A multigroup approach to delayed prion production

arXiv:2307.05524 · math.DS, q-bio.PE · Submitted 2023-07-07 · 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: We're looking at the paper: "A multigroup approach to delayed prion production".

Marcus: The gist The authors generalize a model for prion infection to a network of neurons using a multigroup approach to Delay Differential Equations (DDEs) and derive the basic reproduction number…

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

Paper summary: Ines: To follow up on that, "A multigroup approach to delayed prion production" is essentially taking a model for prion infection and generalizing it for a network of neurons by using this specific multigroup approach to the system of delay differential equations they started with.

Marcus: They then derive the classical threshold quantity R0, which is the basic reproduction number, by using the fact that these delay differential equations behave like ordinary differential equations when you look at the disease-free equilibrium.

Yuki: So they're focusing on finding this R0 value as a key parameter that tells us whether a prion infection can survive in this neuronal network or not. What is the core claim they are making about this R0?

Ines: They prove analytically that the disease naturally goes extinct when R0 is less than one, and it persists when R0 is greater than one. That’s the main conclusion of their work regarding persistence and extinction in this system.

Marcus: So for someone looking at this from a data perspective, it means you can calculate an R0 value based on the network structure and parameters to predict if the infection will die out or become endemic.

Yuki: And they mention that under slightly stricter conditions, they are able to prove the existence of at least one endemic equilibrium, which is when the disease keeps going in a steady state even if R0 isn't strictly above one.

Ines: So they aren't just looking at simple extinction versus persistence; they are also showing conditions where a stable endemic state can actually exist for this prion system. This adds another layer to the picture.

Marcus: It’s interesting how they link this to the original work by Adimy, Babin, Pujo-Menjouet in SIAM Journal on Applied Dynamical Systems (two thousand twenty-two), showing how their multigroup approach extends that earlier model to a network of neurons <ref:2307.05524#pg1,SIAM Journal on Applied Dynamical Systems (2022>.

Yuki: That connection is important because it shows the relevance of these mathematical tools across different fields, from prion dynamics to population genetics. It suggests these kinds of delay systems might be relevant everywhere.

Conclusion: Ines: So, in conclusion, "A multigroup approach to delayed prion production" takes the model for prion infection and applies a multigroup approach to delay differential equations on a network of neurons. The authors derive the basic reproduction number R0.

Marcus: They prove that if this R0 is less than one, the disease goes extinct, but if it's greater than one, it persists in the network. They also show conditions for an endemic equilibrium to exist under certain stricter constraints on the system parameters.

Yuki: What this means for us is that we have a mathematical framework that can describe how a prion infection might spread through a connected structure, like neurons or even populations, based on these calculated R0 values.

Ines: It gives us a way to look at these biological processes mathematically and predict the long-term behavior of the system without needing to run every single simulation from scratch.

Marcus: It’s a practical tool for assessing whether an infectious process will be self-sustaining in a complex system like a neural network. The work by Adimy, Chekroun, Pujo-Menjouet, and Sensi is laying groundwork for how we model these kinds of delays in biological systems.

Yuki: It’s about connecting the abstract math to the actual biology of how things spread within a network structure. That's what makes this paper interesting for everyone listening.

Inria University Lyon Claude Bernard University Lyon 1 CNRS UMR 5208 Institute Camille Jordan · University of Tlemcen Laboratoire d’Analyse Nonlin´eaire et Math´ematiques Appliquées University of Tlemcen MathNeuro Team Inria at Universit´e Cˆote d’Azur Department of Mathematical Sciences G. L. Lagrange Politecnico di Torino

math.DS, q-bio.PE

Submitted: 2023-07-07

Updated: 2023-07-07

Comments: 21 pages, 11 figures

Journal ref: Discrete and Continuous Dynamical Systems - B, 29(7), 2972--2998 (2024)

DOI: 10.3934/dcdsb.2023209

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 78/100

The gist: The gist The authors generalize a model for prion infection to a network of neurons using a multigroup approach to Delay Differential Equations (DDEs) and derive the basic reproduction number R0,

Key concepts

Delay Differential Equations (DDEs): These are a type of differential equation where the rate of change depends not only on the current state of a system but also on its past states at some previous time points. In this model, the prion production in neurons is influenced by how many infected neurons were at earlier times.
Multigroup Approach: This is a mathematical technique used to simplify complex systems with multiple interacting components. By grouping similar variables or processes together, the authors can analyze the overall system behavior more effectively than looking at each component in isolation.
Basic Reproduction Number (R0): R0 is a critical value that determines whether an infection will spread through a population or die out. If R0 is less than 1, the disease cannot sustain itself and will eventually disappear. If R0 is greater than 1, the infection will persist and spread.
Disease Free Equilibrium: This is a specific state in the model where no neurons are infected (the disease has gone extinct). The authors analyze how the system behaves when it starts from this state to determine if the disease can invade and grow.

Terminology

Ordinary Differential Equations (ODEs) — 1× in this paper · explained in 2 episodes

Summary

The gist The authors generalize a model for prion infection to a network of neurons using a multigroup approach to Delay Differential Equations (DDEs) and derive the basic reproduction number R0, proving that disease extinction occurs when R0 1 0, by the following system dx1/dt = K1β(y1(t − T1)) − µ1x1(t) − dx1(t) (y1(t) + ka2y2(t)), dx2/dt = K2β(y2(t − T2)) − µ2x2(t) − dx2(t) (y2(t) + ka1y1(t)), dy1/dt = dx1(t) (y1(t) + ka2y2(t)) − α1y1(t), and dy2/dt = dx2(t) (y2(t) + ka1y1(t)) − α2y2(t) 1, then there exists a constant ε > 0 such that for any initial condition (xi0, φi) ∈ R+ × C([−T, 0], R+), for i = 1,., n we have lim sup t→+∞ yi(t) > ε 1 1<ref:2307.05524.

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Improvements for AI systems

AI systems can predict prion disease extinction or permanence by calculating a threshold quantity R0, derived from the spectral radius of an n×n matrix F defined as (F)ij = (R0 if j = i, κjiαj→iR0i if j ≠ i).
AI models can determine the asymptotic stability of the Disease Free Equilibrium by checking if all eigenvalues of (M − V) have negative real parts when R0 < 1.
AI systems can predict the existence and persistence of an Endemic Equilibrium by evaluating conditions based on the minimum row sum is strictly bigger than 1 for the matrix F.
AI models can forecast strong uniform persistence of prion infection, stating that "there exists a constant ε > 0 such that, for any initial condition (xi0, φi) ∈ R+ × C([−T, 0], R+), for i = 1,..., n, we have lim inf t→+∞ yi(t) > ε."

Sources

A gathering of Barbalat's lemmas and their (unsung) cousins