The effects of tumor angiogenesis factor (TAF) and tumor inhibitor factors (TIFs) on tumor vascularization: A mathematical model

A mathematical model, which examines the effects of Tumor Angiogenesis Factor (TAF) and Tumor Inhibitor Factors (TIFs) on tumor angiogenesis and predicts the onset of vascularization, is presented. The TAF and TIFs are produced within the tumor, while in the prevascular stage, by a layer of viable proliferating cancer cells on the tumor boundary. When the concentrations of TAF and TIFs have reached a critical level, they are released into the surrounding tissue. If TAF and TIFs have penetrated the tissue to the extent that they can reach the tips of the neighboring capillaries, then regulation of the formation of new blood vessels begins. The present model describes this process in three stages, and the appropriate diffusion equations for the production and secretion of TAF and TIFs are solved in spherical geometry. The concentrations of these chemical substances are monitored and the rate of growth of the capillary boundary, which moves towards the tumor surface marking the onset of vascularization, is determined.     

Qualitative analysis of a mathematical model for tissue inflammation dynamics

The non-linear differential eguation $$\dot X = \frac{{rX}}{{I + X}} - \frac{{XY}}{{X + Y}},\dot Y = \alpha (I + \theta X - Y)$$ is studied. It is the main aim of this paper to show the existence of bifurcation of saddle connection type; and to show the creation of limit cycles under certain conditions of the parameters, together with their biological significance.     

Qualitative dynamics, root-loci and asymptotic structure of universal adaptive stabilizers

A problem of universal adaptive stabilization in Rⁿ is approachedin a qualitative manner to relate the form of the trajectoriesin the extended state space Rⁿ⁺¹ to the root locus of the associatedfixed-parameter linear system. Relationships are derived betweenthe values of the limit gain and the initial conditions. Numericalstudies are used to illustrate and support these ideas by computationof generic trajectories and attempted computation of certainnon-generic possibilities. The implications of the study formore general dynamic situations are outlined. These authors are also with the Department of Mathematics,University of Exeter, EX4 4QE.     

Plastic flow in a conical channel: Qualitative features of the solutions under different yield conditions

The problem of the convergence of the solutions of problems of plasticity theory, with a yield condition which depends on the hydrostatic stress, to solutions based on classical plasticity theory with von Mises or Tresea conditions is considered, with a particular choice of the parameters of the material model. For the case of axisymmetric flow of material in a channel with converging and diverging walls, solutions according to two plasticity theories with a yield condition which depends on the hydrostatic stress are compared with the classical solution. It is shown that only the solution using Spencer's model shows all the main features of the classical solution. As the internal criterion of the choice of the preferred plasticity theory when examining a special class of problems, it is suggested that the criterion of the convergence of the solutions to the solutions of classical plasticity theory should be used.     

Mathematical model of optimal chemotherapy strategy with allowance for cell population dynamics in a heterogeneous tumor

A mathematical model of tumor cell population dynamics is considered. The tumor is assumed to consist of cells of two types: amenable and resistant to chemotherapeutic treatment. It is assumed that the growth of the cell populations of both types is governed by logistic equations. The effect of a chemotherapeutic drug on the tumor is specified by a therapy function. Two types of therapy functions are considered: a monotonically increasing function and a nonmonotone one with a threshold. In the former case, the effect of a drug on the tumor is stronger at a higher drug concentration. In the latter case, a threshold drug concentration exists above which the effect of the therapy reduces. The case when the total drug amount is subject to an integral constraint is also studied. A similar problem was previously studied in the case of a linear therapy function with no constraint imposed on the drug amount. By applying the Pontryagin maximum principle, necessary optimality conditions are found, which are used to draw important conclusions about the character of the optimal therapy strategy. The optimal control problem of minimizing the total number of tumor cells is solved numerically in the case of a monotone or threshold therapy function with allowance for the integral constraint on the drug amount.     

