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This paper aims to reveal an interesting phenomenon about the time delays in control systems: stability preservation with variable time delays over large intermittently or unbounded. That is, given a system asymptotically stable for bounded time delays, it could remain stable as the time delays are enhanced to have peak values over large intermittently or unbounded. The investigation is carried out by theoretical analysis plus case studies with the stochastic system models. The stability of systems is the core issue, the explicit involvement of the time delays in the stability criteria is the premise for them being concerned. To these ends, a series of lemmas are established preliminarily, a new type stability theorem with its practical version are established in succession, a stability criterion with the time delays as explicit parameters is induced. The Lyapunov function method is applied for less restriction and conservativeness. The mechanism and approach for the emergence of the phenomenon under probe are analyzed. Implementation is considered for the linear system model. Typical variable time delays are constructed for illustrations. Finally, the theoretical

The newest infection is a novel coronavirus named COVID-19, that initially appeared in December 2019, in Wuhan, China, and is still challenging to control. The main focus of this paper is to investigate a novel fractional-order mathematical model that explains the behavior of COVID-19 in Ethiopia. Within the proposed model, the entire population is divided into nine groups, each with its own set of parameters and initial values. A nonlinear system of fractional differential equations for the model is represented using Caputo fractional derivative. Legendre spectral collocation method is used to convert this system into an algebraic system of equations. An inexact Newton iterative method is used to solve the model system. The effective reproduction number ( R 0 ) is computed by the next-generation matrix approach. Positivity and boundedness, as well as the existence and uniqueness of solution, are all investigated. Both endemic and disease-free equilibrium points, as well as their stability, are carefully studied. We calculated the parameters and starting conditions (ICs) provided for our model using data from the Ethiopian Public Health Institute (EPHI) and the Ethiopian Ministry of Health from 22 June 2020 to 28 February 2021. The model parameters are determined using least squares curve fitting and MATLAB R2020a is used to run numerical results. The basic reproduction number is R 0 = 1.4575 . For this value, disease free equilibrium point is asymptotically unstable and endemic equilibrium point is asymptotically stable, both locally and globally.

Vibrational control seeks to stabilize an unstable system by judiciously injecting a state-free high-frequency dither. This paper presents some robustness properties of vibrational control with respect to additive disturbances in a nonlinear system. We assume that, without disturbances, the appropriately averaged system is regionally asymptotically stable. Using perturbation techniques, our first result shows that the stabilization realised through vibrational control is robust with respect to additive disturbances of sufficiently small amplitude. Indeed, the perturbed system of vibrational control has a robustness feature similar to input-to-state stability in a local region. In the case of periodic disturbances, our second result indicates that vibrational control naturally dampens disturbances of sufficiently high frequency, which allows for relatively high amplitude disturbances. A tight bound for the effect of such periodic disturbances on the ultimate deviation of states from the desired equilibrium is presented. Simulation results from a planar manipulator support the theoretic analysis.

It is well known that the mathematical biology and dynamical systems give very important information for the study and research of viral infection models such as HIV, HBV, HCV, Ebola and Influenza. This paper deals with the global dynamics of generalized virus model with logistic growth rate for target cells, general incidence rate and cellular immunity. The results will be obtained by using Lyapunov’s second method and LaSalle’s invariance principle. We prove the global stability of the rest points of the system by the value of basic reproduction number ( R 0 ) and the immune response reproduction number ( R CTL ) . We have found that if R 0 1 , then the infection-free equilibrium is globally asymptotically stable. For R 0 > 1 and R CTL 1 , under certain conditions on incidence rate function, immune-free equilibrium is globally asymptotically stable. Finally, we prove that if R 0 > 1 and R CTL > 1 , then under certain conditions on incidence rate function the endemic equilibrium is globally asymptotically stable. Since the logistic growth rate for target cells and general incidence rate have been included in this manuscript, our obtained results are the generalization of those in the previous literatures. Moreover, the results have been obtained with weaker assumptions in comparison with the previous ones. Numerical simulations are presented to support and illustrate our analytical results.

A spatio-temporal model for HCV dynamics incorporating the spatial mobility of viruses and immune B cells, with general nonlinear incidence functions for both modes of infection transmission, namely, virus-to-cell as well as cell-to-cell transmission, is proposed and analyzed with the homogeneous Neumann boundary conditions. In addition, the model includes the role of B cell or antibody response and considers the non-cytolytic cure of productively infected hepatocytes. The existential conditions and global stability for the system equilibrium points are examined. The analytical results shows that the three types of equilibria for the model with general incidence functions are globally asymptotically stable under certain conditions along with criteria on the basic reproductive ratio. The numerical illustration of the analytical findings are presented for the case of bilinear as well as Holling type-II incidence function in a one-dimensional spatial domain. For both the incidence functions, the model populations converge to the corresponding levels of the infected or infection-free equilibrium, contingent upon the case of the basic reproductive ratio being greater than unity or not, respectively. Furthermore, it is observed that the diffusive coefficients do not influence the asymptotic behavior of the system. The infection will be eliminated when basic reproductive ratio is less than unity.

This article focuses on investigating the synchronization and position/force tracking performance of the multilateral teleoperation system with asymmetric time delays. First, the synchronization and tracking error signals are proposed to transform the nonlinear dynamic system into a closed-loop system governed by the Markovian jumping parameter. Because of the presence of time delays, the information regarding the system states might become unavailable. Owing to this, a feedback control based on the hidden Markov model is developed through which the information on the current state of the actual system can be accessed through a series of observations. Moreover, the forward and backward delays of the master and slave manipulators are assumed to be asymmetric and varying with time. For the stability analysis, the Lyapunov-Krasovskii technique has been adopted for which its derivatives are dealt by employing Jensens' inequality and extended reciprocal convex matrix inequality. Finally, simulation results were provided to validate the proposed methodology guaranteeing the closed-loop system to be asymptotically stable.

