Triangularity of the Jacobian on siphon faces, the Metzler property of its transversal component and other results

arXiv:2603.06778 · q-bio.MN, math.DS · Submitted 2026-03-06 · Read on arXiv

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Introduction to the show: ident: Genomics Radio. Generated commentary on the latest computational biology and genomics papers.

Ines: Today's paper: "Triangularity of the Jacobian on siphon faces, the Metzler property of its transversal component and other results".

Marcus: This paper synthesizes concepts from Chemical Reaction Networks Theory (CRNT) and mathematical epidemiology (ME) to provide powerful tools for analyzing the stability and bifurcation problems of positive ordinary differential equations…

Ines: First, who's behind it and why it matters.

Title and authors: Ines: So we’re looking at this paper today titled "Triangularity of the Jacobian on siphon faces, the Metzler property of its transversal component and other results." It seems like they're really trying to bridge the gap between chemical reaction networks and mathematical epidemiology to solve stability problems for positive ordinary differential equations.

Marcus: Exactly, Ines. I’m looking at how they use these CRNT tools—specifically siphons—to establish some structural properties about the system dynamics that are relevant for epidemiological models. It sounds like they're building a framework that connects the algebraic structure of the network directly to what happens in the dynamical systems.

Yuki: From a population genetic perspective, I’m thinking about how these concepts might relate to species interactions or disease spread across different populations, where stability hinges on these types of structural constraints. It feels like they are providing a way to look at the underlying network topology and immediately get hints about its long-term behavior.

Ines: Right, Yuki, that’s a good connection. The core idea seems to be defining what makes a set of species or reactions behave in a certain way—the siphons—and using that structure to simplify stability checks on the boundary of the positive orthant. This is pretty fundamental for understanding when an epidemic model will settle down or explode.

Marcus: I agree, and that structural simplification is key because it allows them to prove things like the Next Generation Matrix theorem in a way that feels more grounded in chemical kinetics than just standard matrix algebra. It’s about showing *why* certain results hold based on the network's shape rather than just plugging numbers into a formula.

Yuki: I wonder how these structural constraints translate when you move from a simple two-species model to, say, a four-species SIRWS system that we see in real ecological data. Does this framework offer any transferable insights into complex species interactions?

Ines: That’s what we’re hoping to find out. The paper suggests that by focusing on these siphon faces and the resulting "Child Selection minors," we can systematically classify instability mechanisms, which should help us predict when oscillations might appear in more complicated scenarios like the SIRWS model.

Marcus: I think it’s promising because it moves us away from just simulating every possible parameter combination and towards a symbolic-numeric approach where we can screen vast spaces of possibilities based on the structure itself. That kind of computational efficiency is exactly what we need when dealing with complex genomic or epidemiological datasets.

Yuki: I appreciate that focus on structure over brute force computation, Marcus. When I look at evolutionary dynamics, the underlying network architecture dictates which pathways are most likely to become dominant, and this paper seems to offer a mathematical way to see those dominant pathways structurally.

Title and authors: Ines: And the specific result they present is about how boundary-face invariance forces certain mixed Jacobian blocks—the Jxy term—to vanish identically on those faces, which simplifies the stability analysis significantly by reducing it to analyzing diagonal blocks. This is a big step in making these proofs more tractable for applied math.

Marcus: That vanishing of the mixed Jacobian block is a very strong condition because it immediately reduces the problem to analyzing simpler sub-problems, which makes testing parameter regimes much more feasible computationally. It’s like finding a shortcut through the most complicated part of the calculation.

Yuki: So, if we think about this in terms of population dynamics, does this structural analysis help us understand why some species might persist while others die out, purely based on their interaction structure within the network?

Ines: It certainly helps with that. The authors introduce a dichotomy based on the sign of these determinant minors—negative feedback versus unstable positive feedback—which gives us a clear algebraic way to classify whether a system is poised for instability or not. This classification is what leads to identifying those specific conditions where oscillations are possible.

Marcus: That classification into negative and positive feedback types based on those minors sounds like it provides a concrete predictive tool. We can use these criteria to filter out unstable configurations before we even waste time running long simulations on them, which is super valuable when dealing with large cohort data.

Yuki: I think that predictability is what matters in biology; knowing the structural requirements for a certain dynamic behavior helps us understand the evolutionary constraints on those behaviors within a species. It grounds the abstract math in something tangible about system constraints.

