Localization Of Compact Invariant Sets Research Articles

Ultimate Bounds for a Diabetes Mathematical Model Considering Glucose Homeostasis

This paper deals with a recently reported mathematical model formulated by five first-order ordinary differential equations that describe glucoregulatory dynamics. As main contributions, we found a localization domain with all compact invariant sets; we settled on sufficient conditions for the existence of a bounded positively-invariant domain. We applied the localization of compact invariant sets and Lyapunov’s direct methods to obtain these results. The localization results establish the maximum cell concentration for each variable. On the other hand, Lyapunov’s direct method provides sufficient conditions for the bounded positively-invariant domain to attract all trajectories with non-negative initial conditions. Further, we illustrate our analytical results with numerical simulations. Overall, our results are valuable information for a better understanding of this disease. Bounds and attractive domains are crucial tools to design practical applications such as insulin controllers or in silico experiments. In addition, the model can be used to understand the long-term dynamics of the system.

Personalized Immunotherapy Treatment Strategies for a Dynamical System of Chronic Myelogenous Leukemia.

Simple SummaryAs computer performance continues to grow at more affordable costs, mathematical modelling and in silico experimentation begin to play a larger role in understanding cancer evolution. The aim of our work is to formulate a control strategy for the Adaptive Cellular Therapy (ACT) that can fully eradicates the Chronic Myelogenous Leukemia (CML) cells population in a mathematical model describing interactions between naive T cells, effector T cells and CML cancer cells in the circulatory blood system. Mathematical analysis and numerical simulations allow us to conclude that it is possible to design a personalized administration protocol for the ACT in the form of a pulse train with asymmetrical waves and a fixed amplitude to achieve complete CML cancer cells eradication. The amplitude of the impulse on which the treatment is applied is given by an arithmetical combination of the parameters of the system with at least a duty cycle of 45 min/day.This paper is devoted to exploring personalized applications of cellular immunotherapy as a control strategy for the treatment of chronic myelogenous leukemia described by a dynamical system of three first-order ordinary differential equations. The latter was achieved by applying both the Localization of Compact Invariant Sets and Lyapunov’s stability theory. Combination of these two approaches allows us to establish sufficient conditions on the immunotherapy treatment parameter to ensure the complete eradication of the leukemia cancer cells. These conditions are given in terms of the system parameters and by performing several in silico experimentations, we formulated a protocol for the therapy application that completely eradicates the leukemia cancer cells population for different initial tumour concentrations. The formulated protocol does not dangerously increase the effector T cells population. Further, complete eradication is considered when solutions go below a finite critical value below which cancer cells cannot longer persist; i.e., one cancer cell. Numerical simulations are consistent with our analytical results.

A numerical approach of tumor‐immune model with B cells and monoclonal antibody drug by multi‐step differential transformation method

Monoclonal antibody (mAb) drugs are used to kill malignant tumors by making them able to see into the immune system. In this article, we developed a mathematical model with the help of tumor‐immune system with antibodies effect and efficacy of mAb drug. We defined the existence of solutions of the model, and by the Lyapunov method, we showed that the solutions are bounded; thereafter, we illustrated the local and global stability phenomenon, which is described by the Lyapunov method and localization of compact invariant sets (LCIS) method for disease‐free equilibrium points including the biological significance. In actual fact, this model suggested some ideas about the malignant tumors reciprocate to mAb drug dosage and its efficacy, eventually to overcome malignancy. It is likely to control the steadiness of the tumor, which is very important for any kind of treatment. The numerical calculations for the proposed model have been carried out by multi‐step differential transformation method (Ms‐DTM). The simulation of the model is depicted through Figures 1–5, which shows the effect of interaction rate of tumor cells and antibodies and dose of the drug for various values of efficacy.

Nonlinear Analysis for a Type-1 Diabetes Model with Focus on T-Cells and Pancreatic β-Cells Behavior