By employing off-diagonal matrix Mittag-Leffler functions and stability theory for line fractional functional differential equations, a new technique is proposed to investigate the existence, uniqueness and global asymptotical stability of S -asymptotically periodic solution for a class of semilinear Caputo fractional functional differential equations. Some better results are derived, which improve and extend the existing research findings in recent years. As an application of the general theory, some decision theorems are established for the asymptotically dynamical behaviors for fractional four-neuron BAM neural networks. The methods used in this paper could be applied to the study of other fractional differential systems in the areas of science and engineering.

In this paper, to characterize the limited availability of medical resources, we incorporate a saturated treatment rate into a network-based susceptible-infected-susceptible (SIS) epidemic model with time delay and nonlinear incidence rate. Analytical study shows the boundedness of solutions, the basic reproduction number R 0 and equilibrium points of the proposed system. For any infection delay, we perform both local and global stability analyses for the disease-free equilibrium point by analyzing the characteristic equation and using Lyapunov functional. Furthermore, this system exhibits bifurcation behavior at R 0 = 1 due to the introduction of saturated treatment. More precisely, a backward bifurcation takes place from the disease-free equilibrium point when the saturation constant β is sufficiently large. Under the given conditions, the unique disease-spreading equilibrium point is also proved to be locally asymptotically stable. In addition, we analyze an optimal control problem with consideration of two time-dependent control measures. Several numerical simulations are presented to validate the obtained theoretical results.

The three-axis attitude tracking control of an underactuated spacecraft in the large-angle maneuver was investigated in this study. As a major contribution of this paper, a robust controller was developed to achieve the perfect attitude tracking of the underactuated spacecraft with consideration of saturation and uncertainties. Interestingly, due to non-singularity of this design, it could relax the burden of limiting the initial condition of the quaternions. The stability analysis of the developed controller could be guaranteed by Lyapunov method as shown here. Overall, the simulation results indicate that the proposed controller has robustness against saturation, external perturbations, inertia uncertainties, and internal disturbances of actuators. As result, the controller was asymptotically stable under the combination of the soft saturation and the perturbations so that attitude parameters converged to the desired path within the 230 s. In this case, the saturation level was consumed 0.08 N.m. Also, it was still asymptotic stable under the hard saturation whose level is equal to 0.01 Nm, 12.5% of the soft saturation level.

The interval type-2 Takagi-Sugeno (T-S) fuzzy dynamic output feedback and H-infinity stability analysis is studied for a class of networked control systems with multiple time-varying additive uncertainties, time-varying signal communication delay and external disturbance . Firstly, the interval type-2T-S fuzzy is employed to denote the system plant. Secondly, the multiple time-varying additive uncertainties are introduced in the controller design and the state variables depending on additive uncertainties. Thirdly, the delay-dependent Lyapunov-Krasovskii functional with double integral terms is designed to derive the less conservative stability conditions in terms of linear matrix inequalities (LMIs). The characteristic of the state variables are reflected effectively by employing the controller with multiple time-varying additive uncertainties. The closed-loop system is asymptotically stable with prescribed H-infinity performance index γ by employing the matrices transformation information. The less conservative stability conditions are derived and extended into the networked control system without additive uncertainties. Finally, simulations are presented to show the effectiveness of the proposed methods.

Climate changes are affecting the control of many vector-borne diseases, particularly in Africa. In this work, a dengue fever model with protected travelers is formulated. Caputo-Fabrizio operator is utilized to obtain some qualitative information about the disease. The basic properties and the reproduction number are studied. The two steady states are determined and the local stability of the states are found to be asymptotically stable. The fixed point theory is used to obtain the existence and uniqueness of solutions of the model. The Adams–Bashforth scheme is employed to solve an approximate solution of the fractional dengue model. The numerical simulation suggests that the fractional-order affects the dynamics of dengue fever.

The problem of finite-time attitude-tracking control (ATC) for a rigid spacecraft subject to inertial uncertainties, external disturbances, actuator saturations and faults is addressed. First, a fast nonsingular terminal sliding mode (FNTSM) manifold is constructed to improve robustness. Second, the fuzzy logic system (FLS) is integrated into the manifold derivative to deal with the lumped unknown function. The specific assumptions about uncertainties in most of the existing achievements are no longer needed. Combining the FNTSM and fuzzy approximation techniques, an enhanced fault-tolerant control scheme is developed. Compared to almost all finite-time ATC results based on FLS or neural network, a new Proof line is proposed to prove that the approximation errors are finite-time stable instead of asymptotically stable. Therefore, the attitude controller presented herein guarantees the real finite-time stability in a complete sense. Finally, numerical experiments are conducted to verify the effectiveness of the solution, and comparisons with related works are displayed. National Natural Science Foundation of ChinaNational Natural Science Foundation of China (NSFC) [61773108] This work is supported by National Natural Science Foundation of China under Grant 61773108.

With the rapid growth of renewable energy and power electronic loads, voltage source converters (VSCs) have been commonly applied in different VSCs-based power systems, which also bring in stability issues due to their power electronic characteristics. The impedance-based stability theory and corresponding reshaping method can be used to analyze and solve system stability issue. However, dc stability issue in VSCs-based power systems becomes more complicated when considering ac/dc coupling relationship. This paper proposes an improved reshaping control strategy based on the detailed impedance model considering grid impedance and control delay. The influence of grid impedance, control delay and control parameters of VSC on dc stability are further analyzed to identify the cause of potential resonance risk. The proposed impedance reshaping method considers the trade-off between stability improvement and control performance, and compensates the impacts of grid impedance and control delay at the same time. The validity of stability analysis and reshaping method is further verified through the simulation and experimental results.