Ines: And they culminate this approach by defining an "Oscillatory Core of Class I," which requires both an unstable positive feedback and a stable super-Child Selection minor to occur together for a Hopf bifurcation to happen, which is the ultimate goal for analyzing periodic solutions.

Marcus: That definition of Recipe I provides a clear checklist for when we should expect periodic behavior in an epidemiological model like SIRWS, connecting the abstract mathematical structure back to observable biological phenomena. It’s a very useful bridge between theory and data.

Yuki: So, if we take this paper's findings regarding unstable cores and Hopf witnesses, does this suggest that certain network structures are inherently more prone to complex dynamics across different biological systems?

Ines: The implication is that the specific algebraic configuration of the reaction network dictates the inherent capacity for oscillatory behavior in the resulting ODE system. If you find a specific pattern of Child Selection minors, you can predict if your model will exhibit those oscillations without having to explore every single parameter point.

Marcus: And computationally, this means we can use this symbolic-numeric method to automatically scan huge reaction networks for these unstable cores, which is a massive reduction in the computational burden when trying to find the parameters that cause disease outbreaks or system instability.

Title and authors: Yuki: It sounds like this paper offers a new lens for analyzing complex biological systems by focusing on their underlying network architecture rather than just the kinetic rates themselves. That perspective is valuable for understanding how these networks evolve over time in populations.

Ines: We’ve covered the structure, the proof of the NGM theorem, and how they classify instability using those minors. Moving into improvements, they suggest a way to inherit known local bifurcations from small cores to larger systems like SIRWS models by defining specific recipes for that inheritance.

Marcus: That idea of inheritance rules sounds practical because it allows us to use the results found in simpler subsystems and apply them systematically, which is how many complex biological systems are actually modeled—by breaking them down into manageable parts.

Yuki: I think that systematic transfer of local dynamics is a powerful concept, especially when considering how localized interactions might scale up to affect entire populations or species dynamics. It suggests a hierarchical view of system complexity.

Ines: So, to wrap things up on the improvements, they also propose an algorithmic method for automatically detecting Hopf bifurcation witnesses by analyzing the sign patterns and stability properties of these Child Selection minors in symbolic Jacobians, which should be a very direct computational test.

Marcus: That automated detection is what really makes this tool useful for genomics or epidemiology; we can feed it a model structure and it tells us immediately if that structure has the algebraic potential for an oscillation, bypassing many tedious numerical checks.

Yuki: It seems like the ultimate goal here is to create tools that help us understand the conditions under which complex, non-trivial dynamics arise in biological systems, linking network structure directly to observable phenomena.

Ines: Exactly. We’ve looked at how they connect CRNT and ME to provide a structural classification of instability mechanisms using Child Selection minors in the paper "Triangularity of the Jacobian on siphon faces, the Metzler property of its transversal component and other results."

Marcus: And we’ve seen that this framework offers a way to screen for unstable cores and predict Hopf bifurcations in models like SIRWS by analyzing structural properties rather than just running brute-force simulations.

Yuki: It gives us a new way to think about the constraints imposed by interaction networks on the possible dynamical behaviors of any system built upon them.

Ines: So, we’ve established that this paper provides a rigorous framework for translating network topology into stability predictions and Hopf bifurcation witnesses in positive ODEs.

Marcus: And it gives us the symbolic-numeric tools to actually implement these structural checks efficiently against large models.

Yuki: That's a solid summary of how this work contributes to the field by formalizing the connection between chemical reaction network theory and epidemiological dynamics for stability analysis.

The paper's summary: Ines: So, to recap the main thrust of this paper is that they’re using the structure of chemical reaction networks—specifically those siphons—to prove important stability theorems in mathematical epidemiology, like how you can predict if an epidemic model will settle down or start oscillating.

Marcus: Yeah, and what I find really interesting from a data scientist’s viewpoint is that they aren't just throwing numbers around; they’re deriving these structural rules directly from the stoichiometry of the reaction network itself. It means we can test a model’s behavior before we even run a single simulation on real cohort data, which cuts down on those massive batch effect headaches.

Yuki: From my side, I see this as mapping out the fundamental constraints of interaction. If you understand how species are connected in a network, you can predict their long-term fate, and this paper gives us the mathematical language to do that prediction reliably. It connects the abstract rules of chemistry right to the real history of how species coexist across generations.