Type-1 diabetes mellitus (T1DM) is an autoimmune disease that has an impact on mortality due to the destruction of insulin-producing pancreatic β -cells in the islets of Langerhans. Over the past few years, the interest in analyzing this type of disease, either in a biological or mathematical sense, has relied on the search for a treatment that guarantees full control of glucose levels. Mathematical models inspired by natural phenomena, are proposed under the prey–predator scheme. T1DM fits in this scheme due to the complicated relationship between pancreatic β -cell population growth and leukocyte population growth via the immune response. In this scenario, β -cells represent the prey, and leukocytes the predator. This paper studies the global dynamics of T1DM reported by Magombedze et al. in 2010. This model describes the interaction of resting macrophages, activated macrophages, antigen cells, autolytic T-cells, and β -cells. Therefore, the localization of compact invariant sets is applied to provide a bounded positive invariant domain in which one can ensure that once the dynamics of the T1DM enter into this domain, they will remain bounded with a maximum and minimum value. Furthermore, we analyzed this model in a closed-loop scenario based on nonlinear control theory, and proposed bases for possible control inputs, complementing the model with them. These entries are based on the existing relationship between cell–cell interaction and the role that they play in the unchaining of a diabetic condition. The closed-loop analysis aims to give a deeper understanding of the impact of autolytic T-cells and the nature of the β -cell population interaction with the innate immune system response. This analysis strengthens the proposal, providing a system free of this illness—that is, a condition wherein the pancreatic β -cell population holds and there are no antigen cells labeled by the activated macrophages.

Bounding the Dynamics of a Chaotic-Cancer Mathematical Model

The complexity of cancer has motivated the development of different approaches to understand the dynamics of this large group of diseases. One that may allow us to better comprehend the behavior of cancer cells, in both short- and long-term, is mathematical modelling through ordinary differential equations. Several ODE mathematical models concerning tumor evolution and immune response have been formulated through the years, but only a few may exhibit chaotic attractors and oscillations such as stable limit cycles and periodic orbits; these dynamics are not that common among cancer systems. In this paper, we apply the Localization of Compact Invariant Sets (LCIS) method and Lyapunov stability theory to investigate the global dynamics and the main factors involved in tumor growth and immune response for a chaotic-cancer system presented by Itik and Banks in 2010. The LCIS method allows us to compute what we define as the localizing domain, which is formulated by the intersection of all lower and upper bounds of each cells population in the nonnegative octant, R+,03. Bounds of this domain are given by inequalities in terms of the system parameters. Then, we apply Lyapunov stability theory and LaSalle’s invariance principle to establish existence conditions of a global attractor. The latter implies that given any nonnegative initial condition, all trajectories will go to the largest compact invariant set (a stable equilibrium point, limit cycles, periodic orbits, or a chaotic attractor) located either inside or at the boundaries of the localizing domain. In order to complement our analysis, numerical simulations are performed throughout the paper to illustrate all mathematical results and to better explain their biological implications.

Global dynamics of a colorectal cancer treatment model with cancer stem cells.

We present and analyze a mathematical model of the treatment of colorectal cancer using a system of nonlinear ordinary differential equations. The model describes the effectiveness of immunotherapy and chemotherapy for treatment of tumor cells and cancer stem cells (CSCs). The effects of CD8+T cells, natural killer cells, and interleukin proteins on tumor cells and CSCs under the influence of treatment are also illustrated. Using the method of localization of compact invariant sets, we present conditions on treatment parameters to guarantee a globally attracting tumor clearance state. Numerical simulations using estimated parameters from the literature are included to showcase various global dynamics of the model.

Nonlinear speed sensorless control of a surface-mounted PMSM based on a Thau observer

This paper presents an alternative to solve the speed sensorless control of a surface-mounted synchronous motor based on localization of compact invariant sets (LCIS) and the Thau observer. Through the LCIS, the domain of attraction of the system dynamics is analyzed, allowing to obtain global knowledge about its operational bounds and its associated Lipschitz constant. Necessary and sufficient conditions for existence of a stable observer are fulfilled by two inequalities, providing two different sets of stability conditions and, as a consequence, two observers are proposed. The observer design is based on the availability of stator currents for measurement and stator voltages for feedback in a rotating reference frame. The designed observers are able to work in a wide speed range and also estimate rotor position accurately, even at low speed and zero-crossing speed. Simulations demonstrate that the observers can estimate both rotor speed and position. Additionally, the observers are experimentally validated with the Technosoft\(^{\textregistered }\) MCK28335 platform. Results show that the observers can solve sensorless problem in a real-world scenario.

Localizing sets and trajectory behavior

The problem is considered of finding domains in the phase space in which the trajectories of a system have a fairly simple behavior determined by a typical scenario. The problem is solved by applying the method of localization of compact invariant sets of the system. It is proved that localizing sets separate simple and complex dynamics of nonlinear systems, namely, in the complement of any localizing set, the trajectory behavior of a system admits a standard description in the form of several scenarios, while, in localizing sets and their intersections, the trajectory behavior of the system can be very complex. Specifically, it is shown that the α- and ω-limit sets of any trajectory are contained in localizing sets.