This paper is focused on a strange novel chaotic fractional-order system with fully golden proportion equilibria. By using the stability theory of fractional-order systems, we give sufficient conditions for local stability of such system around equilibrium point. And then we prove the conditions for the existence of Hopf bifurcation. Moreover, delayed feedback control method is used to control the chaotic behavior of system. The results indicate that the fractional system is more stable than the classical system, where the fractional order $$\alpha $$ and time delay $$\tau $$ play an important role in controlling the chaotic dynamics. Finally, by using Adams–Bashforth–Moulton method, we implement some simulations to substantiate the obtained theoretical results.

In this study, we propose and analyse an extended SEIARD model with vaccination. We compute the control reproduction number ℛcof our model and study the stability of equilibria. We show that the set of disease-free equilibria is locally asymptotically stable when ℛc< 1 and unstable when ℛc> 1, and we provide a sufficient condition for its global stability. Furthermore, we perform numerical simulations using the reported data of COVID-19 infections and vaccination in Mexico to study the impact of different vaccination, transmission and efficacy rates on the dynamics of the disease.

In this paper, an SWEIQR epidemic model with transport-related infection is proposed. The model considers inter-patch travel with entry-departure screening. The reproduction number, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{ed}^\phi $$\end{document}Redϕ, is computed and analyzed with respect to awareness and screening parameters. The analytic computations show that the disease-free equilibrium in the absence of travel is globally asymptotically stable when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_\omega \le 1$$\end{document}Rω≤1 and unstable otherwise. The trans-critical bifurcation occurs at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_\omega =1$$\end{document}Rω=1 and the locally stable endemic equilibrium point appears if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_\omega >1$$\end{document}Rω>1 near to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_\omega =1$$\end{document}Rω=1. The numerical simulations are performed to verify the analytical computation and explore the dynamic behavior with respect to different model parameters. The result shows that disseminating awareness through the population reduces the spread of disease. Furthermore, the full model results show that the departure screening may reduce the spread of disease in each patch.

Consider the heterogeneity (e.g., heterogeneous social behaviour, heterogeneity due to different geography, contrasting contact patterns and different numbers of sexual partners etc.) of host population, in this paper, the authors propose an infection age multi-group SEIR epidemic model. The model system also incorporates the feedback variables, where the infectivity of infected individuals may depend on the infection age. In the direction of mathematical analysis of model, the basic reproduction number R0 has been computed. The global stability of disease-free equilibrium and endemic equilibrium have been established in the term of R0. More precisely, for R0 ≤ 1, the disease-free equilibrium is globally asymptotically stable and for R0 > 1, they establish global stability of endemic equilibrium using some graph theoretic techniques to Lyapunov function method. By considering a numerical example, they investigate the effects of infection age and feedback on the prevalence of the disease. Their result shows that feedback parameters have different and even opposite effects on different groups. However, by choosing an appropriate value of feedback parameters, the disease could be eradicated or maintained at endemic level. Besides, the infection age of infected individuals may also change the behaviour of the disease, global stable to damped oscillations or damped oscillations to global stable.

Malaria, one of the longest-known vector-borne diseases, poses a major health problem in tropical and subtropical regions of the world. Its complexity is currently being exacerbated by the emerging COVID-19 pandemic and the threats of its second wave and looming third wave. We formulate and analyze a mathematical model incorporating some epidemiological features of the co-dynamics of both malaria and COVID-19. Sufficient conditions for the stability of the malaria only and COVID-19 only sub-models’ equilibria are derived. The COVID-19 only sub-model has globally asymptotically stable equilibria while under certain condition, the malaria-only could undergo the phenomenon of backward bifurcation whenever the sub-model reproduction number is less than unity. The equilibria of the dual malaria-COVID19 model are locally asymptotically stable as global stability is precluded owing to the possible occurrence of backward bifurcation. Optimal control of the full model to mitigate the spread of both diseases and their co-infection are derived. Pontryagin’s Maximum Principle is applied to establish the existence of the optimal control problem and to derive the necessary conditions for optimal control of the diseases. Though this is not a case study, simulation results to support theoretical analysis of the optimal control suggests that concurrently applying malaria and COVID-19 protective measures could help mitigate their spread compared to applying each preventive control measure singly as the world continues to deal with this unprecedented and unparalleled COVID-19 pandemic.

In this article, a model representing the spread of Hepatitis B disease is constructed as a nonlinear autonomous system. The model divides the considered human population into three classes, namely susceptible, infected, and recovered class. The dynamical analysis shows that there are two equilibrium points in the model, namely a disease-free equilibrium point and an endemic equilibrium point. The existence and stability of the equilibrium points depend on the basic reproduction number (R_0). The disease-free equilibrium point is local asymptotically stable when R_0 1. The five parameters of the model are estimated by applying Downhill Simplex (Nelder-Mead) Algorithm and by using the infected data cases taken from such a hospital in Malang. The estimated parameters are the transmission of infection rate, the saturation rate, the vaccination rate, the recovery rate, and the immunity loss rate. The resulting parameter estimation supports the analytical result and is used to illustrate the analytical results numerically. Based on the considered model and the result of the parameters estimation, it can be concluded that the Hepatitis B spread in Malang is controllable.

