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# Detection of a serious error in the paper: "On identifiability of nonlinear ODE models and applications in viral dynamics"

Agostino Martinelli

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## To cite this version:

Agostino Martinelli. Detection of a serious error in the paper: "On identifiability of nonlinear ODE models and applications in viral dynamics". 2025. hal-04419008v2

HAL Id: hal-04419008

https://inria.hal.science/hal-04419008v2

Preprint submitted on 20 Jul 2025

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# Detection of a serious error in the paper: "On identifiability of nonlinear ODE models and applications in viral dynamics"

Agostino Martinelli*

July 20, 2025

## Contents

<table><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>2</td><td>Non uniqueness of δ, N, and η</td><td>2</td></tr><tr><td>3</td><td>The error in Section 6.2 of [1]</td><td>3</td></tr><tr><td>3.1</td><td>Two typos in Equation (6.23) in [1]</td><td>4</td></tr><tr><td>3.2</td><td>The serious error made by the authors of [1]</td><td>4</td></tr><tr><td>4</td><td>Simplified model</td><td>5</td></tr><tr><td>5</td><td>Feedback from the review process at SIREV</td><td>6</td></tr><tr><td>5.1</td><td>Report of the first reviewer</td><td>6</td></tr><tr><td>5.2</td><td>Report of the second reviewer (most likely one of the authors of [1])</td><td>7</td></tr><tr><td>5.3</td><td>Decision from the editor of the Survey and Review section of SIREV</td><td>7</td></tr><tr><td>5.4</td><td>Discussion</td><td>8</td></tr><tr><td>6</td><td>Conclusion</td><td>9</td></tr><tr><td>A</td><td>Technical details for the derivation of Eq. (5)</td><td>10</td></tr><tr><td>B</td><td>Script MATLAB that performs the computation in Section 3.2</td><td>12</td></tr></table>

## Abstract

This note highlights a critical error in a 2011 paper published in SIAM Review (SIREV), specifically in Section 6.2 of [1]. Addressing this issue is important for two reasons: (i) [1] is one of the most cited contributions in the field of identifiability for viral dynamics models, and (ii) due to the error, a widely used model for HIV dynamics has been incorrectly classified as identifiable. In contrast, we show that three of its parameters are, in fact, not identifiable - even locally. We demonstrate this non-identifiability by explicitly constructing infinitely many distinct parameter sets that yield indistinguishable model outputs. The non-uniqueness is local, not just global. We then analyze the reasoning in [1] and identify the specific methodological oversight that led to the incorrect conclusion of full identifiability.

<footnote>*A. Martinelli is with INRIA Rhone Alpes, Montbonnot, France e-mail: agostino.martinelli@inria.fr</footnote>
An earlier version of this note was submitted to SIAM Review as an erratum, but it was not accepted. A preliminary version was subsequently made available on arXiv [5]. This revised version expands upon that preprint, adding clarifying elements to improve transparency and reproducibility. First, we provide MATLAB scripts that reproduce all the symbolic computations involved in identifying the error, to enable independent verification and ensure full reproducibility. Second, we present a simplified version of the biological model in which a structurally analogous error occurs in a significant simpler context. Each step of the original analysis is mapped to this simplified setting, clarifying the core of the mistake and making the argument more accessible. Third, to document the review process, we include the referee reports and editorial correspondence from the SIREV submission. This material is included to illustrate how the error was interpreted during the review process, and to foster a broader discussion on the challenges of evaluating errata in peer-reviewed settings - particularly when the technical issue is conceptually simple but not widely understood.

Overall, this revised version is intended as a constructive contribution - not only to correct a specific technical error in the literature, but also to encourage reflection on how the publication process might better support transparency and accuracy in similar situations.

# 1 Introduction

Section 6.2 of [1] presents an identifiability analysis of a widely studied model describing HIV/AIDS dynamics. The model involves six parameters ($\lambda, \delta, \eta, \rho, N, c$), and the analysis focuses on the case in which one of these parameters, $\eta$, is time-varying. The model is described by the following equations:

$$ 
\left\{ 
\begin{aligned}
\dot{T}_U &= \lambda - \rho T_U - \eta(t) T_U V \\
\dot{T}_I &= \eta(t) T_U V - \delta T_I \\
\dot{V} &= N \delta T_I - cV \\
y &= [V, T_U + T_I],
\end{aligned}
\right. 
\qquad (1) $$

These correspond to Equations (6.15-6.17) in [1]. We refer the reader to Section 6.2 of [1] for a complete definition of all quantities involved.

However, the conclusion drawn in [1] - namely, that all parameters are locally identifiable when $\eta$ is time-varying - does not hold. A detailed identifiability analysis reveals that only three of the five constant parameters are locally identifiable. The remaining two constant parameters ($\delta$ and $N$), along with the time-varying parameter $\eta$, are not locally identifiable; that is, they cannot be uniquely determined from the model output, even in a local sense.

In Section 2, we demonstrate this non-identifiability by explicitly constructing infinitely many distinct yet observationally equivalent parameter configurations. In Section 3, we examine the reasoning in [1] and identify the specific step that led to the incorrect conclusion. To support independent verification and full reproducibility, we also provide MATLAB scripts that replicate the analysis.

To further illustrate the nature of the mistake, Section 4 introduces a simplified version of the HIV model. In this context, each step of the analysis from [1] is mirrored in a simpler setting, highlighting the structural origin of the error in a more accessible way.

Finally, this note also aims to document the process through which the issue was identified and communicated. An erratum was originally submitted to SIAM Review (SIREV), but it was not accepted for publication. In Section 5, we include the full peer-review feedback received from SIREV, including referee reports and editorial correspondence. This material provides insight into how the issue was assessed during review and may also help inform broader discussions on how errata are evaluated - particularly when involving technical errors that, while conceptually straightforward, are not widely recognized.

