Bélair, Jacques
- Full Professor
-
Faculty of Arts and Science - Department of Mathematics and Statistics
André-Aisenstadt Office 4439
Courriels
Affiliations
- Membre Centre de recherches mathématiques
- Membre CRM Centre de recherches mathématiques
Research area
- Applied mathematics
- Modelisation
- Nonlinear dynamic
- Pharmacometrics
- Haematopoiesis
- Erythrocytes
- Neutropenia
- Delay differential equations
- Chaos theory
- Bifurcation theory
Student supervision Expand all Collapse all
Sur un modèle d’infection virale avec délai distribué
Theses and supervised dissertations / 2024-05
Trahan, Marc-Antoine
Abstract
Abstract
The mathematical modeling of the dynamics of autoimmune diseases contributes to the
understanding of their mechanisms, thus providing better guidance for treatments. In this
context, this thesis analyzes a distributed delay differential equations system modeling the
evolution of HIV in an infected body, describing the interactions between uninfected CD4-T
cells, infected cells, virus particles and the immune response. Aavani [1] studied a similar but
simpler model, incorporating a discrete delay, which we generalize using alternative methods
for the investigation of stability of stationary solutions.
The asymptotic behavior of the solutions is entirely characterized by the delay, denoted
\(\tau \) , representing the time before an infected cell produces virus particles. It is shown that
for a sufficiently large value of \(\tau \) , i.e. above a certain threshold \(\tau_1 \), the infection tends to
die out since the disease-free steady-state is asymptotically stable. Then, for a delay below
this threshold, the infection persists : the disease-free steady-state being unstable. In this
case, the acute steady-state and the chronic stage exchange asymptotic stability according
to another threshold \(\tau_2 \). Numerical simulations finally support the conclusions obtained
analytically.
Analyse de la stabilité d'un système d'équations différentielles à délais modélisant la régulation de cellules sanguines
Theses and supervised dissertations / 2023-12
Desrochers, Steven
Abstract
Abstract
Blood cells play a fundamental role in the proper functioning of the body and are regulated to respond to its immediate needs. Despite their distinct functions, several studies suggest that the regulatory processes of red blood cells and blood platelets interact with each other, notably through the intervention of hormones. We aim to study the conceptual interactions between these two families of blood cells using a coupled two-delay differential equation model. The analysis of the distribution of eigenvalues of the characteristic equation of the linearized model allows us to outline a stability portrait of the system in a suitable parameter plane. We are particularly interested in exploring the possibilities of destabilization and restabilization through the coupling of the two equations. The analysis of stability diagrams
for these different scenarios highlights interesting dynamics such as stability switches due to the influence of delays and various types of destabilizing bifurcations of the equilibrium.
Modèle épidémiologique multigroupe pour la transmission de la COVID-19 dans une résidence pour personnes âgées
Theses and supervised dissertations / 2021-11
Ndiaye, Jean François
Abstract
Abstract
In this thesis, we consider a multiple group epidemiological model in a heterogeneous population to describe COVID-19 outbreaks in an elderly residential population. Age-based heterogeneity reflects higher transmission with enhanced interactions, and higher fatality rates in the elderly. Mathematically, we analyse a SEIR model in the form of a system of integro-differential equations with general distribution function for the infectious period. Lyapunov functions and graph-theoretical methods are employed to establish the role played by the basic reproduction ratio \(\mathcal{R}_0\) : global asymptotic stability of the disease-free equilibrium and no sustained outbreak when \(\mathcal{R}_0 \leq 1\), as opposed to persistent outbreak and globally asymptotic endemic equilibrium when \(\mathcal{R}_0>1\). Numerical simulations are presented to illustrate public health control strategies.
Modélisation de l'interaction entre les virus de la grippe et de la rougeole
Theses and supervised dissertations / 2020-12
Bouthillette, François
Abstract
Abstract
People infected with measles experience immune suppression. This work focuses on the
influence of this characteristic of measles on another pathogen, here the flu. We also have that
the flu will increase the production of other pathogens in a co-infection model. Modeling
the immune response to such virus-virus interaction is currently of signficant relevance, given
the limited knowledge on SARS-CoV-2/influenza interactions. A model of each pathogen will
be developed and analysed. We will look for the fixed points, conditions for their stability
and we will observe some numerical results of their evolution over time. Then a model following
the evolution of the two pathogens having simultaneously infected an individual will
be designed. In this model we will include the interactions of the pathogens on each other to
theoretically determine the effects in individuals infected with both influenza and measles.