Prey-predator system is enormously complex and nonlinear interaction between species. Such complexity regularly requires development of new approaches which involves more factors in analysis of its population dynamics. In this paper, we formulate a modified Lotka-Volterra model that incorporates factors such as refuge prey and immigrants. We investigate the effects of refuge prey and immigrants by varying the refuge factor, with and without immigrants. The results show that with Holling’s type I functional response, the proposed model is asymptotically convergent when a refuge prey factor is introduced. Moreover, with Holling’s type II functional response, the proposed mathematical model is unstable and does not converge. However, with Holling’s type III functional response in a system, the proposed mathematical model is asymptotically stable. These results point out the following remarks: The effects of refuge prey on stability of the dynamical system vary depending on the type of functional response, and when the predator population increases, the likelihood of prey extinction declines when the proportion of preys in refuge population increases. Hence, the factor of refuge prey is crucial for controlling the population of the predator and obtaining balances between prey and predator in the ecosystem. Keywords: Refuge prey, stability, prey-predator, immigrants, Mathematical modelling

The subject of research in the article is sigularly perturbed controllable systems of differential equations containing terms with a small parameters on the right-hand side, which are not completely known, but only satisfy some constraints. The aim of the work is to expand the study of the behavior of solutions of singularly perturbed systems of differential equations to the case when the system is influenced not only by dynamic (small factor at the derivative) but also parametric (small factor at the right side of equations) uncertainties and to determine conditions under which such systems will be asymptotically resistant to any perturbations, estimate the upper limit of the small parameter, so that for all values of this parameter less than the obtained estimate, the undisturbed solution of the system was asymptotically stable. The following problems are solved in the article: singularly perturbed systems of differential equations with regular perturbations in the form of terms with a small parameter in the right-hand sides, which are not fully known, are investigated; an estimate is made of the areas of asymptotic stability of the unperturbed solution of such systems, that is, the class of systems that can be investigated for stability is expanded, the formulas obtained that allow one to analyze the asymptotic stability of solutions to systems even under conditions of incomplete information about the perturbations acting on them. The following methods are used: mathematical modeling of complex control systems; vector Lyapunov functions investigation of asymptotic stability of solutions of systems of differential equations. The following results were obtained: an estimate was made for the upper bound of a small parameter for sigularly perturbed systems of differential equations with fully known parametric (fully known) and dynamic uncertainties, such that for all values of this parameter less than the obtained estimate, such an unperturbed solution is asymptotically stable; a theorem is proved in which sufficient conditions for the uniform asymptotic stability of such a system are formulated. Conclusions: the method of vector Lyapunov functions extends to the class of singularly perturbed systems of differential equations with a small factor in the right-hand sides, which are not completely known, but only satisfy certain constraints., Предметом исследования в статье является сигулярно возмущенные управляемые системы дифференциальных уравнений, содержащих слагаемые с малым множителем в правой части, которые не являются полностью известными, а лишь удовлетворяют некоторым ограничениям. Цель работы – расширить исследование поведения решений сигулярно возмущенных систем дифференциальных уравнений на случай, когда на систему влияют не только динамические (малый множитель при производной), но и параметрические (малый множитель в правой части уравнений) неопределенности и определить условия, при которых решения таких систем будут асимптотически устойчивыми к любым возмущениям , оценить верхнюю границу малого параметра, таким образом что для всех значений этого параметра, меньших чем полученная оценка, невозмущенное решение системы являлось асимптотически устойчивым. В статье решаются следующие задачи: исследуются сингулярно возмущенные системы дифференциальных уравнений, имеющих регулярные возмущения в виде слагаемых с малым множителем в правых частях, которые не являются полностью известными; делается оценка областей асимптотической устойчивости невозмущенного решения таких систем, то есть расширяется класс систем, которые можно исследовать на устойчивость, получены формулы, позволяющие анализировать асимптотической устойчивости решений систем даже в условиях неполной информаций о возмущения, действующие на них. Используются такие методы: математическое моделирование сложных систем управления; векторные функции Ляпунова исследования асимптотической устойчивости решений систем дифференциальных уравнений. Получены следующие результаты: сделана оценка верхней границы малого параметра для сигулярно возмущенных систем дифференциальных уравнений с параметрическими (не полностью известными) и динамическими неопределенностями, такая что для всех значений этого параметра, меньших чем полученная оценка, невозмущенное решение такой является асимптотически устойчивым; доказана теорема, в которой сформулированы достаточные условия равномерной асимптотической устойчивости такой системы. Выводы: метод векторных функций Ляпунова может быть расширен на класс сингулярно возмущенных систем дифференциальных уравнений с малым множителем в правых частях, которые не являются полностью известными, а лишь удовлетворяют некоторым ограничениям., Предметом дослідження в статті є сигулярно збурені керовані системи диференціальних рівнянь, що містять доданки з малим множником у правій частині, які не є повністю відомими, а лише задовольняють деяким обмеженням. Мета роботи — поширити дослідження поведінки розв’язків сигулярно збурених систем диференціальних рівнянь на випадок, коли на систему впливають не тільки динамічні (малий множник при похідній), а ще і параметричні (малий множник у правій частині рівнянь) невизначеності та визначити умови, за яких розв’язки таких систем будуть асимптотично стійкими до будь-яких збурень, оцінити верхню границю малого параметру, таким чином що для всіх значень цього параметру, менших ніж отримана оцінка, незбурений розв’язок системи є асимптотично стійким. В статті вирішуються наступні завдання: досліджуються сингулярно збурені системи диференціальних рівнянь, що мають регулярні збурення у вигляді доданків з малим множником у правих частинах, які не є повністю відомими; робиться оцінка областей асимптотичної стійкості незбуреного розв’язку таких систем, тобто розширюється клас систем, які можна досліджувати на стійкість, отримуються формули, що дозволяють аналізувати асимптотичну стійкість розв’язків систем навіть за умов неповної інформацій про збурення, що діють на них. Використовуються такі методи: математичне моделювання складних систем керування; векторні функції Ляпунова дослідження асимптотичної стійкості розв’язків систем диференціальних рівнянь. Отримано наступні результати: зроблена оцінка верхньої границі малого параметра для сигулярно збурених систем диференціальних рівнянь параметричними (неповністю відомими) і динамічними невизначеностями , така що для всіх значень цього параметру, менших ніж отримана оцінка, незбурений розв’язок такої є асимптотично стійким; доведена теорема, в якій сформульовані достатні умови рівномірної асимптотичної стійкості такої системи. Висновки: метод векторних функцій Ляпунова може бути поширеним на клас сингулярно збурених систем диференціальних рівнянь з малим множником у правих частинах, які не є повністю відомими, а лише задовольняють деяким обмеженням.