# 2 Non uniqueness of $\delta$, $N$, and $\eta$

Let us consider the viral model presented in Section 6.2 of [1]. The constant parameters $\lambda, \delta, \rho, N$, and $c$, the three states $T_U(t), T_I(t), V(t)$, and the time-varying parameter $\eta(t)$ satisfy the three dynamical equations in (1).
Based on these original parameters and states, we define a new set of constant parameters, states, and a time-varying parameter, for any $\tau \in \mathbb{R}$:

$$ \left\{ \begin{array}{lcl} \lambda' & = & \lambda \\ \delta' & = & \delta'(\tau) \\ \rho' & = & \rho \\ N' & = & N'(\tau) \\ c' & = & c, \end{array} \right. \qquad (2) $$

$$ \left\{ \begin{array}{lcl} T'_{U} & = & T'_{U}(t, \tau) = T_{U}(t) + T_{I}(t) - \frac{T_{I}(t)}{\rho} (\delta e^{-\rho\tau} + \rho - \delta) \\ T'_{I} & = & T'_{I}(t, \tau) = \frac{T_{I}(t)}{\rho} (\delta e^{-\rho\tau} + \rho - \delta) \\ V' & = & V'(t) = V(t) \end{array} \right. \qquad (3) $$

$$ \eta' = \eta'(t, \tau) = \frac{\eta(t)T_U(t)V(t)\rho e^{\rho\tau} + [T_I(t)\delta^2 - T_I(t)\delta\rho - \eta(t)T_U(t)V(t)\delta][e^{\rho\tau} - 1]}{V(t)[T_I(t)\delta + T_U(t)\rho]e^{\rho\tau} - V(t)T_I(t)\delta} \qquad (4) $$

We claim that the new constant parameters, states, and time-varying parameter defined above are indistinguishable from the original ones, for any value of $\tau \in \mathbb{R}$.

This can be directly verified by substitution.

First, the outputs $y_1(t) = T_U(t) + T_I(t)$ and $y_2(t) = V(t)$ remain unchanged, since $T'_U(t, \tau) + T'_I(t, \tau) = T_U(t) + T_I(t)$ and $V'(t) = V(t)$.

Second, using (1) together with (2-4), one can verify, by straightforward substitution, that the new states satisfy the following dynamical system (details are provided in Appendix A):

$$ \left\{ \begin{array}{lcl} \frac{d}{dt} T'_{U}(t, \tau) & = & \lambda' - \rho' T'_{U}(t, \tau) - \eta'(t, \tau) T'_{U}(t, \tau) V'(t) \\ \frac{d}{dt} T'_{I}(t, \tau) & = & \eta'(t, \tau) T'_{I}(t, \tau) V'(t) - \delta' T'_{I}(t, \tau) \\ \frac{d}{dt} V'(t) & = & N' \delta' T'_{I}(t, \tau) - c' V'(t) \end{array} \right. \qquad (5) $$

In other words, the transformed states satisfy the same dynamical system as in (1), but with the transformed parameters.

We conclude that $T'_U, T'_I, V', \eta', \lambda', \delta', \rho', N'$, and $c'$ cannot be distinguished from $T_U, T_I, V, \eta, \lambda, \delta, \rho, N$, and $c$, respectively. Since $\eta' \neq \eta, \delta' \neq \delta$, and $N' \neq N$, we conclude that $\eta, \delta, N$ are not identifiable.

Moreover, since $\tau$ can take arbitrarily small values, and in the limit $\tau \to 0$ we have $T'_U \to T_U, T'_I \to T_I, V' \to V, \eta' \to \eta, \delta' \to \delta$, and $N' \to N$, we conclude that the parameters $\eta, \delta, N$ are not even locally identifiable.

In Chapter 6 of [2], we also presented a numerical validation of this result (see Section 6.6 of [2]), using the dataset introduced in [4]. In Section 6.7 of [2], we further characterized the minimal information that must be added to the observed outputs in order to make all model parameters identifiable (and all states observable).

We wish to emphasize that our claim was easily verified by direct substitution. What is non-trivial is the analytic derivation of the expressions in (2-4). These were obtained in [2] using the analytic solution of a fundamental problem closely related to the identifiability of time-varying parameters: the unknown input observability problem. The general analytic solution to this problem was recently introduced in [3]. The solution presented in [2] builds on the results of [3] and provides a systematic framework for assessing the identifiability of any ODE model, including models with time-varying parameters and arbitrary nonlinearities (not necessarily polynomial, as in the methods discussed in [1]). In contrast to those earlier methods, it is fully algorithmic, following a well-defined sequence of analytic steps. Moreover, in the presence of unidentifiability, it automatically produces the locally indistinguishable states and parameters (as in the present case, the non-trivial expressions in (2-4) for infinitesimal $\tau$).

## 3 The error in Section 6.2 of [1]

The derivation provided in Section 6.2 of [1] is based on an intermediate result that is correct. This result states that the time-varying parameter $\eta(t)$ is identifiable if all the constant parameters are identifiable. The error lies in the derivation that allegedly proves the identifiability of the constant parameters.
To explain this significant mistake, we first highlight a minor error that may be a simple typo (Section 3.1). This minor issue was recently acknowledged by the authors of [1] in [6].

