Theory and simulations of delayed stochastic and deterministic models of prion diseases

Authors

• Gangadhara Boregowda, The University of Texas Rio Grande Valley
• Omar Sharif, The University of Texas Rio Grande Valley
• Daniel Gutierrez III, The University of Texas Rio Grande Valley
• Allegra Simmons, University of Texas at Dallas
• Laurent Pujo-Menjouet
• Tamer Oraby, The University of Texas Rio Grande ValleyFollow
• Michael R. Lindstrom, The University of Texas Rio Grande ValleyFollow

Document Type

Article

Publication Date

4-23-2026

Abstract

Neurodegenerative diseases (NDs), such as Alzheimer’s, Parkinson’s, and prion diseases, are characterized by the dynamical spread of toxic proteins through the brain. In prion diseases, cellular prion protein (PrPC), produced by neurons, misfolds into a toxic form, known as scrapie prion protein (PrPSc). PrPSc induces neuronal stress which ultimately leads to cell death. In this paper, we develop mathematical models for the progression of prion diseases, incorporating a cellular defense mechanism that introduces a delay term affecting protein translation and a volatility term accounting for unaccounted biological factors influencing the system. We also extend the model to capture the spatial spread of toxic proteins over the brain connectome. Our first objective is to establish the existence and uniqueness of a global positive solution to the prion disease models. Afterwards, we analyze the asymptotic behavior of the models by identifying regimes of persistence and extinction of toxic proteins. For the deterministic delayed systems, we perform a stability analysis for the persistence and demonstrate that the system undergoes a Hopf bifurcation. We also study the intensity of fluctuations of the equilibrium state of the stochastic model. Additionally, we present numerical simulations to illustrate the model dynamics using biologically relevant parameters.

Comments

This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. 

Recommended Citation

Boregowda, Gangadhara, Omar Sharif, Daniel Gutierrez III, Allegra Simmons, Laurent Pujo-Menjouet, Tamer Oraby, and Michael R. Lindstrom. "Theory and simulations of delayed stochastic and deterministic models of prion diseases: G. Boregowda et al." Journal of Mathematical Biology 92, no. 5 (2026): 75. https://doi.org/10.1007/s00285-026-02390-6

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Publication Title

Journal of Mathematical Biology

DOI

10.1007/s00285-026-02390-6