Оптимальные возмущения систем с запаздывающим аргументом для управления динамикой инфекционных заболеваний на основе многокомпонентных воздействий

In this paper, we apply optimal perturbations to control mathematical models of infectious diseases expressed as systems of nonlinear differential equations with delayed argument. We develop the method for calculation of perturbations of the initial state of a dynamical system with delayed argument producing maximal amplification in the given local norm taking into account weights of perturbation components. For the model of experimental virus infection, we construct optimal perturbation for two types of stationary states, with low or high virus load, corresponding to different variants of chronic virus infection flow.Работа посвящена применению оптимальных возмущений для управления математическими моделями инфекционных заболеваний, сформулированными в виде систем нелинейных дифференциальных уравнений с запаздывающим аргументом. Разработан алгоритм вычисления возмущений начального состояния динамической системы с запаздыванием, обладающих максимальной амплификацией в заданной локальной норме с учетом значимости компонент возмущения. Для модели экспериментальной вирусной инфекции построены оптимальные возмущения для двух типов стационарных состояний, с низким и высоким уровнем вирусной нагрузки, отвечающих различным вариантам течения хронической вирусной инфекции

Infinitely many local higher symmetries without recursion operator or master symmetry: integrability of the Foursov--Burgers system revisited

We consider the Burgers-type system studied by Foursov, w_t &=& w_{xx} + 8 w w_x + (2-4\alpha)z z_x, z_t &=& (1-2\alpha)z_{xx} - 4\alpha z w_x + (4-8\alpha)w z_x - (4+8\alpha)w^2 z + (-2+4\alpha)z^3, (*) for which no recursion operator or master symmetry was known so far, and prove that the system (*) admits infinitely many local generalized symmetries that are constructed using a nonlocal {\em two-term} recursion relation rather than from a recursion operator.Comment: 10 pages, LaTeX; minor changes in terminology; some references and definitions adde

Global stability for a class of virus models with CTL immune response and antigenic variation

We study the global stability of a class of models for in-vivo virus dynamics, that take into account the CTL immune response and display antigenic variation. This class includes a number of models that have been extensively used to model HIV dynamics. We show that models in this class are globally asymptotically stable, under mild hypothesis, by using appropriate Lyapunov functions. We also characterise the stable equilibrium points for the entire biologically relevant parameter range. As a byproduct, we are able to determine what is the diversity of the persistent strains.Comment: 15 page

Hybrid models in biomedical applications

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Erratum: Mathematical immunology: From phenomenological to multiphysics modelling (Russ. J. Numer. Anal. Math. Modelling (2021) 35: 4 (203-213) DOI: 10.1515/rnam-2020-0017)

The inequality (2.2) in the original manuscript should be read as (γVFF∗ + γVMM∗ + γVCC∗) (bCEE∗ + bm) > σC∗ (ν + nbCEE∗). © 2021 De Gruyter. All rights reserved

Mathematical immunology: From phenomenological to multiphysics modelling

Mathematical immunology is the branch of mathematics dealing with the application of mathematical methods and computational algorithms to explore the structure, dynamics, organization and regulation of the immune system in health and disease. We review the conceptual and mathematical foundation of modelling in immunology formulated by Guri I. Marchuk. The current frontier studies concerning the development of multiscale multiphysics integrative models of the immune system are presented. © 2020 Walter de Gruyter GmbH, Berlin/Boston 2020

Reaction–diffusion equations in immunology

The paper is devoted to the recent works on reaction–diffusion models of virus infection dynamics in human and animal organisms. Various regimes of infection propagation in tissues are described. In particular, it is shown that infection can spread in tissues of organs as a reaction–diffusion wave. Methods for studying the conditions of the existence of wave modes of the time and space dynamics of infections is discussed. © Pleiades Publishing, Ltd., 2018

Hybrid models in biomedical applications

The paper presents a review of recent developments of hybrid discrete-continuous models in cell population dynamics. Such models are widely used in the biological modelling. Cells are considered as individual objects which can divide, die by apoptosis, differentiate and move under external forces. In the simplest representation cells are considered as soft spheres, and their motion is described by Newton's second law for their centers. In a more complete representation, cell geometry and structure can be taken into account. Cell fate is determined by concentrations of intra-cellular substances and by various substances in the extracellular matrix, such as nutrients, hormones, growth factors. Intra-cellular regulatory networks are described by ordinary differential equations while extracellular species by partial differential equations. We illustrate the application of this approach with some examples including bacteria filament and tumor growth. These examples are followed by more detailed studies of erythropoiesis and immune response. Erythrocytes are produced in the bone marrow in small cellular units called erythroblastic islands. Each island is formed by a central macrophage surrounded by erythroid progenitors in different stages of maturity. Their choice between self-renewal, differentiation and apoptosis is determined by the ERK/Fas regulation and by a growth factor produced by the macrophage. Normal functioning of erythropoiesis can be compromised by the development of multiple myeloma, a malignant blood disorder which leads to a destruction of erythroblastic islands and to sever anemia. The last part of the work is devoted to the applications of hybrid models to study immune response and the development of viral infection. A two-scale model describing processes in a lymph node and other organs including the blood compartment is presented. © 2019 Nikolai M. Bessonov, Gennady A. Bocharov, Anass Bouchnita, Vitaly A. Volper

Control of models of virus infections with delayed variables, based on optimal disturbances

Abstract: A new method for constructing the multi-modal impacts on the immune system in the chronic phase of a viral infection, based on the mathematical models with delayed argument, was proposed. So called, optimal disturbances, widely used in the aerodynamic stability theory with models without delays, were proposed for perturbing the steady states of the system for maximizing the perturbation-induced response. The concept of optimal disturbances was generalized on the systems with delayed argument. An algorithm for computing the optimal disturbances was proposed for such systems. The developed technology was tested using a system of four nonlinear delay-differential equations with delayed time which represents the model of experimental infection in mice caused by lymphocytic choriomeningitis virus. Steady-state perturbations causing the maximum response were computed with the proposed algorithm for two types of steady states: with low and with high level of viral load. The possibility of correction of the infection dynamics and restoration of function of virus specific lymphocytes of immune system by perturbing the steady states was demonstrated.Note: Research direction:Mathematical modelling in actual problems of science and technic

Optimal Perturbations of Systems with Delayed Independent Variables for Control of Dynamics of Infectious Diseases Based on Multicomponent Actions

In this paper, we apply optimal perturbations to control mathematical models of infectious diseases expressed as systems of nonlinear differential equations with delayed independent variables. We develop the method for calculation of perturbations of the initial state of a dynamical system with delayed independent variable producing maximal amplification in the given local norm taking into account weights of perturbation components. For the model of experimental virus infection, we construct optimal perturbation for two types of stationary states, with low or high viral load, corresponding to different variants of chronic virus infection flow. © 2021, Springer Science+Business Media, LLC, part of Springer Nature