Maximization of viability time in a mathematical model of cancer therapy.

In this paper, we study a dynamic optimization problem for a general nonlinear mathematical model for therapy of a lethal form of cancer. The model describes how the populations of cancer and normal cells evolve under the influence of the concentrations of nutrients (oxygen, glucose, etc.) and the applied therapeutic agent (drug).

[Reasons to reconsider interpretation of densitometry for corneal transparency].

Aim: To assess the effect of light scattering in the corneal epithelium on densitometric brightness of the stroma through mathematical modeling of the interaction between a light beam and the two-layer epithelium-stroma system.

Material And Methods: In order to study the scattering behavior of a plane-parallel non-coherent beam at the epithelium-stroma interface, a multi-age group was formed (87 patients, 174 eyes) that comprised two subgroups with equal number of assign participants: healthy patients with no systemic changes and mixed patients with undisturbed corneal transparency that, nevertheless, were under instillation therapy. In the first subgroup, the assessment of light scattering was done at random times, while in the second subgroup - within the first 200 seconds after the instillation in order to avoid a reaction of the ocular surface structures, including epithelium (rapid response of epithelial cells to instillations).

On Eigen's Quasispecies Model, Two-Valued Fitness Landscapes, and Isometry Groups Acting on Finite Metric Spaces.

A two-valued fitness landscape is introduced for the classical Eigen's quasispecies model. This fitness landscape can be considered as a direct generalization of the so-called single- or sharply peaked landscape. A general, non-permutation invariant quasispecies model is studied, and therefore the dimension of the problem is [Formula: see text], where N is the sequence length.

Exact solutions for the selection-mutation equilibrium in the Crow-Kimura evolutionary model.

We reformulate the eigenvalue problem for the selection-mutation equilibrium distribution in the case of a haploid asexually reproduced population in the form of an equation for an unknown probability generating function of this distribution. The special form of this equation in the infinite sequence limit allows us to obtain analytically the steady state distributions for a number of particular cases of the fitness landscape. The general approach is illustrated by examples; theoretical findings are compared with numerical calculations.

On the behavior of the leading eigenvalue of Eigen's evolutionary matrices.

We study general properties of the leading eigenvalue w¯(q) of Eigen's evolutionary matrices depending on the replication fidelity q. This is a linear algebra problem that has various applications in theoretical biology, including such diverse fields as the origin of life, evolution of cancer progression, and virus evolution. We present the exact expressions for w¯(q),w¯(')(q),w¯('')(q) for q = 0, 0.

Linear algebra of the permutation invariant Crow-Kimura model of prebiotic evolution.

A particular case of the famous quasispecies model - the Crow-Kimura model with a permutation invariant fitness landscape - is investigated. Using the fact that the mutation matrix in the case of a permutation invariant fitness landscape has a special tridiagonal form, a change of the basis is suggested such that in the new coordinates a number of analytical results can be obtained. In particular, using the eigenvectors of the mutation matrix as the new basis, we show that the quasispecies distribution approaches a binomial one and give simple estimates for the speed of convergence.

Numerical and asymptotic approach for evaluating complex wavenumbers of guided modes in viscoelastic plates.

The approximate description of the dispersion curves is obtained using asymptotics of complex wavenumbers for different boundary conditions on the plate surfaces. Their comparison with the exact results shows satisfactory agreement. This approach provides an algorithm to evaluate the infinite spectrum of non-propagating modes more easily and numerically stable even for wavenumbers of big values.

A note on the replicator equation with explicit space and global regulation.

A replicator equation with explicit space and global regulation is considered. This model provides a natural framework to follow frequencies of species that are distributed in the space. For this model, analogues to classical notions of the Nash equilibrium and evolutionary stable state are provided.

Existence and stability of stationary solutions to spatially extended autocatalytic and hypercyclic systems under global regulation and with nonlinear growth rates.

Analytical analysis of spatially extended autocatalytic and hypercyclic systems is presented. It is shown that spatially explicit systems in the form of reaction-diffusion equations with global regulation possess the same major qualitative features as the corresponding local models. In particular, using the introduced notion of the stability in the mean integral sense we prove the competitive exclusion principle for the autocatalytic system and the permanence for the hypercycle system.