Applications and dynamics of delay differential equations with threshold state-dependent delay

Delay differential equations (DDEs) with state-dependent delays defined by threshold conditions are studied.Threshold delays are implicitly determined by history states of the system over a bounded time interval.While threshold delays enhance model realism, they also pose analytical challenges.I study three different systems modeled by threshold DDEs, providing tangible strategies for tackling the threshold delays analytically and numerically to study their dynamics and bifurcations.Specifically, one approach focuses on reformulating the threshold condition through differentiation and stabilization when necessary to solve initial value problems.Another approach involves approximating the threshold delay by discretizing the distributed delay interval to solve a boundary value problem.Additionally, the effect of different delay forms, whether as a discrete constant or a discrete state-dependent delay, is examined in relation to the dynamics of the threshold delay problem.In Chapter 2 we consider an operon model with three differential equations for gene expression, accounting for time delays in DNA transcription, mRNA translation, and also models cell growth.We develop numerical techniques to solve the equations as initial value problems, and to perform numerical continuation and bifurcation detection.This is done by differentiating the threshold condition and discretizing the threshold delay.The implementations allow us to use standard software packages that are unable to handle threshold-type delays directly.Through numerical exploration of the bifurcations and dynamics of this system, we find that the inclusion of threshold delays broadens the possible operon dynamics.Namely, repressible systems may exhibit multistability and higher order bifurcation structures, and inducible systems may have stable periodic solutions.In Chapter 3 we analyze a scalar DDE modeling a transport process with a non-zero completion time, where the transport speed depends on the system's state.In contrast to the previous chapter, this model's scalar nature makes an in-depth analysis feasible.We perform systematic analytical and numerical studies of the various combinations of positive or negative feedback in the monotone nonlinear response function and the transport velocity to identify invariant objects including steady states, periodic orbits and their bifurcations.We identify rich bifurcation structures including Bautin, Bogdanov-Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depends on the ii Abstract iii signs of the first derivative of the nonlinearities.Our analysis confirms that many of these patterns persist in the limit where the nonlinearities are switch-like and change their value abruptly at a threshold.In Chapter 4 we study an immuno-epidemiological model describing pathogen transmission in a large population.The threshold delay in the model, which takes a more complex form than in the previous two models, accounts for the latency period which can be shortened by multiple exposures during the exposed stage.For this system, we analyze the solution behavior near the steady states, and contrast it with the behavior resulting from using the differentiated form and the discrete state-dependent form of the threshold delay respectively.While both forms leave the steady-state value unchanged, they affect stability: the differentiated form introduces a spurious positive eigenvalue, while the discrete state-dependent form alters the eigenvalue set.Since our previous numerical algorithm for solving initial value problems involves differentiating the threshold condition, which is unstable for this model, we remove the instability by adding a penalty term to the differentiated threshold condition, which ensures that the spurious eigenvalue is negative.We also provide details on finding a discrete approximation for the threshold delay, which is essential for solving boundary value problems.Résumé v systématiques des différentes combinaisons de rétroaction positive ou négative dans la fonction de réponse non linéaire monotone et la vitesse de transport afin d'identifier les objets invariants, y compris les états stables, les orbites périodiques et leurs bifurcations.Nous identifions de riches structures de bifurcation, notamment des bifurcations de Bautin, de Bogdanov-Takens, de Hopf ainsi que des bifurcations homocliniques, des fronces et des col-noeuds homocliniques, dont l'existence dépend du signe de la dérivée première des non-linéarités.Notre analyse confirme que nombre de ces modèles persistent dans la limite où les non-linéarités changent brusquement de valeur à une valeur seuil.Dans le chapitre 4, nous étudions un modèle immuno-épidémiologique qui décrit la transmission d'un agent pathogène dans une grande population.Le retard défini par condition de seuil dans le modèle, qui prend une forme plus complexe que dans les deux modèles précédents, tient compte de la période de latence qui peut être raccourcie par des expositions multiples au cours de la phase exposée.Pour ce système, nous analysons le comportement de la solution près des états stables et le comparons au comportement résultant de l'utilisation de la condition de seuil différenciée et de la forme discrétisée du retard (retard discret dépendant de l'état), respectivement.Bien que ces deux formes ne changent pas la valeur de l'état d'équilibre, elles affectent la stabilité de celui-ci: la forme différenciée introduit une valeur propre positive parasite, tandis que le retard discret dépendant de l'état modifie l'ensemble des valeurs propres.Puisque notre algorithme numérique susmentionné pour résoudre les problèmes de valeurs initiales nécessite de différencier la condition de seuil, qui est instable pour ce modèle, nous éliminons l'instabilité en ajoutant un terme de pénalité à la condition de seuil différenciée, garantissant ainsi que la valeur propre parasite est négative.Nous apportons également des précisions sur la recherche d'une approximation discrète pour les retards définis par conditions de seuil qui est essentielle pour résoudre les problèmes aux limites. CONTRIBUTION TO ORIGINAL KNOWLEDGEWe propose numerical techniques for handling threshold state-dependent delays, with implementations that enable the solution of initial value problems in ddesd and numerical bifurcation analysis in DDE-BIFTOOL.For solving initial value problems, we describe a reformulation of the threshold condition by differentiating it to derive an additional discrete delay differential equation, as outlined in Section 2.4.3.In Section 4.3, we present an enhanced implementation that addresses the issue of numerical instability when it arises.For solving boundary value problems, we outline an approach in Sections 2.4 and 4.5, which involves discretizing the threshold condition to approximate the delay.Additionally, in Section 2.4.1, we propose a linearization correction method that, while offering limited access to DDE-BIFTOOL's full functionality, is computationally inexpensive.We also develop a heuristic linearization approach designed to be accessible to a broader scientific audience than Banach space analysis.In Section 2.3 and Section 4.2.1, we provide a detailed derivation of the characteristic equation for the threshold delay model.Using this approach, we investigate how freezing the delay at its steady-state value or treating it as a discrete state-dependent delay affects the dynamics of the model, as discussed in Section 4.2.3.We also apply this approach to address the numerical instability that arises when differentiating the threshold condition, as described in Section 4.2.2.In Chapter 2 we discover novel potential operon dynamics in the presence of statedependent delays in both repressible and inducible gene regulatory systems.These include the existence of relaxation type oscillations and multistability.In Chapter 3, we investigate the complex dynamics of a scalar delay differential equation with a threshold delay.We numerically identify bifurcations which are novel in threshold delay problems including Bogdanov-Takens bifurcation with the associated nearby fold, Hopf and homoclinic bifurcations (see Figure 3.14), fold-Hopf bifurcations and evidence for the existence of a double homoclinic bifurcation (see Figure 3.13).In Chapter 4, we overcome the numerical and analytical obstacles to directly investigate a previously published immuno-epidemiological model with threshold delay.We demonstrate that the dynamics of the model are changed by the simplifications made by previous authors that involved replacing the threshold delay by a discrete state-dependent delay.We emphasize that it is essential to be able to model directly with threshold delays to accurately analyze these problems.vi

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