Ines: That connection is powerful because it moves stability analysis away from just being curve-fitting and toward a more fundamental understanding of system architecture. The core result they’re pushing is that these structural features dictate whether a system can sustain complex dynamics, like periodic outbreaks in an epidemic model.

Marcus: And that's where the computational efficiency comes in; because they've found ways to simplify the Jacobian analysis using these stoichiometric minors, we can apply this to much larger models than before without getting bogged down by intractable matrix calculations. It’s about finding a shortcut through the math so we can get real results faster.

Yuki: That ability to screen massive networks for unstable substructures is what excites me most for population genetics; it allows us to prioritize which interaction pathways are most likely to drive evolutionary change or rapid spread in a given environment.

Ines: Exactly, Yuki, and I think the implications for computational biology are huge because this gives us a systematic way to identify precisely *why* a model might fail—is it just parameter tuning, or is the underlying reaction network structure itself inherently unstable?

Marcus: And for me, it means we can start building diagnostic tools that look at the kinetic parameters and immediately flag potential instability before we even touch the massive datasets. It’s about reducing uncertainty in our statistical models by incorporating this structural knowledge upfront.

Yuki: I see a future where we use these same methods to analyze complex ecological models, not just disease spread, but how entire communities maintain or lose stability based on their interaction topology. It broadens the scope of what we can study through this lens.

Ines: So, looking ahead, the authors are suggesting ways to transfer these localized instability findings from small core systems to larger models like SIRWS, which opens up a pathway for predicting oscillations in more realistic scenarios. That’s a really practical application for modelers.

Marcus: That inheritance rule concept is what makes it feel scalable; it lets us build complexity piece by piece, ensuring that the stability characteristics we find at the local level carry over predictably to the big picture. It’s smart engineering for modeling large-scale phenomena.

Yuki: And I hope this framework helps us see deeper into how these interactions evolve across different species or populations over long timescales, giving us a better historical context for their current dynamic states.

Ines: We've covered the core concepts and the practical implications of using CRNT to predict dynamical behavior in epidemiological systems. Next up, we’re going to look at how they actually implemented these symbolic-numeric tools for real-world testing.

The paper's improvements: Tom: So, we're looking at how this paper goes beyond just presenting results by suggesting ways to make these structural analyses even more useful for complex systems. Basically, they’re talking about improving their framework for predicting instability in larger models like SIRWS by introducing concepts called "recipes."

Ines: Right, and what I find compelling about those recipes is that they offer a systematic way to take the local findings—the small unstable cores—and transfer that understanding up to the whole system. It’s a method for building complexity piece by piece, which is exactly how many biological systems operate.

Marcus: From my data perspective, that ability to inherit stability characteristics means we can use simpler models as building blocks for more complex ones, which drastically cuts down on the computational time needed to test huge cohort datasets for instability. It’s about using structural knowledge to guide our statistical modeling.

Yuki: I think the implication here is that we might be able to trace the evolutionary history of a system's stability by looking at these inherited properties, seeing how fundamental interaction rules shape the long-term dynamics across different ecological contexts.

Ines: That tracing of dynamical inheritance across scales is really significant because it gives us a way to understand how robustness or fragility is built into the network structure itself, not just through parameter tuning.

Marcus: And for cohort analysis, if we know which structural features are inherently unstable regardless of our exact kinetic rates, we can filter out those scenarios from our batch effect analysis much more effectively. It’s a way to move beyond simply finding correlations in the data to understanding the underlying mechanism causing the correlation.

Yuki: I hope this helps us understand why certain interaction patterns are consistently favored by evolution across different species, giving us a structural reason for their persistence or extinction patterns we observe today.

Ines: Exactly, Yuki; it gives us a more principled way to interpret those historical stability patterns we see in the fossil record and contemporary populations.

Marcus: And from an engineering standpoint, these inheritance rules are essentially a form of model reduction that preserves critical stability information, which is exactly what we need when dealing with high-dimensional models that would otherwise crash our simulations.

Yuki: It feels like this work is providing the mathematical vocabulary to discuss the deep constraints on biological persistence and change in terms of network topology.

Ines: So, we’ve seen how they classify instability using minors, and now they’re proposing methods to systematically predict when those instabilities will manifest in larger, more realistic epidemiological scenarios.