Throughout this section, we follow the notation in Section 6.2 of [1], and denote the vector of constant parameters as:

$$ \theta \triangleq [\lambda, \delta, \rho, c, N]^T. \qquad (6) $$

## 3.1 Two typos in Equation (6.23) in [1]

Equation (6.23) was derived from (6.21) and (6.22), which are correct. Based on (6.21)-(6.22), the correct expression is:

$$ \begin{aligned} \ddot{y}_1 y_2 \dot{y}_2 - \dot{y}_1 y_2 \ddot{y}_2 - \delta y_1 y_2 \ddot{y}_2 + \lambda y_2 \ddot{y}_2 - (\delta + c) \dot{y}_1 y_2 \dot{y}_2 + (\delta \rho - \delta^2 - \delta c) y_1 y_2 \dot{y}_2 & \quad (7) \\ & + (\rho + \delta) \dot{y}_1 y_2 \dot{y}_2 + \lambda c y_2 \dot{y}_2 + \rho c \dot{y}_1 y_2^2 + (\rho \delta c - \delta^2 c) y_1 y_2^2 - N \delta y_1 \ddot{y}_1 y_2 \\ & + c \ddot{y}_1 y_2^2 - N \delta (\rho + \delta) y_1 \dot{y}_1 y_2 - N \delta^2 \rho y_1^2 y_2 + N \delta^2 \lambda y_1 y_2 = 0 \end{aligned} $$

Compared to (6.23), the two differences are as follows:

1. The sixth term in (6.23) is incorrectly written as $(\delta\rho + \rho + \delta - \delta^2 - \delta c)y_1y_2\dot{y}_2$, while the correct form is: $(\delta\rho - \delta^2 - \delta c)y_1y_2\dot{y}_2 + (\rho + \delta)\dot{y}_1y_2\dot{y}_2$.

2. The seventh term in (6.23) is given as $cy_2\dot{y}_2$, but it should be $\lambda cy_2\dot{y}_2$.

## 3.2 The serious error made by the authors of [1]

Let us denote the expression in (7) as $\phi = \phi(y_1, \dot{y}_1, \ddot{y}_1, y_2, \dot{y}_2, \ddot{y}_2, \lambda, \delta, \rho, c, N)$. Using this notation, Equation (7) becomes:

$$ \phi = 0. $$

In Section 6.2 of [1], the authors claim the local identifiability of all parameters based on Equation (6.23). Two methods are suggested: (i) applying the third statement of Theorem 3.8 in [1], and (ii) applying the Implicit Function Theorem, following the procedure in Section 3.4 of [1].

We cannot comment on the first method, as the authors provide only the last polynomial (i.e., $B_n(u, y, \theta_1, \dots, \theta_q)$) from Theorem 3.8, omitting the remaining polynomials ($B_1, \dots, B_{n-1}$) needed for its application.

As for the second method, the error can be described as follows. According to the procedure in Section 3.4 of [1], one first computes the first four time derivatives of $\phi$, yielding:

$$ \phi^1 \triangleq \frac{d}{dt}\phi = 0, \quad \phi^2 \triangleq \frac{d^2}{dt^2}\phi = 0, \quad \phi^3 \triangleq \frac{d^3}{dt^3}\phi = 0, \quad \phi^4 \triangleq \frac{d^4}{dt^4}\phi = 0 $$

Next, we construct the vector function $\Phi \triangleq [\phi, \phi^1, \phi^2, \phi^3, \phi^4]^T$. Since the parameters $\lambda, \delta, \rho, c, N$ are constant, $\Phi$ depends on them and on the outputs $(y_1, y_2)$ and their time derivatives up to order 6.

To assess identifiability of $\lambda, \delta, \rho, c, N$, we examine the system:

$$ \Phi = 0 \qquad (8) $$

The Implicit Function Theorem is then applied to the Jacobian of $\Phi$ with respect to $\theta = [\lambda, \delta, \rho, c, N]^T$, yielding the 5 × 5 matrix:

$$ M \triangleq \frac{\partial \Phi}{\partial \theta}. \qquad (9) $$

The serious error committed in [1] is to assume that all quantities involved in $M$ - namely, $\lambda, \delta, \rho, c, N, y_1, y_2$, and the time derivatives of $y_1$ and $y_2$ up to order six - are mutually independent. Under this incorrect assumption, the Jacobian $M$ appears to have full rank (rank 5), leading the authors to conclude that all parameters are locally identifiable.
However, the two output functions $y_1(t)$ and $y_2(t)$, together with their time derivatives, are not free variables - **they are constrained by the model dynamics**. Due to these constraints, all time derivatives of $y_1$ and $y_2$ must be expressed in terms of the model states $(T_U, T_I, V)$, the constant parameters $\lambda, \delta, \rho, c, N, T_U, T_I, V, \eta, \eta^{(1)}, \eta^{(2)}, \eta^{(3)}, \eta^{(4)}, \eta^{(5)}$ (with $\eta^{(k)} \triangleq \frac{d^k \eta}{dt^k}$), the rank of $M$ is strictly less than 5**. In fact, it cannot exceed 4. Consequently, the conclusion that Equation (8) implies local identifiability of all five parameters is incorrect.

The MATLAB script used to verify this rank computation is included in Appendix B.

## 4 Simplified model

We introduce a simplified version of the model defined by (1), characterized by the following dynamics and outputs:

$$ egin{cases} \dot{x}_1 = -w(t) \\ \dot{x}_2 = -p_1 x_2 + w(t) \\ \dot{x}_3 = p_2 x_2 \\ y = [y_1, y_2]^T = [x_3, x_1 + x_2]^T. \end{cases} \quad (10) $$

The state vector is $[x_1, x_2, x_3]$, and the model includes two constant parameters $p_1, p_2$ and one time-varying parameter $w(t)$.

From the outputs, we compute:

$$ \dot{y}_1 = \dot{x}_3 = p_2 x_2, \quad \dot{y}_2 = \dot{x}_1 + \dot{x}_2 = -p_1 x_2. $$

These imply the relation:

$$ p_1 \dot{y}_1 + p_2 \dot{y}_2 = 0. $$

We define the function:

$$ \phi = p_1 \dot{y}_1 + p_2 \dot{y}_2. \quad (11) $$

Note that in the original model the function $\phi$ is more complex (see Eq. (7)), while in the simplified model it takes the simple form in Eq. (11). In both cases, $\phi$ depends on all constant parameters and on measurable quantities (outputs and their time derivatives up to a certain order: 2 for the original model, and 1 for the simplified model).