Then we can compare between the different populations when there is no interaction and
with the different interactions between the two pathogens.
Modèle épidémiologique compartimental à délai pour le virus de la dengue
Theses and supervised dissertations / 2020-12
Bérubé, François
Abstract
Abstract
Dengue is a viral infection affecting from 100 to 400 million people each year. According to the WHO, "severe dengue is a leading cause of serious illness and death in some Asian and Latin American countries". This justifies the modelling of this illness's propagation in a population using mathematical compartmental models. Results of Forshey et al. on dengue fever seem to indicate the possibility that a dengue infection does not yield a long term immunity against the different dengue serotypes, and that an homotypical reinfection could be common. We study a SIRS model for the dengue virus that takes into account this loss of immunity via a system of delay differential equations. We characterize the stationary states and their stability in terms of the different parameters considered, in particular the basic reproduction ratios associated to each dengue serotype. We study the system's bifurcations in its main parameters, especially the Hopf bifurcations arising from the presence of a delay in the system of differential equations. Numerical simulations of the model are presented to represent the model's different regimes.
Bifurcation de Hopf dans un modèle de signalement de NF-κB
Theses and supervised dissertations / 2018-12
Le Sauteur-Robitaille, Justin
Abstract
Abstract
The signaling system for the transcription factor NF-κB is involved in over 150 genes in a mammal cell. This leads scientists to try to analyse this molecule to understand its effect on a cell. Many scientists, including Krishna and al., noticed oscillations in the amount of nucleic NF-κB. Before anyone noticed those oscillations, the quantities were thought to be somewhat stable, and they are, but not in every condition. This change of condition creates this instability and the transition of such stability for the stationary solution is caused by a Hopf bifurcation. To determine the existence of the stationary state in the tridimensional system and to analyse the bifurcation is important to predict the oscillations that might appear in certain conditions. It is then necessary to determine what kind of cycle appears or disappears at the bifurcation to understand the stability of those periodic solutions, of those oscillations. Finally, we simulate numercially the bifurcation diagrams for two models and differents parameters to observe the local similarities and global divergence of the diagrams.
Étude d’équations à retard appliquées à la régulation de la production de plaquettes sanguines
Theses and supervised dissertations / 2018-11
Boullu, Loïs
Abstract
Abstract
The object of this thesis is the study, using mathematical models, of the regulation mechanism
maintaining an optimal quantity of blood platelets. The first chapter presents the biological and mathematical
context of the thesis. In a second chapter, we introduce a model for platelet production assuming a regulation
by the platelet quantity of both the differentiation rate of stem cells to the platelet cell line and the amount
of platelets produced by each megakaryocyte. We show that the dynamic of this model corresponds to a delay
differential equation x'(t) = −γx(t)+f(x(t))g(x(t−τ)), and we obtain for this equation new sufficient conditions
for stability and for the oscillation of solutions. In a third chapter, we analyze a second model for platelet
production in which the regulation is continuous through the maturation speed of megakaryocyte progenitors.
The stability analysis requires to adapt a pre-existing framework to problems where the bifurcation parameter is
not the delay, and allows to show that increasing the death rate of megakaryocyte progenitors leads to the onset
of periodic solutions, in agreement with clinical observation of amegakaryocytic cyclical thrombocytopenia. The
last chapter covers a differential equation with two delays that appears among others in a model of platelet
production which considers that platelet death can both age-independent and age-dependent.
Modélisation pharmacocinétique du rythme circadien
Theses and supervised dissertations / 2016-12
Véronneau-Veilleux, Florence
Abstract
Abstract
Humans are organised according to an internal clock with a period of approximatively 24 hours. The pharmacokinetic of several classes of drugs are then influenced by circadian rhythms. Indeed, the area under the curve (of the drug concentration as a function of time), the maximal concentration and the time to maximal concentration can change according to the time at which the drug is taken. The objective of this present work is to find a model to represent the variations in the maximal drug concentration according to the absorption’s time.
We first study a model presented by Godfrey. It allows to find the drug concentration as a function of time while taking into account circadian rhythms. Unfortunately, this model could not represent the variations in the maximal concentration according to the time at which the drug is taken.
