A note on the replicator equation with explicit space and global regulation
Alexander S. Bratus Vladimir P. Posvyanskii Artem S. Novozhilov
Mathematical Biosciences & Engineering 2011, 8(3): 659-676 doi: 10.3934/mbe.2011.8.659
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. A sufficient condition for a uniform stationary state to be a spatially distributed evolutionary stable state is presented and illustrated with examples.
keywords: evolutionary stable state reaction-diffusion systems. Nash equilibrium Replicator equation
On optimal and suboptimal treatment strategies for a mathematical model of leukemia
Elena Fimmel Yury S. Semenov Alexander S. Bratus
Mathematical Biosciences & Engineering 2013, 10(1): 151-165 doi: 10.3934/mbe.2013.10.151
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.
keywords: cancer model. Optimal therapy control chemotherapy
On viable therapy strategy for a mathematical spatial cancer model describing the dynamics of malignant and healthy cells
Alexander S. Bratus Svetlana Yu. Kovalenko Elena Fimmel
Mathematical Biosciences & Engineering 2015, 12(1): 163-183 doi: 10.3934/mbe.2015.12.163
A mathematical spatial cancer model of the interaction between a drug and both malignant and healthy cells is considered. It is assumed that the drug influences negative malignant cells as well as healthy ones. The mathematical model considered consists of three nonlinear parabolic partial differential equations which describe spatial dynamics of malignant cells as well as healthy ones, and of the concentration of the drug. Additionally, we assume some phase constraints for the number of the malignant and the healthy cells and for the total dose of the drug during the whole treatment process.
    We search through all the courses of treatment switching between an application of the drug with the maximum intensity (intensive therapy phase) and discontinuing administering of the drug (relaxation phase) with the objective of achieving the maximum possible therapy (survival) time. We will call the therapy a viable treatment strategy.
keywords: viable therapy strategy chemotherapy. Spatial cancer model optimal therapy control

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