Some computational features of two-dimensional problems of gas dynamics on irregular triangular grids

We consider a number of computational features of two-dimensional Lagrangian schemes on triangular grids: loss of approximation near the axis (cylindrical symmetry) and measures to counteract this effect; approximation and operator interpretation at corner points where the type of the boundary conditions changes; loss of stability with free approximation of type II boundary conditions; distortion of the Lagrangian grid associated with first-order accuracy on the boundary of an irregular grid.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 10–20, 1985.     

New features of soliton dynamics in 2+1 dimensions

Exponentially localized soliton solutions have been found recently for all the equations of the hierarchy related to the Zakharov-Shabat hyperbolic spectral problem in the plane. In particular theN ²-soliton solution of the Davey-Stewartson I equation is considered. It is shown that the boundaries fix the kinematics of solitons, while the dynamics of their mutual interaction is determined by the chosen initial condition. The interacting solitons can have, asymptotically, zero mass and can simulate quantum effects as inelastic scattering, fusion and fission, creation and annihilation.Work supported in part by M.U.R.S.T.     

Effects of time-lagged niche construction on metapopulation dynamics and environmental heterogeneity

Time-delayed responses to environmental changes and disturbance can beget profound effects on the spatiotemporal dynamics of metapopulations. Here, we first examined the effect of three forms of time-lag (that is, equal-weight, recency and primacy effects) on population dynamics, using a spatially structured lattice model. The time-lag was incorporated in the niche construction process of the system (an organism–environment feedback). Using bifurcations diagrams and numerical simulations, we found that the time-lag can form a phase-locked oscillation. Three typical spatial patterns emerged: spiral wave, spiral-broken wave and circular wave. These spatial patterns gradually become immobile as a result of the self-organized ecological imprinting due to niche construction. Therefore, the phase-locked oscillation and the ecological imprinting process together determine the spatial structure of metapopulations and the environmental heterogeneity.     

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→∞.     

Transmission dynamics of Trichomonas vaginalis: A mathematical approach

Despite the availability of treatment that is effective, Trichomonas vaginalis infections are still high. A deterministic model for transmission dynamics of Trichomonas vaginalis is presented as a system of non-linear differential equations. Analysis of the reproduction number has shown that an increase in the number of straight women (non-lesbians) infected result in an increase in the number of lesbians infected. This suggests that straight women are turning into lesbians already infected. The disease-free equilibrium is shown to be globally asymptotically stable when the corresponding reproduction number is less than unity. Analytical results and numerical simulations both show that treatment is able to control Trichomonas vaginalis infections. This suggests an effective control of trichomoniasis rests in encouraging and persuading sexual partners of those displaying symptoms to seek treatment. Failure for the asymptomatic to seek treatment (mostly males given that the majority of males does not show symptoms) will continue to fuel the infection.     

Characteristic features of the dynamics of the Ginzburg-Landau equation in a plane domain

We study the boundary value problem w_t=ℵ₀Δw+ℵ₁w-ℵ₂w|w|²,w|∂Ω₀=0 in the domain Ω₀={(x,y):0 ≤ x ≤ l₁,0 ≤ y ≤ l₂}. Here, w is a complex-valued function, Δ is the laplace operator, and ℵ_j, j=0,1,2, are complex constants withRe ℵ_j > 0. We show that under a rather general choice of the parameters l₁ and l₂, the number of stable invariant tori in the problem, as well as their dimensions, grows infinitely asRe ℵ₀ → 0 andRe ℵ₀ → 0. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 2, pp. 205–220, November, 2000.     

Wave features of a hyperbolic model for the within-season dynamics of insect pathogens

Hyperbolic models are suitable for describing invasive phenomena with a well-defined boundary. In fact, for a class of hyperbolic reaction–diffusion models derived in the context of extended thermodynamics (ET), the non-existence of smooth travelling waves has been proved under suitable assumptions on the wave speed. In this paper a hyperbolic model for the within-season dynamics of insect pathogens is derived and smooth and discontinuous travelling wave solutions are investigated. Validation of the model in point is also accomplished by searching for numerical solutions of the system of PDEs.