Alexander S. Bratus

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. Another consequence of the suggested approach is a parametric solution to the system of equations determining the quasispecies. Using this parametric solution we show that our approach leads to exact asymptotic results in some cases, which are not covered by the existing methods. In particular, we are able to present not only the limit behavior of the leading eigenvalue (mean population fitness), but also the exact formulas for the limit quasispecies eigenvector for special cases. For instance, this eigenvector has a geometric distribution in the case of the classical single peaked fitness landscape. On the biological side, we propose a mathematical definition, based on the closeness of the quasispecies to the binomial distribution, which can be used as an operational definition of the notorious error threshold. Using this definition, we suggest two approximate formulas to estimate the critical mutation rate after which the quasispecies delocalization occurs.

Solution of the Feedback Control Problem in the Mathematical Model of Leukaemia Therapy

A mathematical model of leukaemia therapy based on the Gompertzian law of cell growth is investigated. The effect of the medicine on the leukaemia and normal cells is described in terms of therapy functions. A feedback control problem with the purpose of minimizing the number of the leukaemia cells while retaining as much as possible the number of normal cells is considered. This problem is reduced to solving the nonlinear Hamilton–Jacobi–Bellman partial differential equation. The feedback control synthesis is obtained by constructing an exact analytical solution to the corresponding Hamilton–Jacobi–Bellman equation.

Optimal radiation fractionation for low-grade gliomas: Insights from a mathematical model

We study optimal radiotherapy fractionations for low-grade glioma using mathematical models. Both space-independent and space-dependent models are studied. Two different optimization criteria have been developed, the first one accounting for the global effect of the tumor mass on the disease symptoms and the second one related to the delay of the malignant transformation of the tumor. The models are studied theoretically and numerically using the method of feasible directions. We have searched for optimal distributions of the daily doses dj in the standard protocol of 30 fractions using both models and the two different optimization criteria. The optimal results found in all cases are minor deviations from the standard protocol and provide only marginal potential gains. Thus, our results support the optimality of current radiation fractionations over the standard 6 week treatment period. This is also in agreement with the observation that minor variations of the fractionation have failed to provide measurable gains in survival or progression free survival, pointing out to a certain optimality of the current approach.

An optimal strategy for leukemia therapy: a multi-objective approach

In this work we introduce a multi-objective optimization problem using the example of a leukemia treatment model. We believe that treatment affects not only leukemia cells, but also the healthy cells. The treatment effect is modelled as a therapy function. The optimization problem con- sists of two objective functions that are in conflict: on the o ne hand, minimizing the leukemia cells and on the other hand, maximizing the number of healthy cells. We reduce this multi-objective prob- lem by using the e-constraint method. With the aid of Pontryagins Maximum Pr inciple we give an analytical solution to this reduced problem. In order to sol ve the problem using the epsilon-constraint- method, the restriction of a threshold value for the number of healthy cells is separately considered as an optimization problem with a new extended objective function. For the most relevant parameters the maximum dose of chemotherapeutics should be administered as long as the predetermined restrictions are not violated. Furthermore, the case in which singular control may occur during the therapy process is analysed. In this case, the optimal control is also determ ined.

Evolutionary Games with Randomly Changing Payoff Matrices

Evolutionary games are used in various fields stretching from economics to biology. In most of these games a constant payoff matrix is assumed, although some works also consider dynamic payoff matrices. In this article we assume a possibility of switching the system between two regimes with different sets of payoff matrices. Potentially such a model can qualitatively describe the development of bacterial or cancer cells with a mutator gene present. A finite population evolutionary game is studied. The model describes the simplest version of annealed disorder in the payoff matrix and is exactly solvable at the large population limit. We analyze the dynamics of the model, and derive the equations for both the maximum and the variance of the distribution using the Hamilton–Jacobi equation formalism.

On assessing quality of therapy in non-linear distributed mathematical models for brain tumor growth dynamics

In this paper a mathematical model for glioma therapy based on the Gompertzian law of cell growth is presented. In the common case the model is considered with non-linear spatially varying diffusion depending on a parameter. The case of the linear spatially-varying diffusion arose as a special case for a particular value of the parameter. Effectiveness of the medicine is described in terms of a therapy function. At any given moment the amount of the applied chemotherapeutic agent is regulated by a control function with a bounded maximum. Additionally, the total quantity of chemotherapeutic agent which can be used during the treatment process is bounded. The main goal of the work is to compare the quality of the optimal strategy of treatment with the quality of another one, proposed by the authors and called the alternative strategy. As the criterion of the quality of the treatment, the amount of the cancer cells at the end of the therapy is chosen. The authors concentrate their efforts on finding a good estimate for the lower bound of the cost-function. Thus it becomes possible to compare the quality of the optimal treatment strategy with the quality of the alternative treatment strategy without explicitly finding the optimal control function.

On the reaction–diffusion replicator systems: spatial patterns and asymptotic behaviour

The replicator equation is ubiquitous for many areas of mathematical biology. One of major shortcomings of this equation is that it does not allow for an explicit spatial structure. Here we review analytical approaches to include spatial variables to the system. We also provide a concise exposition of the results concerning the appearance of spatial patterns in replicator reaction-diffusion systems.

On optimal and suboptimal treatment strategies for a mathematical model of leukemia.

In this work an optimization problem for a leukemia treatment model based on the Gompertzian law of cell growth is considered. The quantities of the leukemic and of the healthy cells at the end of the therapy are chosen as the criterion of the treatment quality. In the case where the number of healthy cells at the end of the therapy is higher than a chosen desired number, an analytical solution of the optimization problem for a wide class of therapy processes is given. If this is not the case, a control strategy called alternative is suggested.

Implementing a GPU-based numerical algorithm for modelling dynamics of a high-speed train

ABSTRACT This paper discusses the initiative of implementing a GPU-based numerical algorithm for studying various phenomena associated with dynamics of a high-speed railway transport. The proposed numerical algorithm for calculating a critical speed of the bogie is based on the first Lyapunov number. Numerical algorithm is validated by analytical results, derived for a simple model. A dynamic model of a carriage connected to a new dual-wheelset flexible bogie is studied for linear and dry friction damping. Numerical results obtained by CPU, MPU and GPU approaches are compared and appropriateness of these methods is discussed.

Optimal bounded noisy feedback control for damping random vibrations

We consider a stochastic optimal feedback control problem for a single-degree-of-freedom vibrational system, where uncertainty is described by two independent noises. The first of them is induced by the control actions and called internal, whereas the second one acts externally. The drift vector also depends on the control function. The set of pointwise control constraints is assumed to be bounded. The minimization functional is taken as the mean system response energy. The Cauchy problem for the corresponding Hamilton–Jacobi–Bellman (HJB) equation without the control constraints is first investigated. This allows us to find the sought-for feedback control strategy in a specific domain of the space of state and time variables. Then a proper extension to the remaining parts of the space is constructed, and the optimality of the resulting global feedback control strategy is proved. The obtained control law is compared with the dry friction and saturated viscous friction control laws.