Asymptotic stability analysis of autonomous systems by applying the method of localization of compact invariant sets

The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is proposed for verifying the fulfillment of these conditions. Illustrative examples are given.

Dynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapy.

Understanding the global interaction dynamics between tumor and the immune system plays a key role in the advancement of cancer therapy. Bunimovich-Mendrazitsky et al. (2015) developed a mathematical model for the study of the immune system response to combined therapy for bladder cancer with Bacillus Calmette-Guérin (BCG) and interleukin-2 (IL-2) . We utilized a mathematical approach for bladder cancer treatment model for derivation of ultimate upper and lower bounds and proving dissipativity property in the sense of Levinson. Furthermore, tumor clearance conditions for BCG treatment of bladder cancer are presented. Our method is based on localization of compact invariant sets and may be exploited for a prediction of the cells populations dynamics involved into the model.

Localization of compact invariant sets and global stability in analysis of one tumor growth model

In this paper, we study some features of global behavior of the four‐dimensional superficial bladder cancer model with Bacillus Calmette‐Guérin (BCG) immunotherapy described by Bunimovich‐Mendrazitsky et al. in 2007 with the help of localization analysis of its compact invariant sets. Its dynamics is defined by the BCG treatment and by densities of three cells populations: effector cells, tumor infected cells by BCG, and tumor uninfected cells. We find upper bounds for ultimate dynamics of the whole state vector in the positive orthant and also under condition that there are no uninfected tumor cells. Further, we prove the existence of the bounded positively invariant domain in both of these two situations. Finally, by using these assertions, we derive our main result: sufficient conditions of global asymptotic stability of the tumor‐free equilibrium point in the positive orthant. Copyright © 2013 John Wiley & Sons, Ltd.

Localization of compact invariant sets of continuous-time systems with disturbance

Localization of compact invariant sets of continuous-time systems with disturbance

Velocity Estimation via Thau Observer for a PMSM in Chaotic Regime and its Physical Validation

In this paper, we design a Thau observer for the permanent-magnet synchronous motor (PMSM) model, which estimates the angular velocity from the measurement of currents. We apply some Bounds obtained from the localization of compact invariant sets of the system describing dynamics of a PMSM. From these bounds, based on the dissipative property of the model, we calculate parameters of a Thau observer. In our work, we take the case where the PMSM presents chaotic like behavior in order to validate the effectiveness of the proposed observer. Simulations are realized to verify our results. Additionally, some circuit experiments are implemented validating the observer convergence in a physical environment.

Localization of compact invariant sets of a new nonlinear finance chaotic system

The application of the nonlinear chaotic dynamic system in economics and finance has expanded rapidly in the last decades. This paper considers the localization of all compact invariant sets of a new three-dimensional autonomous nonlinear finance chaotic system. The boundedness of the new finance chaotic system is the first time being investigated. Based on the iteration theorem and the first-order extremum theorem, a new method is proposed, too. The comparison of our method with the traditional method is presented as well. More specifically, the compact invariant sets are analyzed in three aspects: First of all, a localization of the new finance chaotic system by two frusta and an ellipsoidal used by traditional methods is discussed. Second, a localization of the new finance chaotic system by two frusta and a parabolic cylinder is provided. Third, localization of the new finance chaotic system according to superposition of the ellipsoidal, parabolic cylinder, and two frusta are presented, and the boundedness of the chaotic attracter is smaller than in the classical methods. Numerical simulations are given to indicate the effectiveness of the proposed method.

Localization of compact invariant sets of discrete-time control systems

Localization of compact invariant sets of discrete-time control systems

Localization of compact invariant sets of discrete-time systems with disturbances

Localization of compact invariant sets of discrete-time systems with disturbances

Bounds for compact invariant sets of the system describing dynamics of the nuclear spin generator

This paper is concerned with the localization problem of compact invariant sets of the system describing dynamics of the nuclear spin generator. We establish that all compact invariant sets of this system are located in the intersection of a ball with two frusta and compute its parameters. In addition, localization by using the two-parameter set of parabolic cylinders is described. Our results are obtained with help of the iteration theorem concerning a localization of compact invariant sets. One numerical example illustrating a localization of a chaotic attractor is presented as well.

Localization of compact invariant sets of the Lorenz system

The problem of finding domains in the state space of a nonlinear system which contain all compact invariant sets is considered. Such domains are computed for the Lorenz system by using different localizing functions.