In this article, the problem of stochastic Cohen–Grossberg Bidirectional Associative Memory (CGBAM) neural networks with probabilistic time-varying delay is analyzed by input-to-state stability theory. The stochastic variable with Bernoulli distribution gives the information of probabilistic time-varying delay and it is transformed into one with deterministic time-varying delay in the stochastic manner. Further, by constructing a novel Lyapunov–Krasovskii functional and utilizing Ito’s and Dynkin’s formula with stochastic analysis theory, the sufficient criterion is derived for the input-to-state stability of stochastic CGBAM neural networks. Finally, numerical examples are provided to examine the merits of the given method.

In this paper, an uncertain nonlinear switched system is a nonlinear switched system disturbed by subjective uncertainties, which can be illustrated by uncertain differential equations. Stability issues have been deeply studied on switched systems while few results about stability analysis for uncertain switched systems were published before. In order to fill this gap, three different types of stabilities called stability in measure, almost sure stability and stability in mean concerning uncertain nonlinear switched systems with infinite-time domain and countable switches are investigated in order. The internal property of the uncertain switched systems will be described and captured from diverse perspectives on the basis of the above stability analysis. The corresponding criteria to judge these stabilities are obtained according to uncertainty theory and stability theory. A numerical example concerning stability in measure is provided to show the effectiveness of the results derived.

In this paper, we investigate the global properties of two general models of pathogen infection with immune deficiency. Both pathogen-to-cell and cell-to-cell transmissions are considered. Latently infected cells are included in the second model. We show that the solutions are nonnegative and bounded. Lyapunov functions are organized to prove the global asymptotic stability for uninfected and infected steady states of the models. Analytical expressions for the basic reproduction number $\mathcal{R}_{0}$ and the necessary condition under which the uninfected and infected steady states are globally asymptotically stable are established. We prove that if $\mathcal{R}_{0}$ < 1 then the uninfected steady state is globally asymptotically stable (GAS), and if $\mathcal{R}_{0}$ > 1 then the infected steady state is GAS. Numerical simulations are performed and used to support the analytical results.

In this paper, we study a diffusive HCV infections epidemic model with nonlinear incidence rate and analyze the stability of the two kinds of equilibria. By constructing various Lyapunov functions, we prove that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number $$R_{0}< 1$$ and the endemic equilibrium is globally asymptotically stable when the basic reproduction number $$R_{0} >1$$ . Finally, some numerical simulations are given to confirm the theoretical analysis. The results show that when other parameters are the same, the linear infection rate and the non-linear infection rate have different effects on disease spread.

Although novel coronavirus pneumonia (COVID-19) was widely spread in mainland China in early 2020, it was soon controlled. To study the impact of government interventions on the spread of disease during epidemics, a differential equation system is established to simulate the process of virus propagation in this paper. We first analyze its basic properties, basic reproduction number $$R_{0}$$ and existence of equilibria. Then we prove that the disease-free equilibrium (DFE) is Globally Asymptotically Stable when $$R_{0}$$ is less than 1. Through the analysis of the daily epidemic data from January 10, 2020 to March 11, 2020, combined with the implementation of the national epidemic policy, we divide the whole process into three stages: the first stage (natural state), the second stage (isolation state), the third stage (isolation, detection and treatment). By using the weighted nonlinear least square method to fit the data of three stages, the parameters are obtained, and three basic reproduction numbers are calculated, which are: $$R_{01} =2.6735$$ , $$R_{02} = 0.85077$$ , $$R_{03} = 0.18249$$ . Sensitivity analysis of threshold parameters and corresponding graphical results were also performed to examine the relative importance of various model parameters to the spread and prevalence of COVID-19. Finally, we simulate the trend of three stages and verify the theory of Global Asymptotic Stability of DFE. The conclusion of this paper proves theoretically that the Chinese government’s epidemic prevention measures are effective in the fight against the spread of COVID-19. This study can not only provide a reference for research methods to simulate COVID-19 transmission in other countries or regions, but also provide recommendations on COVID-19 prevention measures for them.

In this paper, we propose a susceptible-infected-susceptible (SIS) epidemic model with demographics on heterogeneous metapopulation networks. We analytically derive the basic reproduction number, which determines not only the existence of endemic equilibrium but also the global dynamics of the model. The model always has the disease-free equilibrium, which is globally asymptotically stable when the basic reproduction number is less than unity and otherwise unstable. We also provide sufficient conditions on the global stability of the unique endemic equilibrium. Numerical simulations are performed to illustrate the theoretical results and the effects of the connectivity and diffusion. Furthermore, we find that diffusion rates play an active role in controlling the spread of infectious diseases.

We develop two discrete models to study how supplemental releases affect the Wolbachia spreading dynamics in cage mosquito populations. The first model focuses on the case when only infected males are released at each generation. This release strategy has been proved to be capable of speeding up the Wolbachia persistence by suppressing the compatible matings between uninfected individuals. The second model targets the case when only infected females are released at each generation. For both models, detailed model formulation, enumeration of the positive equilibria and their stability analysis are provided. Theoretical results show that the two models can generate bistable dynamics when there are three positive equilibrium points, semi-stable dynamics for the case of two positive equilibrium points. And when the positive equilibrium point is unique, it is globally asymptotically stable. Some numerical simulations are offered to get helpful implications on the design of the release strategy.

