Irina Volinsky

The stabilization of solutions by distributed feedback control functions for second- and third-order ordinary differential equations (ODEs) has been presented in earlier studies. The present paper extends these results to the stabilization of n-th order ODEs using a distributed control function expressed in integral form with first-order derivatives. The problem of stabilizing n-th order ODE solutions by distributed control functions is significantly more complex and nontrivial. This work introduces a method for selecting the parameter set within the distributed control function. Furthermore, the connection between palindromic polynomials, log-concavity, and stability with respect to initial conditions (Lyapunov stability) in n-th order ODEs with distributed feedback control functions is established. We use the symmetry property of palindromic polynomials.

Integro‐differential equations (IDE) have been applied in a variety of areas of research, including epidemiology. Recently, IDE systems were applied to study dengue fever transmission dynamics at the population level. In this study, we extend the approach presented in a previous study for describing the epidemiological model of dengue fever. In this paper, we find the exact solutions of the corresponding linearized system model by constructing the Cauchy and Green's matrices. Furthermore, we extend our investigation towards estimating the solutions of a closely related nonlinear system. Given that the coefficients of the model are only known approximately, we address the problem of the stability of a system with uncertain coefficients. Furthermore, we estimate the influence of errors in the coefficient determination on solution behavior.

The deterministic AI system under review is an alternative to neural-network-based machine learning. In its application fields, which are science, technology, engineering, and business, the implementation of rule-based AI systems leads to benefits such as accuracy and correctness of design, and personalization of the process itself and the results. An algorithmic AI suite is based on design and logical imitation models alone, without creating and/or using Big Data and knowledge bases. Excessive complexity of configuration and high design resource capacity, which are inherent in deterministic systems, are balanced by a special methodology. A hierarchical modeling approach gives a quasi-dynamic network effect, symmetric to the analogous effect in neural networks. System performance is improved by deterministic reference training capable of modifying imitation models in online interaction with users. Such training, which serves as an alternative to neural machine learning, can be implemented by means of experimental partially empirical algorithms and system–user dialogues to build reference model libraries (portfolios). Partially empirical algorithms based on experimental design methods and system user dialogues are used to create reference model libraries (portfolios) that form a deterministic training system, which can be an alternative to neural machine learning. Estimated resources can be saved by using modified optimization techniques and by controlling the computational complexity of the algorithms. Since the proposed system in the considered layout has no analogues, and the relevant research and practical knowledge are extremely limited, special methods are required to implement this project. A gradual, phased implementation process involves the step-by-step formation of sets of algorithms with verification tests at each stage. Each test is performed using an iteration method, and each test includes test, tweak, and modification cycles. Final testing should lead to the development of an AI algorithm package, including related methodological and working papers.

In the current paper, we demonstrate a new approach for an stabilization criteria for n-order functional-differential equation with distributed feedback control in the integral form. We present a correlation between the order of the functional-differential equation and degree of freedom of the distributed control function. We present two cases of distributed control function in the integral form. Such a case of stabilization control functions plays a very important role in physics, aeronautics, aerospace, ship navigation and traffic network control management. Structure of functional-differential equations is based on the symmetry properties.

In the current paper, the classification of the equilibrium points of an HBV mathematical model with combined therapy is presented. The influence of right-hand side changes on solution behavior is estimated, and regulation with delays in upper- and lower-bound integral limits that presents a time period with IL-2 support therapy are researched.

We present a theoretical study of bladder cancer treatment with Bacillus Calmette-Guerin (BCG) and interleukin 2 (IL-2) using a system biology approach to translate the treatment process into a mathematical model. We investigated the influence of IL-2 on effector cell proliferation, presented as a distributed feedback control in integral form. The variables in the system of Ordinary Differential Equations (ODE) are the main participants in the immune response after BCG instillations: BCG, immune cells, tumor cells infected with BCG, and non-infected with BCG. IL-2 was involved in the tumor-immune response without adding a new equation. We use the idea of reducing the system of integro-differential equations (IDE) to a system of ODE and examine the local stability analysis of the tumor-free equilibrium state of the model. A significant result of the model analysis is the requirements for the IL-2 dose and duration, depending on the treatment regimen and tumor growth. We proved that the BCG+IL-2 treatment protocol is more effective in this model, using the spectral radius method. Moreover, we introduced a parameter for individual control of IL-2 in each injection using the Cauchy matrix for the IDE system, and we obtained conditions under which this system would be exponentially stable in a tumor-free equilibrium.