We developed a new two-compartmental model for the three ways of absorption (oral, intravenous and intravenous bolus). The resulting systems of ordinary differential equations will be studied. The effect of the phase parameters on the maximal concen- tration will also be studied. Finally, the proof of well-poseness of the model will be developed in the Annex.
Sur un modèle d'érythropoïèse comportant un taux de mortalité dynamique
Theses and supervised dissertations / 2015-01
Paquin-Lefebvre, Frédéric
Abstract
Abstract
This thesis addresses erythropoiesis mathematical modeling, which is the process of erythrocytes production and its regulation by erythropeitin. We propose an erythropoiesis model extension which includes aging of mature cells. First, we consider an age-structured model with moving boundary condition, whose dynamics are represented by advection equations. Biologically, the moving boundary condition means that the maximal lifespan varies to account for a constant degraded cells flux. Then, hypotheses are introduced to simplify and transform the model into a system of three delay differential equations for the total population, the hormone concentration and the maximal lifespan. An alternative model composed of two equations with two constant delays is obtained by supposing that the maximal lifespan is constant. Finally, a new model is introduced, which includes an exponential death rate depending on erythrocytes maturity level. A linear stability analysis allows to detect simple and double Hopf bifurcations emerging from variations of the gain in the feedback loop and from parameters associated to the survival function. Numerical simulations also suggest a loss of stability caused by interactions between two linear modes and the existence of a two dimensional torus in the phase space close to the stationary solution.
Modélisation mathématique de la propagation de la malaria
Theses and supervised dissertations / 2014-12
Niyukuri, Fidèle
Abstract
Abstract
A mathematical model for the spread of malaria has been developed to determine the influence that a population shift from rural to urban areas may have on the persistence or reduction of the disease. This discrete-time model, a system of fourteen finite-difference equations, is then compared with a continuous time model, a system of ordinary differential equations. A comparative study of recently published models allows a determination of the strengths and weaknesses of our model.
Sur un système de deux oscillateurs FitzHugh-Nagumo couplés
Theses and supervised dissertations / 2012-05
Molinié, Marcela
Abstract
Abstract
We study the dynamical behaviour of a pair of identical, coupled FitzHugh-Nagumo oscillators. We determine the parameter values leading to the existence of up to five equilibrium solutions, and analyze the asymptotic stability of each one. A combination of analytical and numerical techniques is used to analyze the numerous bifurcations (saddle-node, Hopf, period-doubling, heteroclinic) occurring as parameters, most notably the coupling strength, are varied, attention being paid to the rôle played by symmetries in the system.
Stabilité d'un réseau de neurones à délai distribué modélisant la mémoire spatiale
Theses and supervised dissertations / 2006
Grégoire-Lacoste, François
Abstract
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.
Dynamique de colonisation par bactéries résistantes de porcs d'élevage
Theses and supervised dissertations / 2005
Sanche, Steven
Abstract
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.
Équations différentielles à retard et leur application en hématopoïèse, avec étude du cas de la neutropénie cyclique
Theses and supervised dissertations / 2003
Bernard, Samuel
Abstract
Abstract
Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.
Analyse des bifurcations dans un modèle du flutter auriculaire
Theses and supervised dissertations / 2003
Doyon, Nicolas
Abstract
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.
Modélisation de la cinétique de médicaments à action rapide
Theses and supervised dissertations / 2001
Lafrance, Patrick
Abstract
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.
Périodicité des cycles épidémiques de la rougeole au Québec
Theses and supervised dissertations / 1998
Vanier, Jean-François
Abstract
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.
L'étude du système couplé de deux oscillateurs neuronaux de Wilson-Cowan
Theses and supervised dissertations / 1995
Stabilité dans les réseaux neuronaux avec délais temporels
Theses and supervised dissertations / 1994
Dufour, Steven
Abstract
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.
Estimation numérique de la vitesse de décroissance des spectres de puissance
Theses and supervised dissertations / 1994
Étude d'un oscillateur biologique
Theses and supervised dissertations / 1992
Dumas, Pierre
Abstract
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.
Étude d'une équation différentielle à retard modélisant des processus de contrôle physiologique
Theses and supervised dissertations / 1990
Babaï, Dariouch
Abstract
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.