Following the procedure in Section 3.4 of [1], to identify all parameters, one computes time derivatives of $\phi$ up to a certain order. For the original model with 5 constant parameters, derivatives up to the fourth order are required. For the simplified model with only 2 constant parameters, only one additional derivative suffices:

$$ \phi^{(1)} \triangleq \dot{\phi} = p_1 \ddot{y}_1 + p_2 \ddot{y}_2. \quad (12) $$

Thus, in the simplified model the vector function $\Phi$ is:

$$ \Phi = \begin{bmatrix} \phi \\ \phi^{(1)} \end{bmatrix}. \quad (13) $$

The associated matrix $M$, defined as the Jacobian of $\Phi$ with respect to the constant parameters, is:

$$ M = \begin{bmatrix} \frac{\partial \Phi}{\partial p_1} & \frac{\partial \Phi}{\partial p_2} \\ \end{bmatrix} = \begin{bmatrix} \dot{y}_1 & \dot{y}_2 \\ \ddot{y}_1 & \ddot{y}_2 \end{bmatrix}. \quad (14) $$

If we (incorrectly) assume that $y_1, y_2$ and their time derivatives up to second order can take arbitrary values, the rank of $M$ is generically 2, suggesting local identifiability. This is precisely the error made in Section 6.2 of [1].

However, the outputs and their derivatives cannot be arbitrarily chosen; they must satisfy the dynamical constraints of the system. Accounting for these constraints, we rewrite $M$ using the system equations:

$$ M = \begin{bmatrix} \dot{y}_1 & \dot{y}_2 \\ \ddot{y}_1 & \ddot{y}_2 \end{bmatrix} = \begin{bmatrix} p_2 x_2 & -p_1 x_2 \\ p_2(-p_1 x_2 + w) & -p_1(-p_1 x_2 + w) \end{bmatrix}. \quad (15) $$
Observe that the second column is a scalar multiple of the first (specifically, it equals $-\frac{p_1}{p_2}$ times the first column). Therefore, the rank of $M$ cannot exceed 1, which is strictly less than 2. Hence, we cannot conclude that the model is locally identifiable.

In summary, applying the identifiability criterion while ignoring the internal dynamical constraints leads to a false conclusion. This simplified model makes explicit the logical flaw in the identifiability analysis of [1]. The mistake lies in the assumption that outputs and their time derivatives can be varied independently, whereas they are fundamentally linked by the system dynamics.

# 5 Feedback from the review process at SIREV

This note was initially submitted to the SIAM Review Journal (SIREV) and subsequently rejected. In this section, we present the feedback received during the SIREV review process and offer a critical discussion on the technical competence of both the editor and the two reviewers appointed by the editor.

Sections 5.1 and 5.2 report the detailed comments from the first and second reviewers, respectively. Notably, the second reviewer is identified - based on the content of their review - as one of the authors of [1]<fnref n="1" />. Section 5.3 presents the editor's final decision. Finally, Section 5.4 provides a comprehensive analysis of the review feedback, highlighting significant shortcomings in the technical assessments by the reviewers (particularly the second) as well as inadequacies in the editor's handling of the submission - both in terms of technical understanding and editorial judgment.

## 5.1 Report of the first reviewer

This note proposes a correction of a well-known paper on "Identifiability of nonlinear ODE models and applications in viral dynamics", by Miao et al. (reference [1] from now on). In [1], it was claimed that a model of host HIV/AIDS dynamics containing a time-varying parameter is structurally unidentifiable. The submitted note argues that the opposite is true. This note is relevant because the original paper [1] is one of the most popular and cited references in the topic of structural identifiability analysis, and the HIV model has been used as a case study in later works. Hence it is important to clarify the correctness of those results. The present note attempts to do so by means of three main claims:

(1) That three parameters of the model have infinitely many equivalent solutions. This is argued in Section 1 of the note, by showing a set of parameter transformations, depending on a "tau" parameter, that allegedly yields a solution with dynamics indistinguishable from the original one. (2) That there is an error in equation (6.23) of [1], which may have been the cause of the wrong result. This is shown in Section 2 of the note. (3) That the implicit function Theorem, when applied to the model under study, yields a 5x5 matrix $M$ which has a rank of at most 4 < 5, which indicates non-identifiability.

Claim (2) above is indeed true. However, I have not been able to confirm claims (1) and (3). As a matter of fact, when attempting to replicate the results reported in the present note, I obtained instead results that seem to support the conclusions in [1]: in regard to claim (1), I was unable to recover the original model equations by direct substitution of the transformed parameters; instead I obtained different ones. As for claim (3), when applying the similarity transform method, I obtained a 5x5 matrix with rank 5, which would indicate full identifiability.

While my results contradict those presented in the correction, the correction does provide a convincing rationale and it raises some - at least - intriguing concerns. Therefore, I think that the underlying claims deserve more investigation. To obtain these results I did some calculations by hand, and others on a computer. Although I double-checked them, I might have made a mistake somewhere; likewise, there might be an error in the calculations performed by the author of the submitted note. To avoid any possibility of confusion, it would be very helpful if the author provided some code that performs the necessary calculations, so that it would be easier to spot any differences in methodology as well as possible errors.

<footnote n="1">¹This is apparent from their remarks.</footnote>
To sum up, my suggestion would be that the correction is resubmitted along with further details and code to reproduce the results.

## 5.2 Report of the second reviewer (most likely one of the authors of [1])

We thank Dr. Martinelli for pointing out an error in Eq. (6.23) of Miao et al. (2011). However, there are several misunderstandings in the derivations and arguments in this submission such that the conclusions on parameter identifiability are not justified. We thus do not recommend publication of this submission in SIAM REV, but all the authors of Miao et al. (2011) are willing to work with the SIAM REV editorial office to publish an erratum on Eq. (6.23).