In this paper, we investigate the problem of robust H∞ filter design for a class of discrete-time nonlinear systems. The systems under consider involves time-varying delays and parameters uncertainties. The main objective is to design a linear full-order filter to ensure that the resulting filtering error system is asymptotically stable with a prescribed H∞ performance level. By constructing an appropriate Lyapunov-Krasovskii functional, some novel sufficient conditions are established to guarantee the filter error dynamics system is robust asymptotically stable with H∞ performance γ , and the H∞ filter is designed in term of linear matrix inequalities. Finally, a numerical example is provided to illustrate the efficiency of proposed method.

This article researches the stable deployment of a triangular tethered satellite formation (TTSF) with velocities unmeasurable. By state space transformation, the system is transformed to four subsystems. Controllers with two kinds of robust terms are proposed and proved to make the system asymptotically stable. Then a high gain observer is given to estimate unmeasurable velocity variables in the system, where the design principles of the observer gains are found as well, and the system is locally asymptotically stable for the nonlinearities in the observer. Furthermore, the factors affecting the convergence of the system in the presence of state observer are found by virtue of Lyapunov function. Numerical simulation validates the effectiveness of the proposed controller and the stability of the system with state observer.

This paper is studied with the hierarchical type stability and stabilization of networked control systems (NCSs) with event-triggered mechanism (ETM). In the cause of reducing the amount of data transmission and saving the limited network bandwidth, ETM is introduced into NCSs, and the closed-loop time-delay NCSs model with ETM is presented. An improved Lyapunov–Krasovskii functional (LKF), containing delay-product-type terms and being appropriate for the canonical BesselLegendre inequality (BLI), is first constructed. Then, by utilizing the canonical BLI and the extended reciprocally convex matrix inequality (ERCMI) to deal with the single integral terms of the derivative of LKF, a sufficient condition on asymptotically stable is derived for NCSs. Based on above N-dependent stability criteria, a co-design method is developed, which can be capable of calculating the control gain of controller and the weighting matrix of the ETM. Finally, the feasibility and superiority of the results are verified by two examples.

In this paper, the terminal guidance problem of unpowered lifting reentry vehicle to stationary target is studied. Based on the requirement of attacking the target with high precision and high impact angle constraint, a fractional-order theory combined sliding mode guidance law is proposed. Its sliding surface is specially designed to satisfy the requirements in the terminal guidance phase. The novel fractional-order sliding mode guidance law is established in both two-dimensional environment and three-dimensional environment; then, the systems are proved to be asymptotically stable according to the Lyapunov stability principle. Finally, compared with the one without fractional-order term, experiments show the novel guidance law has better stability. Monte Carlo simulation verifies that the designed guidance law is more robust against the disturbance of random noise and ensures higher precision in terms of impact angle error and miss distance.

Let A and B be two closed linear relations acting between two Banach spaces X and Y , and let λ be a complex number. We study the stability of the nullity and deficiency of A when it is perturbed by λ B . In particular, we show the existence of a constant ρ > 0 for which both the nullity and deficiency of A remain stable under perturbation by λ B for all λ inside the disk λ < ρ .

Asymptomatic carriers play an important role in modelling infectious diseases. Asymptomatic infected people have no symptoms but can infect other people and spread the disease among other people. The laboratory confirmed that asymptomatic hepatitis B may infect individuals and may generate other infected cases. Due to this significant role of asymptomatic carriers, we are considering a new mathematical model for hepatitis B virus with asymptomatic carriers to study its dynamic analysis. First, we briefly discussed the formulation of the model, and then used the Caputo derivative to generalize the model. Using the definition of fractional stability analysis, we study the results of the model, and show that the model is locally asymptotically stable for disease-free cases when R 0 1 . We also give the result so that the model is globally asymptotically stable when R 0 1 . The endemic equilibrium of the model is obtained when R 0 > 1, which demonstrates the existence of a unique endemic equilibrium. Additionally, we use a new numerical scheme that is introduced for fractional differential equations to get numerical results. We use a number of fractional order values and present the graphical results of the model.

The problem of event-triggered fixed-time adaptive neural dynamic surface control (DSC) for stochastic non-triangular structure nonlinear systems is discussed in this article. Combined with the fixed-time stability theory, DSC technique and event-triggered control (ETC) technique, a novel event-triggered fixed-time adaptive controller is designed, under which both the closed-loop stability and the tracking performance can be guaranteed simultaneously in a fixed time. At the same time, the problems of “explosion of complexity” and “singularity” under the traditional backstepping design framework are avoided. Moreover, the design of event-triggered control mechanism can save the network resources effectively. In addition, the unknown nonlinear functions are approximated by some radial basis function neural networks (RBFNNs), and the filtering errors are compensated by the novel error compensating signals. Rigorous theoretical derivation and two simulations are included to illustrate the effectiveness of the proposed method.

Taking two susceptible groups into account, we formulate a modified subhealthy-healthy-infected-recovered (SHIR) model with time delay and nonlinear incidence rate in networks with different topologies. Concretely, two dynamical systems are designed in homogeneous and heterogeneous networks by utilizing mean field equations. Based on the next-generation matrix and the existence of a positive equilibrium point, we derive the basic reproduction numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{1}$$\end{document}R01 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{2}$$\end{document}R02 which depend on the model parameters and network structure. In virtue of linearized systems and Lyapunov functions, the local and global stabilities of the disease-free equilibrium points are, respectively, analyzed when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{1}1$$\end{document}R01>1. Finally, numerical simulations are performed to conduct sensitivity analysis and confirm theoretical results. Moreover, some conjectures are proposed to complement dynamical behavior of two systems.

