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In this paper, we study three stochastic two-species models. We construct the stochastic models corresponding to its deterministic model by introducing stochastic noise into the equations. For the first model, we show that the system has a unique global solution starting from the positive initial value. In addition, we discuss the extinction and the existence of stationary distribution under some conditions. For the second system, we explore the existence and uniqueness of the solution. Then we obtain sufficient conditions for global asymptotic stability of the equilibrium point and the positive recurrence of solution. For the last model, the existence and uniqueness of solution, the sufficient conditions for extinction and asymptotic stability and the positive recurrence of solution and weak persistence are derived. And numerical simulations are performed to support our results. [ABSTRACT FROM AUTHOR]

In this paper, an SEAI epidemic model with asymptomatic infection is studied under the background of mass transmission of COVID-19. First, we use the next-generation matrix method to obtain the basic reproductive number R 0 and calculate the equilibrium point. Secondly, when R 0 < 1 , the local asymptotic stability of the disease-free equilibrium is proved by Hurwitz criterion, and the global asymptotic stability of the disease-free equilibrium is proved by constructing the Lyapunov function. When R 0 > 1 , the system has a unique endemic equilibrium point and is locally asymptotically stable, and it is also proved that the system is uniformly persistent. Then, the application of optimal control theory is carried out, and the expression of the optimal control solution is obtained. Finally, in order to verify the correctness of the theory, the stability of the equilibrium point is numerically simulated and the sensitivity of the parameters of R 0 is analyzed. We also simulated the comparison of the number of asymptomatic infected people and symptomatic infected people before and after adopting the optimal control strategy. This shows that the infection of asymptomatic people cannot be underestimated in the spread of COVID-19 virus, and an isolation strategy should be adopted to control the spread speed of the disease. [ABSTRACT FROM AUTHOR]

This paper investigates a two-species amensalism model that includes the fear effect on the first species and the Beddington–DeAngelis functional response. The existence and stability of possible equilibria are investigated. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, global dynamics analysis of the model is performed. It is observed that under certain parameter conditions, when the intensity of the fear effect is below a certain threshold value, as the fear effect increases it will only reduce the density of the first species population and will have no influence the extinction or existence of the first species population. However, when the fear effect exceeds this threshold, the increase of the fear effect will accelerate the extinction of the first species population. Finally, numerical simulations are performed to validate theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a cross-diffusion tumor invasion model with double-taxis effect. We first investigate the global existence of classical solutions of this model in two-dimensional space. The essential difficulty lies in the second-level taxis effect of immune cells on tumor cells, where chemotactic factor (tumor cells) exhibit their own taxis behavior, the double-taxis effect makes us have to use more detailed analysis and calculation, and some new estimation techniques are used. Subsequently, we also investigate the stability of some equilibria. For small proliferation coefficient, we prove the global asymptotic stability or local asymptotic stability of a semitrivial equilibrium point. While, for the other equilibria, the stability analysis is complicated even for some special cases, and both the double chemotactic coefficients will affect the stability of the solution. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear mathematical model for tuberculosis transmission, which incorporates multiple saturated exogenous reinfections, is proposed and explored. The existence of disease-free and endemic steady states is investigated. Disease-free equilibrium (DFE) is locally asymptotically stable (LAS) but not globally asymptotically stable (GAS) when the basic reproduction number, ℛ 0 < 1. However, it is GAS only when there is no exogenous reinfection. The local asymptotic stability and global asymptotic stability of the unique endemic equilibrium point (EEP) are established under certain conditions when ℛ 0 > 1. Further, the EEP is GAS when ℛ 0 > 1 , provided there is no exogenous reinfection. When ℛ 0 is below unity, the presence of multiple endemic equilibria is found which leads to backward bifurcation. It is demonstrated that the system encounters a Hopf-bifurcation when the transmission rate β crosses a critical value, resulting in the formation of limit cycles, i.e. periodic solutions bifurcate around the EEP when β passes a critical value. The stability and direction of Hopf-bifurcation are also studied. The results of the analytical work are validated through numerical simulations. A numerical simulation illustrates that EEP losses its stability via Hopf-bifurcation for specific parameters. However, when the bifurcation parameter β is increased further, the EEP regains its stability. In addition, Hopf-bifurcation occurs due to exogenous reinfection rates p and . Thus, our model shows some important nonlinear dynamical behaviors. [ABSTRACT FROM AUTHOR]