1. We appreciate that Dr. Martinelli pointed out an error in the coefficients of Eq. (6.23). While Eq. (2.1) derived by Dr. Martinelli is correct, it can be further simplified by combining the 5th and the 7th terms of Eq. (2.1) to yield:

$$ \ddot{y}_1 y_2 \dot{y}_2 - \dot{y}_1 y_2 \ddot{y}_2 - \delta y_1 y_2 \ddot{y}_2 + \lambda y_2 \ddot{y}_2 + (\rho - c) \dot{y}_1 y_2 \dot{y}_2 + (\delta \rho - \delta^2 - \delta c) y_1 y_2 \dot{y}_2 \\ + \lambda c y_2 \dot{y}_2 + \rho c \dot{y}_1 y_2^2 + (\rho \delta c - \delta^2 c) y_1 y_2^2 - N \delta y_1 \ddot{y}_1 y_2 + c \ddot{y}_1 y_2^2 \\ - N \delta (\rho + \delta) y_1 \dot{y}_1 y_2 - N \delta^2 \rho y_1^2 y_2 + N \delta^2 \lambda y_1 y_2 = 0. $$

It is easy to show from the revised Eq. (2.1) by the methods in our paper that the model parameters are all identifiable as claimed in our original paper. Specifically, let $\Phi$ denote the left side of Eq. (2.1) and let $\theta_c = (\lambda, \rho, N, \delta, c)^T$ denote the constant parameter vector, we have $\Phi = \Phi(\theta_c, y_1, \dot{y}_1, \ddot{y}_1, y_2, \dot{y}_2, \ddot{y}_2)$ and $\Phi^{(n)} = \frac{d^n}{dt^n} \Phi(y_1, \dot{y}_1, \ddot{y}_1, y_2, \dot{y}_2, \ddot{y}_2)$, where n is a positive integer. The Jacobian matrix we obtained is as follows:

$$ \begin{bmatrix} \frac{\partial \Phi}{\partial \lambda} & \frac{\partial \Phi}{\partial \rho} & \frac{\partial \Phi}{\partial N} & \frac{\partial \Phi}{\partial \delta} & \frac{\partial \Phi}{\partial c} \\ \frac{\partial \Phi^{(1)}}{\partial \lambda} & \frac{\partial \Phi^{(1)}}{\partial \rho} & \frac{\partial \Phi^{(1)}}{\partial N} & \frac{\partial \Phi^{(1)}}{\partial \delta} & \frac{\partial \Phi^{(1)}}{\partial c} \\ \frac{\partial \Phi^{(2)}}{\partial \lambda} & \frac{\partial \Phi^{(2)}}{\partial \rho} & \frac{\partial \Phi^{(2)}}{\partial N} & \frac{\partial \Phi^{(2)}}{\partial \delta} & \frac{\partial \Phi^{(2)}}{\partial c} \\ \frac{\partial \Phi^{(3)}}{\partial \lambda} & \frac{\partial \Phi^{(3)}}{\partial \rho} & \frac{\partial \Phi^{(3)}}{\partial N} & \frac{\partial \Phi^{(3)}}{\partial \delta} & \frac{\partial \Phi^{(3)}}{\partial c} \\ \frac{\partial \Phi^{(4)}}{\partial \lambda} & \frac{\partial \Phi^{(4)}}{\partial \rho} & \frac{\partial \Phi^{(4)}}{\partial N} & \frac{\partial \Phi^{(4)}}{\partial \delta} & \frac{\partial \Phi^{(4)}}{\partial c} \end{bmatrix} $$

Note that $y_1$ and $y_2$ are observed and their arbitrary-order time derivatives ($y_1^{(n)}$ and $y_2^{(n)}$) are deemed as known. Also, according to the persistently exciting input assumption (see Page 5 in Miao et al. 2011), $y_1^{(n)}$ and $y_2^{(n)}$ will not vanish for the purpose of identifiability analysis. That is, $y_1$, $y_2$ and their derivatives up to the sixth order in $\Phi^{(n)}$ are known and they will not vanish (e.g., degenerate to zeros); therefore, $\Phi^{(i)} \neq \Phi^{(j)}$ ($i \neq j$) is expected at certain time points in theory and the rank of the Jacobian matrix can be 5.

2. The derivations in Section 1 showing the model parameters are not all identifiable are not convincing for two reasons:

a. An additional parameter $\tau$ was introduced in Eqns. (1.1)~(1.3) to re-parameterize the original model. In general, it is not recommended to introduce additional (redundant) parameters into a model in the context of identifiability analysis as such practice is likely to change the given model structure and subsequently change the identifiability of existing model parameters.

b. The parameter identifiability problem here is to verify whether $\theta = (\eta, \lambda, \rho, N, \delta, c)^T$ is identifiable given the model structure in Eqns. (6.15)~(6.17) and given the observations $y_1 = T_U + T_I, y_2 = V$. This is different from the problem of verifying whether $(T_U, T_I, V, \eta, \lambda, \rho, N, \delta, c)^T$ is unique given the model structure and the observations. In short, directly including $(T_U, T_I, V)$ in identifiability analysis is not appropriate, and $\theta = (\eta, \lambda, \rho, N, \delta, c)^T$ will not be unique when $\theta$ is locally identifiable.

## 5.3 Decision from the editor of the Survey and Review section of SIREV

Dear Agostino Martinelli,
I am emailing with regard to the manuscript "Correction to" On identifiability of nonlinear ODE models and applications in viral dynamics" submitted to the Survey and Review section of SIAM Review. We have now received sufficient referee feedback to allow me to make an editorial decision. Please find this feedback attached and/or appended below. Based on this material and my own reading, and keeping in mind the high standards of SIAM Review, I must reject your submission.

I will nevertheless ask the authors of the original manuscript to write a correction and acknowledge your hint to the error.

Thank you for considering SIAM Review as an outlet for your work.

Best wishes,

Regards,

Marlis Hochbruck,

Section Editor, SIREV Survey and Review

sirev@math.kit.edu

## 5.4 Discussion

The comments from the editor and the two reviewers bring to light several important issues.

1. The derivation in Section 2 is elementary and clear, yet it seems it was not fully understood by the reviewers. Reviewer 1 mentioned (see Section 5.1): "I was unable to recover the original model equations by direct substitution of the transformed parameters; instead I obtained different ones." These calculations, relying on basic differentiation and substitution, are straightforward and are detailed in Appendix A.