In the present study, a time-delayed SIR epidemic model with a logistic growth of susceptibles, Crowley–Martin type incidence, and Holling type III treatment rates is proposed and analyzed mathematically. We consider two explicit time-delays: one in the incidence rate of new infection to measuring the impact of the latent period, and another in the treatment rate of infectives to analyzing the effect of late treatment availability. The stability behavior of the model is analyzed for two equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). We derive the threshold quantity, the basic reproduction number $$R_0$$ , which determines the eradication or persistence of infectious diseases in the host population. Using the basic reproduction number, we show that the DFE is locally asymptotically stable when $$R_0< 1$$ , linearly neutrally stable when $$R_0= 1$$ , and unstable when $$R_0> 1$$ for the time-delayed system. We analyze the system without a latent period, revealing the forward bifurcation at $$R_0= 1$$ , which implies that keeping $$R_0$$ below unity can diminish the disease. Further, the stability behavior for the EE is investigated, demonstrating the occurrence of oscillatory and periodic solutions through Hopf bifurcation concerning every possible grouping of two time-delays as the bifurcation parameter. To conclude, the numerical simulations in support of the theoretical findings are carried out.

Existing dynamical systems for day-to-day departure time choice are either unstable, or stable but assuming drivers to possess complete information and make decisions on both arrival and departure times. In this paper, we present a new dynamical system with local shifting of departure times, such that a driver only defers or advances his/her departure time to a time interval later or earlier with lower costs. We establish the asymmetrical upper bounds of the deferral and advance coefficients for the discrete model to be well-defined. We then derive the continuous version as a kinematic wave model and present some examples of symmetrical deferral and advance coefficients. We demonstrate that the stationary state of the dynamical system is the same as the user equilibrium, and the user equilibrium is proved with Lyapunov's second method to be stable for the symmetrical deferral and advance coefficients. With numerical examples, we verify the analytical results and examine the model's sensitivity to different factors with different combinations of heuristic asymmetrical coefficients and theoretically stable symmetrical coefficients. Both analytical and numerical results confirm that the new dynamical system is asymptotically stable in a stability region. This study provides some guidelines on how to derive new day-to-day dynamical system models of departure time user equilibrium. Such a dynamical system can potentially be applied to solve the general dynamic traffic assignment problem in the future.

We analyze the stability of a unique coexistence equilibrium point of a system of ordinary differential equations (ODE system) modelling the dynamics of a metapopulation, more specifically, a set of local populations inhabiting discrete habitat patches that are connected to one another through dispersal or migration. We assume that the inter-patch migrations are detailed balanced and that the patches are identical with intra-patch dynamics governed by a mean-field ODE system with a coexistence equilibrium. By making use of an appropriate Lyapunov function coupled with LaSalle’s invariance principle, we are able to show that the coexistence equilibrium point within each patch is locally asymptotically stable if the inter-patch dispersal network is heterogeneous, whereas it is neutrally stable in the case of a homogeneous network. These results provide a mathematical proof confirming the existing numerical simulations and broaden the range of networks for which they are valid.

A mathematical model with distributed time delay describing the labour market is investigated, focusing on the asymptotic stability of the unique positive equilibrium point. The positivity and boundedness of the solutions are proved and the local stability analysis reveals that the positive equilibrium point is asymptotically stable, regardless of the distributed time delay considered in the model. Moreover, the construction of a suitable Lyapunov function leads to global asymptotic stability results. Numerical simulations are presented with the aim of substantiating the theoretical statements.

This paper is devoted to non-fragile L 2 − L ∞ filtering for switched neural networks with time-variant delay. The aim is to design a L 2 − L ∞ filter subject to either additive or multiplicative gain perturbations, such that the filter-error system not only is asymptotically stable when there is no external disturbance but also has a predefined disturbance attenuation index under the zero initial condition. A criterion of the stability and L 2 − L ∞ performance for the filter-error system is proposed by applying mode-dependent Lyapunov functionals, the Bessel–Legendre inequality, as well as the reciprocally convex combination technique. Then, a design method for the non-fragile L 2 − L ∞ filter is developed by getting rid of some nonlinear coupling terms. The method is formulated as a problem of finding a feasible solution to a collection of linear matrix inequalities, which are computationally tractable. At last, two numerical examples are employed to illustrate the applicability of the L 2 − L ∞ filtering design method.

The problem of event-triggered fixed-time control for state-constrained stochastic nonlinear systems is discussed in this article. Different from the Barrier Lyapunov Function (BLF)-based and Integral BLF-based schemes which rely on feasibility conditions (FCs), by introducing nonlinear state-dependent functions , the asymmetric time-varying state constraints are handled without FCs .Combining with the fixed-time stable theory and dynamic surface control technique with fixed-time filter, the fixed-time stability in probability of the closed-loop system is ensured and the problems of “explosion of complexity” and “singularity” are overcome. Furthermore, the novel fixed-time error compensation signals are designed to compensate filtering errors, and event-triggered control technique is used to save network resources. Simulations also illustrate the effectiveness of the proposed method.

Ebola virus is among the most dangerous and devastating threats to human health, causing a large number of fatalities. In this paper, a mathematical modelling for the dynamics of Ebola virus infectious of an individual is presented as a system of nonlinear differential equations. The model has two equilibrium states namely, virus free equilibrium (VFE) and virus persistence equilibrium (VPE) states. The Effective reproduction number was obtained. The conditions under which the virus-free-equilibrium is globally asymptotically stable with the approach of linear Lyapunov function are shown when the effective reproduction numbers is less than unity. The nonlinear Lyapunov approach is employed to show the global stability of the endemic equilibrium only when the effective reproduction number is greater than unity. It was found that VFE is globally asymptotically stable if effective reproduction numbers is less than unity and VPE is globally asymptotically stable if MN.

