Analysis of a time-delayed mathematical model for tumour growth with inhibitors

In this article, we study the fully non-stationary version of a mathematical model for tumour growth under indirect effect of inhibitor with time delay in proliferation. The quasi-stationary version has been studied by our previous work [S. Xu and Z. Feng, Analysis of a mathematical model for tumour growth under indirect effect of inhibitors with time delay in proliferation, J. Math. Anal. Appl. 374 (2011), pp. 178–186]. The existence and uniqueness of a global solution are proved and the asymptotic behaviour of the solution is studied. The results show that the dynamical behaviour of solutions of the fully non-stationary and the quasi-stationary version are similar under some conditions.     

Analysis of a mathematical model for tumor growth under indirect effect of inhibitors with time delay in proliferation

In this paper, a mathematical model for tumor growth with time delay in proliferation under indirect effect of inhibitor is studied. The delay represents the time taken for cells to undergo mitosis. Nonnegativity of solutions is investigated. The steady-state analysis is presented with respect to the magnitude of the delay. Existence of Hopf bifurcation is proved for some parameter values. Local and global stability of the stationary solutions are proved for other ones. The analysis of the effect of inhibitor's parameters on tumor's growth is presented. The results show that dynamical behavior of solutions of this model is similar to that of solutions for corresponding non-retarded problems for some parameter values.     

On a mathematical model of immune competition

This work deals with the qualitative analysis of a nonlinear integro-differential model of immune competition with special attention to the dynamics of tumor cells contrasted by the immune system. The analysis gives evidence of how initial conditions and parameters influence the asymptotic behavior of the solutions.     

On the null controllability of a one-dimensional fluid–solid interaction model

We analyze the null controllability of a one-dimensional nonlinear system which models the interaction of a fluid and a particle. The fluid is governed by the Burgers equation and the control is exerted on the boundary points. We present two main results: the global null controllability of a linearized system and the local null controllability of the nonlinear original model. The proofs rely on appropriate global Carleman inequalities and fixed point arguments. To cite this article: A. Doubova, E. Fernández-Cara, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Exact internal controllability of a one-dimensional aeroelastic plate

This paper considers a control problem associated with a one-dimensional elastic plate with self-induced aerodynamic pressure. When the support of control lies in a prescribed subinterval, exact controllability is established by deriving a key energy estimate. By means of this estimate, it is also proved that the energy of a beam with partial damping decays exponentially fast.This research was supported by AFOSR-89-0268.     

On a mathematical model for laser-induced thermotherapy

We study a mathematical model for laser-induced thermotherapy, a minimally invasive cancer treatment. The model consists of a diffusion approximation of the radiation transport equation coupled to a bio-heat equation and a model to describe the evolution of the coagulated zone. Special emphasis is laid on a refined model of the applicator device, accounting for the effect of coolant flow inside. Comparisons between experiment and simulations show that the model is able to predict the experimentally achieved temperatures reasonably well.     

Analysis of a delayed mathematical model for tumor growth

In this paper, a delayed mathematical model of a nonlinear reaction–diffusion equations modeling the growth of tumors is studied. The establishment of the model is based on the diffusion of nutrient and mass conservation for the two-process proliferation and apoptosis (cell death due to aging). It is assumed that the process of proliferation is delayed compared to apoptosis.Nonnegativity of the solutions and stability of stationary solutions are studied in the paper. The results show that the dynamical behavior of the solutions of the model is similar to that of the solutions for the corresponding non-retarded problem under some assumptions.     

Nonlinear mathematical model of pulsed-therapy of heterogeneous tumors

In this paper, we deal with a nonlinear impulsive differential equations modelling the chemotherapy of a heterogeneous tumor. We consider the case of several drugs with instantaneous effects. We take into account the interactions between sensitive cells and drug resistant cells. We are interested in the stability of the disease. We also study the loss of stability and the bifurcation of nontrivial solutions.     

