scholarly journals Response of Immunotherapy to Tumour-TICLs Interactions: A Travelling Wave Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Joseph Malinzi ◽  
Precious Sibanda ◽  
Hermane Mambili-Mamboundou
Keyword(s):  
Phase Space ◽  
Mathematical Modelling ◽  
Wave Speed ◽  
Interaction Model ◽  
Travelling Wave ◽  
Tumour Invasion ◽  
Wave Analysis ◽  
Stable And Unstable Manifolds ◽  
Minimum Wave Speed ◽  
Unstable Manifolds

There are several cancers for which effective treatment has not yet been identified. Mathematical modelling can nevertheless point out to clinicians tumour invasion properties that should be targeted to mitigate these cancers. We present a travelling wave analysis of a tumour-immune interaction model with immunotherapy. We use the geometric treatment of an apt-phase space to establish the intersection between stable and unstable manifolds. We calculate the minimum wave speed and numerical simulations are performed to support the analytical results.

scholarly journals Travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion

2021 ◽  
Vol 477 (2256) ◽  
Author(s):  
Chloé Colson ◽  
Faustino Sánchez-Garduño ◽  
Helen M. Byrne ◽  
Philip K. Maini ◽  
Tommaso Lorenzi
Keyword(s):  
Wave Solution ◽  
Degradation Rate ◽  
Wave Speed ◽  
Propagation Speed ◽  
Travelling Wave ◽  
Travelling Wave Solution ◽  
Tumour Invasion ◽  
Wave Analysis ◽  
Shooting Argument ◽  
Minimal Wave Speed

In this paper, we carry out a travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion. We consider two types of invasive fronts of tumour tissue into extracellular matrix (ECM), which represents healthy tissue. These types differ according to whether the density of ECM far ahead of the wave front is maximal or not. In the former case, we use a shooting argument to prove that there exists a unique travelling-wave solution for any positive propagation speed. In the latter case, we further develop this argument to prove that there exists a unique travelling-wave solution for any propagation speed greater than or equal to a strictly positive minimal wave speed. Using a combination of analytical and numerical results, we conjecture that the minimal wave speed depends monotonically on the degradation rate of ECM by tumour cells and the ECM density far ahead of the front.

scholarly journals Travelling wave solutions in a negative nonlinear diffusion–reaction model

2020 ◽  
Vol 81 (6-7) ◽  
pp. 1495-1522
Author(s):  
Yifei Li ◽  
Peter van Heijster ◽  
Robert Marangell ◽  
Matthew J. Simpson
Keyword(s):  
Nonlinear Diffusion ◽  
Wave Speed ◽  
Geometric Approach ◽  
Travelling Wave ◽  
Reaction Model ◽  
Diffusion Reaction ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Minimum Wave Speed ◽  
Nonlinear Diffusion Reaction Equation

AbstractWe use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion–reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of interest. We determine the minimum wave speed, $$c^*$$ c ∗ , and investigate its relation to the spectral stability of a desingularised linear operator associated with the travelling wave solutions.

The Role of Time-Dependent Phase Space Structures in Reaction Dynamics and the No-Recrossing Property of Dividing Surfaces

2021 ◽  
Vol 31 (04) ◽  
pp. 2150064
Author(s):  
Cate Mandell ◽  
Stephen Wiggins
Keyword(s):  
Phase Space ◽  
Time Dependence ◽  
Reaction Dynamics ◽  
Time Dependent ◽  
Space Structures ◽  
Stable And Unstable Manifolds ◽  
Normally Hyperbolic Invariant Manifold ◽  
Unstable Manifolds ◽  
Dividing Surface

We analyze benchmark models for reaction dynamics associated with a time-dependent saddle point. Our model allows us to incorporate time dependence of a general form, subject to an exponential growth restriction. Under these conditions, we analytically compute the time-dependent normally hyperbolic invariant manifold; its time-dependent stable and unstable manifolds; and a time-dependent dividing surface that has the no-recrossing property. Consideration of the time dependence of these phase space structures is necessary in order to precisely capture reacting and nonreacting trajectories. Moreover, we show that a time-dependent dividing surface is necessary in order to eliminate recrossing in the time-dependent setting. In other words, if the dividing surface is not time-dependent, recrossing may occur.

scholarly journals Mathematical modelling for the transmission of dengue: Symmetry and travelling wave analysis

2018 ◽  
Vol 41 ◽  
pp. 269-287 ◽  
Author(s):  
Felipo Bacani ◽  
Stylianos Dimas ◽  
Igor Leite Freire ◽  
Norberto Anibal Maidana ◽  
Mariano Torrisi
Keyword(s):  
Mathematical Modelling ◽  
Travelling Wave ◽  
Wave Analysis

Study on the Geometrical Properties and Existence of Orbit Homoclinic to a Saddle Point in n-Dimensional Autonomous Vector Field with Polynomials

2018 ◽  
Vol 28 (14) ◽  
pp. 1850169
Author(s):  
Lingli Xie
Keyword(s):  
Vector Field ◽  
Saddle Point ◽  
Equilibrium Point ◽  
Necessary Conditions ◽  
Continuous System ◽  
Homoclinic Bifurcation ◽  
Stable And Unstable Manifolds ◽  
Geometrical Properties ◽  
Unstable Manifolds

According to the theory of stable and unstable manifolds of an equilibrium point, we firstly find out some geometrical properties of orbits on the stable and unstable manifolds of a saddle point under some brief conditions of nonlinear terms composed of polynomials for [Formula: see text]-dimensional time continuous system. These properties show that the orbits on stable and unstable manifolds of the saddle point will stay on the corresponding stable and unstable subspaces in the [Formula: see text]-neighborhood of the saddle point. Furthermore, the necessary conditions of existence for orbit homoclinic to a saddle point are exposed. Some examples including homoclinic bifurcation are given to indicate the application of the results. Finally, the conclusions are presented.

scholarly journals Hadamard–Perron theorems and effective hyperbolicity

2014 ◽  
Vol 36 (1) ◽  
pp. 23-63 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
YAKOV PESIN
Keyword(s):  
Closing Lemma ◽  
Stable And Unstable Manifolds ◽  
Local Dynamics ◽  
Local Diffeomorphisms ◽  
Finite Amount ◽  
Amount Of Information ◽  
Unstable Manifolds

We prove several new versions of the Hadamard–Perron theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard–Perron theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of ‘effective hyperbolicity’ and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite-orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.

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