The COVID-19 is an infection caused by corona virus which has been designated as pandemic in the word. In this paper, we proposed a model to study the effect of distancing and wearing of mask on transmission of Covid-19 infection dynamics. The invariant region of the model is established. The Covid-19 free equilibrium and the reproduction number of the model were obtained. The local and global stability of the model is determined using linearization technique method and Lyapunov method. It was found that, Covid-19 free equilibrium state is locally asymptotically stable in feasible region Ω if R0<1 and globally asymptomatically stable if R0<1, otherwise unstable if R0>1. More so, numerical analysis and simulations of the dynamics of the Covid-19 infection are presented. It was found that as effectiveness of face mask usage increased it resulted in a decrease in secondary cases and as physical distancing is maintained shown reduction in reproduction number. It is also found that, as infection rate increases, it translated to increase in secondary cases and as number of isolation individual increases, it decreased the reproduction number. More so, varying the treatment rate, means as treatment rate increases, it shown that an individuals in the isolation decreases.

In this paper, we discuss the basic reproduction number of stochastic epidemic models with random perturbations. We define the basic reproduction number in epidemic models by using the integral of a function or survival function. We study the systems of stochastic differential equations for SIR, SIS, and SEIR models and their stability analysis. Some results on deterministic epidemic models are also obtained. We give the numerical conditions for which the disease-free equilibrium point is asymptotically stable.

This paper proposes to utilize intentional time delays as part of controllers to improve the damping of electromechanical oscillations of power systems. Through stability theory, the control parameter settings for which these delays in Power System Stabilizers (PSSs) improve the small signal stability of a power system are systematically identified, including the key parameter settings for which stability regions in the parameter plane remain connected for effective operation. The paper shows that PSSs with two control channels can be effectively designed to achieve best damping characteristics for a wide range of delays. Analytical results are presented on the One-Machine Infinite-Bus (OMIB) electromechanical power system model. To demonstrate the opportunities in more realistic dynamic models, our results are then implemented via numerical analysis on the IEEE standard 14-bus system.

A complex SIR epidemic dynamical model using nonlinear incidence rate and nonlinear recovery rate is established to consider the impact of available hospital beds and interventions reduction on the spread of infectious disease. Rigorous mathematical results have been established for the model from the point of view of stability and bifurcation. The model has two equilibrium points when the basic reproduction number R 0 > 1 ; a disease-free equilibrium E 0 and a disease endemic equilibrium E 1 . We use LaSalle’s invariance principle and Lyapunov’s direct method to prove that E 0 is globally asymptotically stable if the basic reproduction number R 0 1 , and E 1 is globally asymptotically stable if R 0 > 1 , under some conditions on the model parameters. The existence and nonexistence of limit cycles are investigated under certain conditions on model parameters. The model exhibits Hopf bifurcation near the disease endemic equilibrium. We further show the occurring of backward bifurcation for the model when there is limited number of hospital beds. Finally, some numerical results are represented to validate the analytical results.

This paper presents a mathematical model that describes the transmission dynamics of schistosomiasis for humans, snails, and the free living miracidia and cercariae. The model incorporates the treated compartment and a preventive factor due to water sanitation and hygiene (WASH) for the human subpopulation. A qualitative analysis was performed to examine the invariant regions, positivity of solutions, and disease equilibrium points together with their stabilities. The basic reproduction number, R 0 , is computed and used as a threshold value to determine the existence and stability of the equilibrium points. It is established that, under a specific condition, the disease-free equilibrium exists and there is a unique endemic equilibrium when R 0 > 1 . It is shown that the disease-free equilibrium point is both locally and globally asymptotically stable provided R 0 < 1 , and the unique endemic equilibrium point is locally asymptotically stable whenever R 0 > 1 using the concept of the Center Manifold Theory. A numerical simulation carried out showed that at R 0 = 1 , the model exhibits a forward bifurcation which, thus, validates the analytic results. Numerical analyses of the control strategies were performed and discussed. Further, a sensitivity analysis of R 0 was carried out to determine the contribution of the main parameters towards the die out of the disease. Finally, the effects that these parameters have on the infected humans were numerically examined, and the results indicated that combined application of treatment and WASH will be effective in eradicating schistosomiasis.

Event-triggered optimal stabilization problem is investigated for macro–micro composite stage system with dynamical perturbation and interconnections. By using a novel cost function and considering the interconnected subsystems as a whole system, stabilization problem of the unknown interconnected system converts to control problem with optimality for the corresponding nominal system. On the side, the update frequency of the controller can effectively be decreased by employing an event-triggered condition. Furthermore, an optimal controller is derived by solving the modified Hamilton–Jacobi–Bellman equation via single critic neural network with asymptotically stable structure. And then the uniformly ultimately bounded stability issue of closed-loop system is analyzed in this article. Finally, the effectiveness of the developed event-triggered stabilization control scheme is verified by using simulation results for macro–micro composite stage system.

The series-paralleling technique of dc-dc converters is utilized in various domains of electrical engineering for improved power conversion. Previous studies have proposed and classified the control schemes for the series-paralleled converters. However, they have several restrictions and lack diversity. The purpose of this paper is to propose passivity-based control (PBC) for the diverse output series-parallel connection of dc-dc converters. It is proved that the output series-paralleled converters regulated by PBC are asymptotically stable. The output series-paralleled converters are numerically simulated to confirm the asymptotic stability. PBC is shown to maintain the stability of the output series-paralleled converters with diverse circuit topologies, parameters, and steady-states. The result of this paper theoretically supports the numerical and experimental considerations in the previous studies and justifies the further extension of the series-parallel connection., Comment: 10 pages, accepted for publication in IEEE Transactions on Circuits and Systems I: Regular Papers