Research projects Expand all Collapse all
Dynamical Regulation in Biological Systems CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2024 - 2030
Centre de recherches mathématiques (CRM) FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2022 - 2029
Réseaux Mathematics for public health (MfPH)_Funds for the hire of an HQP CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2022 - 2024
Réseaux Mathematics for public health (MfPH)_Funds for the hire of an HQP CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2022 - 2023
Réseaux Mathematics for public health (MfPH)_Project 10. Dynamic Bifurcation and Scenario Analyses CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2022 - 2023
Réseaux Mathematics for public health (MfPH)_Project 10. Dynamic Bifurcation and Scenario Analyses CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2021 - 2024
Modélisation pharmacologique des systèmes appliquée à plusieurs domaines thérapeutiques: une approche intégrative pour les fondements de l'interaction entre la pathologie, le devenir et laction du médicament FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2020 - 2025
Modélisation pharmacologique des systèmes appliquée à plusieurs domaines thérapeutiques: une approche intégrative pour les fondements de l'interaction entre la pathologie, le devenir et laction du médicament FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2020 - 2024
Agent-based and multi-scale mathematical modelling of COVID-19 for assessments of sustained transmission risk and effectiveness of countermeasures IRSC/Instituts de recherche en santé du Canada / 2020 - 2022
Supplément COVID-19 CRSNG_Dynamical Regulation in Physiological Systems CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2020 - 2021
Dynamical Regulation in Physiological Systems CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2018 - 2025
Dynamical Regulation in Physiological Systems CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2018 - 2024
CENTRE DE RECHERCHES MATHEMATIQUES (CRM) FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2015 - 2023
Caractérisation et mesure de la variabilité médicamenteuse et son impact thérapeutique par une pharmacologie probabiliste pour une utilisation rationnelle du médicament FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2015 - 2019
DYNAMICAL REGULATION IN PHYSIOLOGICAL SYSTEMS CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2012 - 2018
CENTRE DE RECHERCHES MATHEMATIQUES (CRM) FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2008 - 2016
Selected publications Expand all Collapse all
Threshold dynamics in an SEIRS model with latency and temporary immunity
Yuan, Yuan et Bélair, Jacques, Threshold dynamics in an SEIRS model with latency and temporary immunity 69, 875--904 (2014), , J. Math. Biol.
Stability and Hopf bifurcation analysis for functional differential equation with distributed delay
Yuan, Yuan et Bélair, Jacques, Stability and Hopf bifurcation analysis for functional differential equation with distributed delay 10, 551--581 (2011), , SIAM J. Appl. Dyn. Syst.
Delay induced oscillations in autonomous drug delivery systems
Bélair, J. et Buono, P-L., Delay induced oscillations in autonomous drug delivery systems 11, 491--507 (2004), , Nonlinear Stud.
Erratum to: ``Oscillations in cyclical neutropenia: new evidence based on mathematical modeling''
Bernard, Samuel, Bélair, Jacques et Mackey, Micahel C., Erratum to: ``Oscillations in cyclical neutropenia: new evidence based on mathematical modeling'' 228, 143 (2004), , J. Theoret. Biol.
Oscillations in cyclical neutropenia: new evidence based on mathematical modeling
Bernard, Samuel, Bélair, Jacques et Mackey, Michael C., Oscillations in cyclical neutropenia: new evidence based on mathematical modeling 223, 283--298 (2003), , J. Theoret. Biol.
Restrictions and unfolding of double Hopf bifurcation in functional differential equations
Buono, Pietro-Luciano et Bélair, Jacques, Restrictions and unfolding of double Hopf bifurcation in functional differential equations 189, 234--266 (2003), , J. Differential Equations
Numerical bifurcation analysis of delay differential equations arising from physiological modeling
Engelborghs, K., Lemaire, V., Bélair, J. et Roose, D., Numerical bifurcation analysis of delay differential equations arising from physiological modeling 42, 361--385 (2001), , J. Math. Biol.
Sufficient conditions for stability of linear differential equations with distributed delay
Bernard, Samuel, Bélair, Jacques et Mackey, Michael C., Sufficient conditions for stability of linear differential equations with distributed delay 1, 233--256 (2001), , Discrete Contin. Dyn. Syst. Ser. B
Resonant codimension two bifurcation in the harmonic oscillator with delayed forcing
Campbell, Sue Ann et Bélair, Jacques, Resonant codimension two bifurcation in the harmonic oscillator with delayed forcing 7, 217--238 (1999), , Canad. Appl. Math. Quart.