Marcus: And that systematic prediction is what makes this incredibly useful for anyone working with large-scale genomic or clinical data—it provides a structured way to look for the 'red flags' in the network before we get overwhelmed by noise.

Yuki: It suggests a hierarchical view of system complexity, where understanding local constraints informs our predictions about global, long-term population dynamics.

Ines: So, moving forward, this paper’s contribution is less about just proving theorems and more about providing an actionable blueprint for how to use network structure to systematically predict the complex dynamics that drive biological phenomena.

Conclusion: Ines: So, to wrap up our discussion on "Triangularity of the Jacobian on siphon faces, the Metzler property of its transversal component and other results," we’ve seen how this paper uses chemical network theory to give us powerful algebraic tools for predicting stability in epidemiological models.

Marcus: Yeah, and what really stands out is that they're not just doing theoretical math; they're providing a concrete framework for identifying the structural 'fault lines' in these models using stoichiometric analysis. It’s about moving from guessing parameters to understanding the fundamental architecture of the system itself.

Yuki: I think this work provides a new language for population genetics because it allows us to link the history of species interaction—the network structure—directly to their current dynamical stability, which is a pretty deep connection for me.

Ines: Exactly, Yuki, and as computational biologists, we get to recover insights into how system topology dictates the emergence of complex behaviors like oscillations without having to run countless simulations just to find them.

Marcus: I'm still thinking about how this applies practically to our work with large datasets; if we can use these structural invariants to pre-screen models, it could drastically reduce the computational load when analyzing those massive cohort comparisons.

Yuki: That systematic screening capability is important because it helps us understand the constraints on species persistence over long evolutionary timescales, which is something we always struggle to model accurately.

Ines: So, in short, this paper gives us a way to use the CRNT perspective to systematically analyze the stability of ODEs and connect that structure directly to biological phenomena like periodic outbreaks.

Marcus: And it provides a robust set of symbolic-numeric tools that make these structural analyses computationally feasible for real-world applications in genomics and epidemiology.

Yuki: It's really about providing a structural foundation for understanding the history and future dynamics of interacting populations, which is fascinating to see formalized this way.

Ines: And the implication is that we can start asking not just what happens under certain conditions, but *why* those conditions lead to specific dynamical outcomes based on network rules.

Marcus: That structural 'why' helps us build better predictive models because we're incorporating knowledge about the underlying constraints of the reaction network into our statistical framework.

Yuki: I think this approach is going to be vital for anyone studying how these interaction networks evolve and adapt over time in real-world biological systems.

Ines: It’s been a really insightful deep dive into how abstract algebraic structures translate into tangible predictions about the stability of dynamic systems in biology.

Marcus: We’re leaving this paper with a much clearer path for using structural analysis to reduce uncertainty and improve our ability to model complex, high-dimensional biological data.

Yuki: I'm excited to see how this framework helps us connect these network constraints to the broader evolutionary context we study.

Ines: Alright team, that concludes our look at "Triangularity of the Jacobian on siphon faces, the Metzler property of its transversal component and other results." Next up, we're shifting gears and looking at those AI papers in NeuroAI and Beyond.

Laboratoire de Mathématiques Appliquées, Université de Pau · Laboratoire d’Analyse, Géométrie et Applications, Département des Mathématiques, Université Ibn-Tofail · Department of Mathematics and Computer Science, University of Bucharest

q-bio.MN, math.DS

Submitted: 2026-03-06

Updated: 2026-09-30

Comments: Major rewriting of the paper

Code: https://github.com/florinav/EpidCRNmodels

License: http://creativecommons.org/licenses/by-sa/4.0/

Importance score: 92/100

The gist: This paper synthesizes concepts from Chemical Reaction Networks Theory (CRNT) and mathematical epidemiology (ME) to provide powerful tools for analyzing the stability and bifurcation problems of

Key concepts

CRN stoichiometric representation: This is a structural way to define a chemical reaction network using a triple (S, Γ, r). It ensures that the system's dynamics stay within the non-negative region of its state space. This algebraic structure links the network's reactions directly to how populations change over time.
Siphons: In this context, siphons are geometric features on a reaction network's boundary faces. They correspond to specific conditions where certain parts of the system's dynamics are constrained. The paper uses their existence to prove crucial properties about the Jacobian matrix, simplifying stability checks.
Child Selection Minors: These are combinatorial tools derived from the stoichiometric matrix used to analyze the characteristic polynomial of a system's Jacobian. They allow researchers to systematically identify and classify different types of instability mechanisms, such as negative or positive feedback loops, within the model.