Reviewer 2's misunderstanding of the role of $\tau$ in Eqs (2) and (4) (see Item 2.a in Section 5.2) further illustrates a missed opportunity to appreciate the subtleties of the model. For any value of $\tau$, the transformed parameters represent a set distinct from, yet indistinguishable from, the original ones. This is a fundamental concept in identifiability theory and should be well within the grasp of researchers in this field.

It is regrettable that such foundational aspects were not fully grasped, perhaps indicating a focus more on formalities and credentials than on the deep engagement with the scientific material itself.

2. Both reviewers and the editor correctly spotted the typographical error in Section 3.1, which was acknowledged in the erratum [6]. However, it is important to note that:

(a) Additional typographical errors remain unaddressed in the original paper.

(b) Despite acknowledging the error, the original authors maintained their conclusion of local identifiability, which contradicts the corrected analysis.

3. A critical misunderstanding concerns the assumption that outputs $y_1(t), y_2(t)$ and their time derivatives can vary freely. Both reviewers assumed such freedom (see Item 1 in Section 5.2 and comments in Section 5.1), overlooking the inherent constraints imposed by the system's dynamics. Recognizing these dependencies is essential for rigorous identifiability analysis.

4. Reviewer 1 proposed constructive dialogue and offered to further investigate the matter with the author, even requesting reproducible code: "While my results contradict those presented in the correction, the correction does provide a convincing rationale and it raises some - at least - intriguing concerns... To avoid any possibility of confusion, it would be very helpful if the author provided some code that performs the necessary calculations." Such openness to collaboration and transparency is commendable and represents the ideal spirit of scientific inquiry.

Unfortunately, this constructive engagement was not embraced by the editor, who proceeded with a rejection. This decision may reflect a missed opportunity to prioritize scientific rigor and collegial discussion over procedural finality.
Ultimately, this case serves as a reminder that beyond titles and formal roles, a genuine passion for understanding and advancing the subject matter is indispensable in research. Engaging deeply with the technical details and maintaining openness to dialogue are the cornerstones of scientific progress.

# 6 Conclusion

This note highlighted a very serious error in a paper published by SIAM Review in 2011. The error occurs in Section 6.2 of [1].

This mistake is significant because, due to it, a very popular viral model (arguably the most referenced in the field of HIV dynamics) has been incorrectly classified as fully identifiable, whereas three of its parameters are in fact not identifiable, even locally.

We first demonstrated the non-uniqueness of these three parameters by explicitly constructing infinitely many distinct yet indistinguishable values (Section 2). This non-uniqueness holds even locally. We then analyzed in detail the crucial error made by the authors of [1], which led to their false conclusion of full local identifiability (Section 3.2).

The note was initially submitted to SIAM Review (SIREV) and subsequently rejected, before being published on arXiv [5]. Included here is the feedback from the editor and the two reviewers selected for the manuscript. This feedback unfortunately reveals a concerning lack of technical rigor by both reviewers and editor and exposes a failure in the editorial management of what should have been a straightforward erratum. Such shortcomings not only hinder scientific progress but also risk undermining the integrity and trustworthiness of the peer review process.

It is with a sense of professional disappointment that we reflect on how the focus on titles and reputations can sometimes overshadow a genuine dedication to scientific accuracy and thoroughness - values that must remain at the core of our community.
# A Technical details for the derivation of Eq. (5)

## First equation

Let us prove the validity of the first equation in (5), i.e., the validity of

$$ \frac{d}{dt} T'_{U} = \lambda' - \rho' T'_{U} - \eta' T'_{U} V' $$

We prove the equality of the two members.

### First member

From the first equation in (3), we obtain:

$$ \frac{d}{dt} T'_{U} = \frac{d}{dt} T_{U} + \frac{d}{dt} T_{I} - \frac{\frac{d}{dt} T_{I}}{\rho} (\delta e^{-\rho\tau} + \rho - \delta). $$

By using the first two equations in (1), we obtain:

$$ \begin{aligned} \frac{d}{dt} T'_{U} &= \lambda - \rho T_{U} - \eta T_{U} V + \eta T_{U} V - \delta T_{I} - \frac{\eta T_{U} V - \delta T_{I}}{\rho} (\delta e^{-\rho\tau} + \rho - \delta) = \\ &= \lambda - \rho T_{U} - \delta T_{I} + \frac{\delta T_{I} - \eta T_{U} V}{\rho} (\delta e^{-\rho\tau} + \rho - \delta). \end{aligned} \quad (16) $$

### Second member

The second member is $\lambda' - \rho' T'_{U} - \eta' T'_{U} V'$. By using (2), we obtain:

$$ \lambda' - \rho' T'_{U} - \eta' T'_{U} V' = \lambda - \rho T'_{U} - \eta' T'_{U} V' $$

By using the expression of $T'_{U}$ and $V'$ in (3), and by using the expression of $\eta'$ in (4), after some simplifications, we obtain:

$$ \lambda' - \rho' T'_{U} - \eta' T'_{U} V' = - \frac{T_{I} \delta^{2} - \lambda \rho + T_{U} \rho^{2} - T_{I} \delta^{2} e^{-\rho \tau} - T_{U} V \delta \eta + T_{U} V \eta \rho + T_{U} V \delta \eta e^{-\rho \tau}}{\rho} \quad (17) $$

By a direct computation it is easy to check that the two members (given in (16) and (17), respectively) coincide, i.e., the following equality holds:

$$ \lambda - \rho T_{U} - \delta T_{I} + \frac{\delta T_{I} - \eta T_{U} V}{\rho} (\delta e^{-\rho\tau} + \rho - \delta) = \frac{T_{I} \delta^{2} - \lambda \rho + T_{U} \rho^{2} - T_{I} \delta^{2} e^{-\rho\tau} - T_{U} V \delta \eta + T_{U} V \eta \rho + T_{U} V \delta \eta e^{-\rho\tau}}{\rho} $$

## Second equation

Let us prove the validity of the second equation in (5), i.e., the validity of

$$ \frac{d}{dt} T'_{I} = \eta' T'_{U} V' - \delta' T'_{I} $$

We prove the equality of the two members.

