input reconstructability matrix
• Definition 2.2: Unknown input degree of reconstructability
• Definition 3.1: Highest UI Degree of Reconstructability
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Remark 4.1
• Remark 4.2
• Remark 4.3