Terminology

Core — 46× in this paper · explained in 2 episodes
Mathematical Epidemiology — 16× in this paper
Bifurcation Analysis — 11× in this paper · explained in 2 episodes
Next Generation Matrix — 9× in this paper
Generalization — 6× in this paper · explained in 13 episodes
Detailed Routh-Hurwitz — 5× in this paper
Chemical Reaction Networks Theory — 3× in this paper
Structural Constraints — 2× in this paper · explained in 2 episodes
Adaptation — 1× in this paper · explained in 1 episode
Admissibility — 1× in this paper · explained in 2 episodes
Computational Biology — 1× in this paper · explained in 2 episodes
Diagnostic Tools — 1× in this paper · explained in 2 episodes
Ordinary Differential Equations (ODEs) — 1× in this paper · explained in 2 episodes
Robustness — 1× in this paper · explained in 5 episodes
Trace — 1× in this paper · explained in 2 episodes

Summary

This paper synthesizes concepts from Chemical Reaction Networks Theory (CRNT) and mathematical epidemiology (ME) to provide powerful tools for analyzing the stability and bifurcation problems of positive ordinary differential equations (ODEs). By introducing CRN-flavored proofs for established results in ME, such as the Next Generation Matrix theorem, and by developing a symbolic-numeric approach based on Child Selection minors of the stoichiometric matrix, the authors establish a framework that connects algebraic structures in reaction networks to dynamical phenomena like Hopf bifurcations. This work is significant because it offers model-independent insights into when epidemic models exhibit periodic solutions and provides computational methods for detecting these instabilities.

General Framework: CRNT and Positive ODEs

The paper begins by establishing the foundational concepts linking positive ODEs to CRNT, defining a CRN stoichiometric representation as a triple (S, Γ, r) that ensures forward invariance of the non-negative orthant. A key structural element is the notion of siphons, which are in one-to-one correspondence with forward invariant boundary faces. The paper formalizes this connection by stating that If w is an ω-limit of a strictly positive initial point for some choice of kinetics/rates, and belongs to a boundary face w ∈ LΣ:= [X ∈ Rn≥0: xi = 0, i ∈ Σ], then Σ is a siphon (Proposition 1). This establishes the geometric link between the algebraic structure of the reaction network and the dynamics on the boundary of the positive orthant.

The Next Generation Matrix (NGM) Theorem via Boundary-Face Invariance

A primary contribution is an elegant chemical reaction networks-flavored proof of the most cited result in ME, the next generation matrix (NGM) theorem. This proof relies on a structural property derived from siphons: Boundary-face invariance forces Jxy = 0 for positive ODEs. Theorem 1 formalizes this, stating that if a face Fx is invariant (i.e., x is a siphon), then The mixed Jacobian block Jxy:= Dyfx(0, y) vanishes identically on the face Fx: Dyfx(0, y) = 0 for all y ∈ Rn≥0. This structural result simplifies stability analysis by implying that the Jacobian has block lower-triangular form J(0, y∗) = Jx 0∗ Jy, reducing the problem to analyzing diagonal blocks.

Symbolic-Numeric Analysis via Child Selection Minors

The authors review the symbolic-numeric approach of Vassena and Stadler, which views the characteristic polynomial of the Jacobian at fixed points as a formal polynomial in symbolic reactivities. They identify its coefficients as Child Selection minors of the stoichiometric matrix. A k-child selection (k-CS) is defined combinatorially as a triple κ = (κ, Eκ, J) where J is a bijection between species and reactions in which they appear as reactants. The Cauchy–Binet expansion of the determinant of the symbolic Jacobian det(G[κ]) is rewritten as a sum over child selections: det(G[κ]) = X (κ,Eκ,J)∈CS(κ) det(Γ[κ, J])Y i∈κ RJ(i),i, where the signs of the determinants are absorbed into the reordered stoichiometric determinants.

Classification of Instability Mechanisms

The paper introduces a dichotomy based on the sign of these determinant minors to classify instability mechanisms:

A k-child-selection κ (or A = Γ[κ]) is called a negative feedback (NF) if sign(−1)k det A = +1.
It is called an unstable positive feedback (UPF) if sign(−1)k det A = −1, meaning the matrix A is Hurwitz unstable.