### First member

From the second equation in (3), we obtain:

$$ \frac{d}{dt} T'_{I} = \frac{\frac{d}{dt} T_{I}}{\rho} (\delta e^{-\rho\tau} + \rho - \delta). $$

From the second equation in (1), we obtain:

$$ \frac{d}{dt} T'_{I} = \frac{\eta T_{U} V - \delta T_{I}}{\rho} (\delta e^{-\rho\tau} + \rho - \delta). \quad (18) $$
## Second member

The second member is $\eta' T'_U V' - \delta' T'_I$. By using (2), we obtain:

$$ \eta' T'_U V' - \delta' T'_I = \eta' T'_U V' - \frac{\delta \rho}{(\rho - \delta) e^{\rho \tau} + \delta} T'_I $$

By using the expression of $T'_U$, $T'_I$, and $V'$ in (3), and by using the expression of $\eta'$ in (4), after some simplifications, we obtain:

$$ \eta' T'_U V' - \delta' T'_I = - \frac{e^{-\rho \tau} (T_I \delta - T_U V \eta) (\delta - \delta e^{\rho \tau} + \rho e^{\rho \tau})}{\rho} \quad (19) $$

By a direct computation it is easy to check that the two members (given in (18) and (19), respectively) coincide, i.e., the following equality holds

$$ \frac{\eta T_U V - \delta T_I}{\rho} (\delta e^{-\rho \tau} + \rho - \delta) = - \frac{e^{-\rho \tau} (T_I \delta - T_U V \eta) (\delta - \delta e^{\rho \tau} + \rho e^{\rho \tau})}{\rho} $$

## Third equation

Let us prove the validity of the third equation in (5), i.e., the validity of

$$ \frac{d}{dt} V' = N' \delta' T'_I - c' V' $$

We prove the equality of the two members.

## First member

From the third equation in (3), we obtain:

$$ \frac{d}{dt} V' = \frac{d}{dt} V $$

From the third equation in (1), we obtain:

$$ \frac{d}{dt} V' = N \delta T_I - c V \quad (20) $$

## Second member

The second member is $N' \delta' T'_I - c' V'$. By using (2), we obtain:

$$ N' \delta' T'_I - c' V' = N e^{\rho \tau} \frac{\delta \rho}{(\rho - \delta) e^{\rho \tau} + \delta} T'_I - c V' $$

By using (3), we obtain:

$$ N' \delta' T'_I - c' V' = N e^{\rho \tau} \frac{\delta \rho}{(\rho - \delta) e^{\rho \tau} + \delta} \frac{T_I}{\rho} (\delta e^{-\rho \tau} + \rho - \delta) - c V \quad (21) $$

By a direct computation it is easy to check that the two members (given in (20) and (21), respectively) coincide, i.e., the following equality holds

$$ N \delta T_I - c V = N e^{\rho \tau} \frac{\delta \rho}{(\rho - \delta) e^{\rho \tau} + \delta} \frac{T_I}{\rho} (\delta e^{-\rho \tau} + \rho - \delta) - c V $$
<figure><img src="image_4.png" /><figcaption>Listing 1: Script MATLAB that validates Eq. (5)</figcaption></figure>

