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.     

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.     

Stability of solutions to a mathematical model for necrotic tumor growth with time delays in proliferation

In this paper, a mathematical model for a solid avascular tumor growth is studied. The model describes tumor growth with a necrotic core and a time delay in proliferation process. The model was proposed by Byrne and Chaplain, and was studied by M. Bodnar and U. Fory? (see [2]). Sufficient conditions which guarantee existence, uniqueness and stability of steady state are given. 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. Our results partially improve the corresponding results given by M. Bodnar and U. Fory?. The results make the research for this model more perfect.     

样条曲线光顺的数学模型分析

采用函数三次样条光顺曲线,证明在样条曲线局部转角小,总转角不超过120°情况下,曲线的光顺指示函数y″(1+y′2)3/2可以简化为二阶导数曲线y″(x).由于y″(x)对x是分段折线函数,对y是线性泛函,因而定出不光顺之处及用叠加原理计算调整公式均变得很简单.此样条函数曲线光顺能够采用电脑自动化进行.     

Analysis of a mathematical predator-prey model with delay

We study the behavior of dynamic processes in a mathematical predator-prey model and show that the dynamical system may have a periodic solution whose period coincides with the delay. By the bifurcation method for stability analysis of periodic solutions, we establish that this periodic solution is unstable.     

Positive pseudo almost periodic solutions for a delayed differential neoclassical growth model

This paper is concerned with a non-autonomous delayed differential neoclassical growth model with an oscillating death rate. Under proper conditions, we employ a novel argument to establish a criterion on the existence and global exponential stability of positive pseudo almost periodic solution, which improves and extends some known relevant results. Finally, we present an example along with its numerical simulations to demonstrate the validity of the proposed result.     

Global stability of a delayed ratio-dependent predator-prey model with Gompertz growth for prey

A delayed ratio-dependent predator-prey model with Gompertz growth for prey is investigated. The local stability of a predator-extinction equilibrium and a coexistence equilibrium is discussed. Furthermore, the existence of Hopf bifurcation at the coexistence equilibrium is established. By constructing a Lyapunov functional, sufficient conditions are obtained for the global stability of the coexistence equilibrium.     

Global dynamics of a mathematical model for HTLV-I infection of CD4 T cells with delayed CTL response

Human T-cell leukaemia virus type I (HTLV-I) preferentially infects the CD4⁺ T cells. The HTLV-I infection causes a strong HTLV-I specific immune response from CD8⁺ cytotoxic T cells (CTLs). The persistent cytotoxicity of the CTL is believed to contribute to the development of a progressive neurologic disease, HTLV-I associated myelopathy/tropical spastic paraparesis (HAM/TSP). We investigate the global dynamics of a mathematical model for the CTL response to HTLV-I infection in vivo. To account for a series of immunological events leading to the CTL response, we incorporate a time delay in the response term. Our mathematical analysis establishes that the global dynamics are determined by two threshold parameters R₀ and R₁, basic reproduction numbers for viral infection and for CTL response, respectively. If R₀≤1, the infection-free equilibrium P₀ is globally asymptotically stable, and the HTLV-I viruses are cleared. If R₁≤1<R₀, the asymptomatic-carrier equilibrium P₁ is globally asymptotically stable, and the HTLV-I infection becomes chronic but with no persistent CTL response. If R₁>1, a unique HAM/TSP equilibrium P₂ exists, at which the HTLV-I infection is chronic with a persistent CTL response. We show that the time delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations and stable periodic oscillations. Implications of our results to the pathogenesis of HTLV-I infection and HAM/TSP development are discussed.     

Analysis of numerical schemes of a mathematical model for sickle cell depolymerization

The purpose of this paper is to present and discuss numerical schemes for a mathematical model that describes carbon monoxide mediated sickle cell polymer melting. Two Runge-Kutta methods are analyzed and shown to be unstable by calculating the first failure value of step size and displaying the bifurcation diagram of RK4. Two nonstandard finite difference (NSFD) schemes are proposed and analyzed; one is shown to be stable subject to a predictable bound on step size, while the second one is unconditionally stable.     

Permanence and extinction of a stochastic hybrid model for tumor growth

A tumor growth model perturbed by both white noise and Markov switching is formulated and explored. The threshold between permanence and extinction is obtained. Some effects of environmental stochasticity on permanence and extinction of the model are revealed.     

Uniform convergence for the incompressible limit of a tumor growth model

We study a model introduced by Perthame and Vauchelet [19] that describes the growth of a tumor governed by Brinkman's Law, which takes into account friction between the tumor cells. We adopt the viscosity solution approach to establish an optimal uniform convergence result of the tumor density as well as the pressure in the incompressible limit. The system lacks standard maximum principle, and thus modification of the usual approach is necessary.     

Qualitative analysis of a delayed free boundary problem for tumor growth under the effect of inhibitors

In this paper we study a delayed free boundary problem for the growth of tumors under the effect of inhibitors. The establishing of the model is based on the diffusion of nutrient and inhibitors, and mass conservation for the two processes proliferation and apoptosis. It is assumed that the process of proliferation is delayed compared to apoptosis. We mainly study the asymptotic behavior of the solution, and prove that under some assumptions, in the case where c₁ and c₂ are sufficiently small, the volume of the tumor cannot expand without limit; it will either disappear or evolve to a dormant state as t→∞.     

Temporal and spatiotemporal variations in a mathematical model of macrophage–tumor interaction

In the present paper we study a three-component mathematical model of tumor–immune system interaction. A number of solid tumors contain a high proportion of macrophages and these immune cells are known to have a remarkable impact on the progression and dormancy of such tumors. We assume these macrophages as the main immune system component facilitating tumor destruction. Stability criteria of the basic model around the steady state of coexistence are derived. Next, we consider the process of macrophage activation as non-instantaneous by using a distributed delay with a weak kernel and obtain a range for the macrophage death rate that ensures system stability. Finally, we incorporate the spatial irregularity of solid tumors by making the delay nonlocal. Analysis of the resulting spatiotemporal model gives a number of thresholds in terms of different system parameters that guarantee tumor stability. Numerical simulations are performed to justify analytical findings.     

An analysis of a mathematical model describing acid‐mediated tumor invasion

We present a mathematical analysis of a reaction‐diffusion model describing acid‐mediated tumor invasion. The model describes the spatial distribution and temporal evolution of tumor cells, normal cells, and excess lactic acid concentration. The model assumes that tumor‐induced alteration of microenvironmental pH provides a simple but complete mechanism for cancer invasion. We provide results on the existence and uniqueness of a solution considering Neumann boundary condition and comments about the same results with Dirichlet boundary conditions. We also provide numerical simulations to the solutions considering both boundary conditions.     

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.     

Qualitative analysis for a mathematical model of aids

In this paper a mathematical model of AIDS is investigated. The conditions of the existence of equilibria and local stability of equilibria are given. The existences of transcritical bifurcation and Hopf bifurcation are also considered, in particular, the conditions for the existence of Hopf bifurcation can be given in terms of the coefficients of the characteristic equation. The method extends the application of the Hopf bifurcation theorem to higher differential equations which occur in biological models, chemical models, and epidemiological models etc.This project is supported by the National Science Foundation Tian Yuan Terms and LNM Institute of Mechanics Academy of Science.This project is supported by the National and Yunnan Province Natural Science Foundation of China.