Furthermore, an unstable core is defined as a k-child-selection where the matrix A is unstable and every proper principal submatrix of A is Hurwitz stable. Lemma 3 specifies that for an unstable core, if sign((−1)k det A) = -1 (UPF case), then "A has an odd number of real eigenvalues with RΛ > 0."

Bifurcation Analysis and Hopf Witnesses

The symbolic-numeric approach is used to analyze specific epidemiological models, such as SIRWS. The analysis reveals that the existence of an UPF implies instability, and the authors confirm the structural capacity for Hopf bifurcation by identifying Oscillatory Core of Class I. They demonstrate that a Hopf bifurcation occurs if Recipe I is present: "Recipe I: An Oscillatory Core of Class I is a pair (UPF, stable super-CS) where the super CS is minimal with the property of being Hurwitz-stable and possessing an unstable-positive feedback as a principal submatrix.

Improvements for AI systems

Here are the specific improvements to AI systems that can be derived from this research, focusing on leveraging Chemical Reaction Network Theory (CRNT) and Mathematical Epidemiology (ME) for stability analysis:

Developing a CRNT-flavored Next Generation Matrix (NGM) Theorem implementation within existing epidemiological models.
Creating symbolic-numeric bifurcation analysis tools based on the Vassena-Stadler approach for ODE systems, specifically targeting positive dynamical systems like SIR/SIRS models.
Implementing Child Selection expansion algorithms to automatically identify unstable substructures (Unstable Positive Feedbacks, UPFs) within complex biochemical or epidemiological interaction networks.
Integrating Inheritance Rules to systematically transfer known local bifurcations (e.g., Hopf bifurcations from small cores) to the dynamics of larger, more realistic systems, such as the four-species SIRWS model.
Developing an algorithmic method for automatic detection of Hopf bifurcation witnesses by analyzing the sign patterns and stability properties of Child Selection minors in symbolic Jacobians.

The improved AI system can perform the following specific tasks:

A researcher could use this system to analyze a new epidemic model (e.g., a novel disease spread mechanism) and immediately determine if it is structurally capable of exhibiting oscillations (Hopf bifurcations) without needing extensive numerical simulation across all parameter spaces.
The system can automatically screen massive libraries of reaction networks or epidemiological models to find unstable cores or autocatalytic clusters that are the primary sources of instability, drastically reducing the computational search space for bifurcation analysis.
By applying the NGM theorem generalization, an AI could instantly determine if a disease-free equilibrium (DFE) on a boundary face is stable or unstable based purely on its stoichiometric and kinetic structure, providing rapid diagnostic insights for public health modeling.
The system can use the symbolic analysis to predict which parameter regimes will lead to oscillations (e.g., Recipe I oscillatory cores) based on the sign of Jacobian minors, guiding experimental design toward conditions known to produce complex dynamics.
The AI could act as a sophisticated model reduction tool, using the identified siphon structures and regular splittings to simplify high-dimensional models into low-dimensional core systems while preserving the essential stability characteristics relevant for disease persistence.

Abstract

We establish two structural properties of the Jacobian of a positive ODE on any invariant boundary face, i.e. on the face defined by a siphon of an associated Chemical Reaction Network. First, the Jacobian is block lower-triangular at every point of such a face, with a transversal diagonal block, which governs the species that vanish on the face, and a tangential one. Second, the transversal block is always a Metzler matrix. These two facts yield an elegant proof, in the spirit of Chemical Reaction Network theory, and a generalization of the most cited result in Mathematical Epidemiology (ME), the Next Generation Matrix (NGM) theorem, and they clarify its hypotheses. The Metzler property is exactly what makes the threshold ρ(FV-1)<1 work, it implies that regular splitting of the transversal block always exist, and, via Perron--Frobenius theory, that a boundary fixed point can never lose its stability through a Hopf bifurcation in the directions transversal to its face. Among other results, we review the ``symbolic-numeric" approach of Vassena and Stadler, which tackles bifurcation problems by viewing the characteristic polynomial of the Jacobian at fixed points as a formal polynomial in the "symbolic reactivities", and identifies its coefficients as ``Child Selection minors of the stoichiometric matrix". We also review two applications of this approach, to an SIRWS model and to a Capasso-Ruan-Wang family of SIRS models, using the Mathematica package Epid-CRN, which implements tools from both Chemical Reaction Network and Mathematical Epidemiology.