Stability analysis of an age-structured model with a state-dependent delay
Bélair, Jacques, Stability analysis of an age-structured model with a state-dependent delay 6, 305--319 (1998), , Canad. Appl. Math. Quart.
Bifurcations, stability, and monotonicity properties of a delayed neural network model
Olien, Leonard et Bélair, Jacques, Bifurcations, stability, and monotonicity properties of a delayed neural network model 102, 349--363 (1997), , Phys. D
Stability in a three-dimensional system of delay-differential equations
Bélair, Jacques et Dufour, Steven, Stability in a three-dimensional system of delay-differential equations 4, 135--156 (1996), , Canad. Appl. Math. Quart.
Frustration, stability, and delay-induced oscillations in a neural network model
Bélair, Jacques, Campbell, Sue Ann et van den Driessche, P., Frustration, stability, and delay-induced oscillations in a neural network model 56, 245--255 (1996), , SIAM J. Appl. Math.
Limit cycles, tori, and complex dynamics in a second-order differential equation with delayed negative feedback
Campbell, Sue Ann, Bélair, Jacques, Ohira, Toru et Milton, John, Limit cycles, tori, and complex dynamics in a second-order differential equation with delayed negative feedback 7, 213--236 (1995), , J. Dynam. Differential Equations
Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback
Campbell, Sue Ann, Bélair, Jacques, Ohira, Toru et Milton, John, Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback 5, 640--645 (1995), , Chaos
Stability and bifurcations of equilibria in a multiple-delayed differential equation
Bélair, Jacques et Campbell, Sue Ann, Stability and bifurcations of equilibria in a multiple-delayed differential equation 54, 1402--1424 (1994), , SIAM J. Appl. Math.
Stability in a model of a delayed neural network
Bélair, Jacques, Stability in a model of a delayed neural network 5, 607--623 (1993), , J. Dynam. Differential Equations
Chaos, noise, and extinction in models of population growth
Milton, John G. et Bélair, Jacques, Chaos, noise, and extinction in models of population growth 37, 273--290 (1990), , Theoret. Population Biol.
A circle map in a human heart
Courtemanche, M., Glass, L., Bélair, J., Scagliotti, D. et Gordon, D., A circle map in a human heart 40, 299--310 (1989), , Phys. D
Consumer memory and price fluctuations in commodity markets: an integrodifferential model
Bélair, Jacques et Mackey, Michael C., Consumer memory and price fluctuations in commodity markets: an integrodifferential model 1, 299--325 (1989), , J. Dynam. Differential Equations
Itinerary of a discontinuous map from the continued fraction expansion
Bélair, Jacques et Milton, John G., Itinerary of a discontinuous map from the continued fraction expansion 1, 339--342 (1988), , Appl. Math. Lett.
Growth properties of the solutions of a linear functional-differential equation
Bélair, Jacques et Giroux, André, Growth properties of the solutions of a linear functional-differential equation 134, 125--128 (1988), , J. Math. Anal. Appl.
Sur le calcul de la dimension fractale
Bélair, Jacques, Sur le calcul de la dimension fractale 11, 7--23 (1987), , Ann. Sci. Math. Québec
Periodic pulsatile stimulation of a nonlinear oscillator
Bélair, Jacques, Periodic pulsatile stimulation of a nonlinear oscillator 24, 217--232 (1986), , J. Math. Biol.
Universality and self-similarity in the bifurcations of circle maps
Bélair, Jacques et Glass, Leon, Universality and self-similarity in the bifurcations of circle maps 16, 143--154 (1985), , Phys. D
On linearly coupled relaxation oscillations
Bélair, Jacques et Holmes, Philip, On linearly coupled relaxation oscillations 42, 193--219 (1984), , Quart. Appl. Math.
Global bifurcations of a periodically forced biological oscillator
Glass, Leon, Guevara, Michael R., Bélair, Jacques et Shrier, Alvin, Global bifurcations of a periodically forced biological oscillator 29, 1348--1357 (1984), , Phys. Rev. A (3)
Self-similarity in periodically forced oscillators
Belair, J. et Glass, Leon, Self-similarity in periodically forced oscillators 96, 113--116 (1983), , Phys. Lett. A
Sur une équation différentielle fonctionnelle analytique
Bélair, Jacques, Sur une équation différentielle fonctionnelle analytique 24, 43--46 (1981), , Canad. Math. Bull.