On a mathematical model of a coupled fluid-structure problem

An existence theorem is proved relative to a mathematical model associated to the fluid circulation in an elastic domain.     

Bifurcation analysis of amathematical model for the growth of solid tumors in the presence of external inhibitors

We study bifurcations from radial solution of a free boundary problem modeling the dormant state of nonnecrotic solid tumors in the presence of external inhibitors. This problem consists in three linear elliptic equations with two Dirichlet and one Neumann boundary conditions and a fourth boundary condition coupling surface tension effects on free boundary. In this paper, surface tension coefficient γ plays the role of bifurcation parameter. We prove that in certain situations there exists a positive null point sequence for γ where bifurcation occurs from radial solution, while in the other situations, either bifurcation occurs at only finite many points of γ or even it does not occur for any γ > 0. Copyright © 2014 John Wiley & Sons, Ltd.     

On the controllability of a robot arm

Considered is the rotation of a robot arm or rod in a horizontal plane about an axis through the arm's fixed end and driven by a motor whose torque is controlled. The model was derived and investigated computationally by Sakawa and co-authors in [7] for the case that the arm is described as a homogeneous Euler beam. The resulting equation of motion is a partial differential equation of the type of a wave equation which is linear with respect to the state, if the control is fixed, and non-linear with respect to the control. Considered is the problem of steering the beam, within a given time interval, from the position of rest for the angle zero into the position of rest under a certain given angle. At first we show that, for every L²-control, there is exactly one (weak) solution of the initial boundary value problem which describes the vibrating system without the end condition. Then we show that the problem of controllability is equivalent to a non-linear moment problem. This, however, is not exactly solvable. Therefore, an iteration method is developed which leads to an approximate solution of sufficient accuracy in two steps. This method is numerically implemented and demonstrated by an example. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On the solvability of a mathematical model for prion proliferation

We show that a model describing the interaction between normal and infectious prion proteins admits global solutions. More precisely, supposing the involved degradation rates to be bounded, we prove global existence and uniqueness of classical solutions. Based on this existence theory, we provide sufficient conditions for the existence of global weak solutions in the case of unbounded splitting rates. Moreover, we prove global stability of the disease-free steady state.     

Null controllability in a fluid-solid structure model

We consider a system coupling the Stokes equations in a two-dimensional domain with a structure equation which is a system of ordinary differential equations corresponding to a finite dimensional approximation of equations modeling deformations of an elastic body or vibrations of a rigid body. For that system we establish a null controllability result for localized distributed controls acting only in the fluid equations and there is no control in the solid part. This controllability result follows from a Carleman inequality that we prove for the adjoint system.     

On a socioeconomic mathematical model and its calibration

This paper is concerned with an operational issue arising in the use of spatial interaction models for solving actual planning problems. In particular it introduces an axiom about the concept of distance detterence and hence a calibration procedure. Such procedure is then used to calibrate two models actually being used to plan the development of a large Italian town and the results obtained on the adherence of the model to the problem being modelled improve by about 50% those given by a currently used calibration procedure.     

Bifurcation analysis of a delayed mathematical model for tumor growth

In this study, we present a modified mathematical model of tumor growth by introducing discrete time delay in interaction terms. The model describes the interaction between tumor cells, healthy tissue cells (host cells) and immune effector cells. The goal of this study is to obtain a better compatibility with reality for which we introduced the discrete time delay in the interaction between tumor cells and host cells. We investigate the local stability of the non-negative equilibria and the existence of Hopf-bifurcation by considering the discrete time delay as a bifurcation parameter. We estimate the length of delay to preserve the stability of bifurcating periodic solutions, which gives an idea about the mode of action for controlling oscillations in the tumor growth. Numerical simulations of the model confirm the analytical findings.     

On the topological structure of a mathematical model of human unconscious

On the basis of two our previous works, in this paper, following Jacques Lacan psychoanalytic theory, we wish to outline some further remarks on the topological structure of a mathematical model of human unconscious.