B Script MATLAB that performs the computation in Section 3.2

<figure><img src="image_5.png" /></figure>
```R
4 % The computation times on an Apple M3 Max with 64GB are as follows:
5 % (i) For the rank without dynamics constraints: 2 seconds
6 % (ii) For the rank with dynamics constraints: 8 seconds
7 % (iii) For the null space of M: 120 seconds
8 % The computation of the ranks is performed both analytically and numerically.
9 clear all
10 clc
11
12 syms lambda delta rho N c real
13 syms y1 y1_1 y1_2 y1_3 y1_4 y1_5 y1_6 y2 y2_1 y2_2 y2_3 y2_4 y2_5 y2_6
14 real
15 Eq6_23=y1_2*y2*y2_1-y1_1*y2*y2_2-delta*y1*y2*y2_2+lambda*y2*y2_2-(delta+c
16 )*y1_1*y2*y2_1+(delta*rho-delta^2-delta*c)*y1*y2*y2_1+(rho+delta)*y1_1
17 *y2*y2_1+lambda*c*y2*y2_1+rho*c*y1_1*y2*y2+(rho*delta*c-delta*delta*c)
18 *y1*y2*y2-N*delta*y1*y1_2*y2+c*y1_2*y2*y2-N*delta*(rho+delta)*y1*y1_1*
19 y2-N*delta*delta*rho*y1*y1*y2+N*delta*delta*lambda*y1*y2;
20 Y=[y1 y1_1 y1_2 y1_3 y1_4 y1_5 y2 y2_1 y2_2 y2_3 y2_4 y2_5 y2_6]';
21 Y1=[y1_1 y1_2 y1_3 y1_4 y1_5 y1_6 y2_1 y2_2 y2_3 y2_4 y2_5 y2_6]';
22 Phi=Eq6_23;
23 Phi1=simplify(jacobian(Phi,Y)*Y1);
24 Phi2=simplify(jacobian(Phi1,Y)*Y1);
25 Phi3=simplify(jacobian(Phi2,Y)*Y1);
26 Phi4=simplify(jacobian(Phi3,Y)*Y1);
27 Theta=[lambda delta rho N c]';
28 PHI=[Phi Phi1 Phi2 Phi3 Phi4]';
29 %WITHOUT DYNAMICAL CONSTRAINT
30 fprintf('RESULT WITHOUT TAKING INTO ACCOUNT FOR THE DYNAMICAL CONSTRAINT
31 (it needs about 2 sec)\n')
32 M=jacobian(PHI,Theta);
33 fprintf('Analytical computation. The rank of M is %d\n',rank(M))
34 X=[y1 y1_1 y1_2 y1_3 y1_4 y1_5 y1_6 y2 y2_1 y2_2 y2_3 y2_4 y2_5 y2_6
35 lambda delta rho N c];
36 Mnum=eval(subs(M, X, rand(1, length(X))));
37 fprintf('Numerical computation. The rank of M is %d\n',rank(Mnum))
38 SVD=svd(Mnum);
39 fprintf('Numerical computation. The singular values of M: %d, %d, %d, %d, %d\n',
40 SVD(1), SVD(2), SVD(3), SVD(4), SVD(5));
41 fprintf('Numerical computation. The determinant of M is: %d\n\n\n\n', det(Mnum));
42
43 fprintf('RESULT THAT TAKES INTO ACCOUNT FOR THE DYNAMICAL CONSTRAINT (it
44 needs about 8 sec)\n')
45 %WITH DYNAMICAL CONSTRAINT
46 syms Tu Ti V eta eta_1 eta_2 eta_3 eta_4 eta_5 real
47 ETA=[eta eta_1 eta_2 eta_3 eta_4]';
48 ETA1=[eta_1 eta_2 eta_3 eta_4 eta_5]';
49 g0=[lambda-rho*Tu, -delta*Ti, N*delta*Ti-c*V]';
50 g=[-Tu*V, Tu*V,0]';
51 f=g0+eta*g;
52 x=[Tu Ti V]';
53
54 Tu_1=jacobian(Tu,x)*f+jacobian(Tu,ETA)*ETA1;
55 Tu_2=jacobian(Tu_1,x)*f+jacobian(Tu_1,ETA)*ETA1;
56 Tu_3=jacobian(Tu_2,x)*f+jacobian(Tu_2,ETA)*ETA1;
57 Tu_4=jacobian(Tu_3,x)*f+jacobian(Tu_3,ETA)*ETA1;
58 Tu_5=jacobian(Tu_4,x)*f+jacobian(Tu_4,ETA)*ETA1;
59 Tu_6=jacobian(Tu_5,x)*f+jacobian(Tu_5,ETA)*ETA1;
60
62 Ti_1=jacobian(Ti,x)*f+jacobian(Ti,ETA)*ETA1;
63 Ti_2=jacobian(Ti_1,x)*f+jacobian(Ti_1,ETA)*ETA1;
64 Ti_3=jacobian(Ti_2,x)*f+jacobian(Ti_2,ETA)*ETA1;
```
```
55 Ti_4=jacobian(Ti_3,x)*f+jacobian(Ti_3,ETA)*ETA1;
56 Ti_5=jacobian(Ti_4,x)*f+jacobian(Ti_4,ETA)*ETA1;
57 Ti_6=jacobian(Ti_5,x)*f+jacobian(Ti_5,ETA)*ETA1;
58
59 V_1=jacobian(V,x)*f+jacobian(V,ETA)*ETA1;
60 V_2=jacobian(V_1,x)*f+jacobian(V_1,ETA)*ETA1;
61 V_3=jacobian(V_2,x)*f+jacobian(V_2,ETA)*ETA1;
62 V_4=jacobian(V_3,x)*f+jacobian(V_3,ETA)*ETA1;
63 V_5=jacobian(V_4,x)*f+jacobian(V_4,ETA)*ETA1;
64 V_6=jacobian(V_5,x)*f+jacobian(V_5,ETA)*ETA1;
65
66 y1=Tu+Ti;
67 y1_1=Tu_1+Ti_1;
68 y1_2=Tu_2+Ti_2;
69 y1_3=Tu_3+Ti_3;
70 y1_4=Tu_4+Ti_4;
71 y1_5=Tu_5+Ti_5;
72 y1_6=Tu_6+Ti_6;
73
74 y2=V;
75 y2_1=V_1;
76 y2_2=V_2;
77 y2_3=V_3;
78 y2_4=V_4;
79 y2_5=V_5;
80 y2_6=V_6;
81
82 M=simplify.eval(M));
83 fprintf('Analytical computation. The rank of M is %d\n',rank(M));
84 X=[Tu Ti V eta eta_1 eta_2 eta_3 eta_4 eta_5 lambda delta rho N c];
85 Mnum=eval(subs(M, X, rand(1, length(X))));
86 fprintf('Numerical computation. The rank of M is %d\n',rank(Mnum));
87 SVD=svd(Mnum);
88 fprintf('Numerical computation. Singular values of M: %d, %d, %d, %d, %d\n',SVD(1),SVD(2),SVD(3),SVD(4),SVD(5));
89 fprintf('Numerical computation. Determinant of M is: %d\n\n\n\n\n',det(Mnum));
90
91
92 fprintf('THE NULLSPACE OF M IS (it needs about 120 sec)\n');
93 null(M)
```

<caption>Listing 2: Script MATLAB that performs the computation in Section 3.2</caption>

## References

[1] H. Miao, X. Xia, A. S. Perelson, and H.Wu, On identifiability of nonlinear ODE models and applications in viral dynamics, SIAM Review, vol. 53, no. 1, pp. 3–39, 2011.

[2] A. Martinelli, Identifiability of nonlinear ODE Models with time varying Parameters: the General Analytical Solution and Applications in Viral Dynamics, arXiv:2211.13507 [eess.SY]

[3] A. Martinelli, Nonlinear unknown input observability and unknown input reconstruction: The general analytical solution, Information Fusion, 2022, ISSN 1566-2535, https://doi.org/10.1016/j.inffus.2022.03.004.

[4] A.F. Villaverde, N. Tsiantis, and J.R. Banga, Full observability and estimation of unknown inputs, states, and parameters of nonlinear biological models. J. R. Soc. Interface 16, 2019

[5] A. Martinelli, Detection of a very serious error in the paper: "On identifiability of nonlinear ODE models and applications in viral dynamics", arXiv:2304.14288 [math.OC]
[6] H. Miao, A. S. Perelson, and H. Wu, Erratum: On Identifiability of Nonlinear ODE Models and Applications in Viral Dynamics, SIAM Review, 2023, 65(3), 732–732