In this paper, a generalized fractional three-species food chain model with delay is investigated. First, the existence of a positive equilibrium is discussed, and the sufficient conditions for global asymptotic stability are given. Second, through selecting the delay as the bifurcation parameter, we obtain the sufficient condition for this non-control system to generate Hopf bifurcation. Then, a nonlinear delayed feedback controller is skillfully applied to govern the system's Hopf bifurcation. The results indicate that adjusting the control intensity or the control target's age can effectively govern the bifurcation dynamics behavior of this system. Last, through application examples and numerical simulations, we confirm the validity and feasibility of the theoretical results, and find that the control strategy is also applicable to eco-epidemiological systems. [ABSTRACT FROM AUTHOR]

It is beyond dispute that cytotoxic T-lymphocytes (CTLs) exert a vital function in the host's antiviral defense mechanism. With the idea of the above factor and the logistic proliferation of CD4 + T-cells, we establish a HTLV-I (human T-cell leukemia virus type-I) mathematical model. First, two threshold parameters ℛ 0 and ℛ c (the basic reproduction numbers for viral infection and CTL immune response, respectively) are obtained. Second, sufficient criteria for local and global asymptotic stabilities of the feasible equilibria of the model are deduced, respectively. Third, the sensitivity analyses of ℛ 0 and ℛ c are performed to better understand the effective strategies for HTLV-I infection. Finally, not only numerical simulations are given to illustrate the stability conclusions, but also the biological significance is stated. [ABSTRACT FROM AUTHOR]

This paper focuses on the stability problem of networked control systems (NCSs) under denial‐of‐service (DoS) attacks and bandwidth limitation. Firstly, the periodic dynamic quantization mechanism is extended to insecure systems, and a periodic‐like quantization strategy is proposed. Due to the designable period, such mechanism allows a trade‐off between better system performance and less energy consumption. With finite quantization levels, a reasonable parametric design is given to avoid quantizer saturation under general energy‐constrained DoS attacks. Secondly, a suitable quantized controller is designed for the system affected by DoS attacks. The relationship between the quantized controller and the DoS attacks is analyzed, and the criterion for NCSs achieving global asymptotic stability is derived. Finally, two numerical examples are given to verify the correctness of the theoretical result. [ABSTRACT FROM AUTHOR]

A nonlinear infectious disease model with information-influenced vaccination behavior and contact patterns is proposed in this paper, and the impact of information related to disease prevalence on increasing vaccination coverage and reducing disease incidence during the outbreak is considered. First, we perform the analysis for the existence of equilibria and the stability properties of the proposed model. In particular, the geometric approach is used to obtain the sufficient condition which guarantees the global asymptotic stability of the unique endemic equilibrium E e when the basic reproduction number R 0 > 1. Second, mathematical derivation combined with numerical simulation shows the existence of the double Hopf bifurcation around E e . Third, based on the numerical results, it is shown that the information coverage and the average information delay may lead to more complex dynamical behaviors. Finally, the optimal control problem is established with information-influenced vaccination and treatment as control variables. The corresponding optimal paths are obtained analytically by using Pontryagin's maximum principle, and the applicability and validity of virous intervention strategies for the proposed controls are presented by numerical experiments. [ABSTRACT FROM AUTHOR]

Hepatitis B is a viral infection that can cause inflammation of the liver and lead to severe liver damage and even death. The study of hepatitis in Burkina Faso is crucial for several reasons. Indeed, understanding the epidemiology of hepatitis in Burkina Faso can help develop effective prevention and control strategies. Its study can also contribute to a better understanding of the global burden of the disease and the development of effective interventions in other parts of the world. To this aim, a new differential susceptibility and infectivity mathematical model of Hepatitis B transmission was developed in order to simulate the potential spread of the Hepatitis B virus in the population of Burkina Faso. Once the mathematical model is presented, the existence and uniqueness of non-negative solutions are proved. The model has a globally asymptotically stable disease-free equilibrium when the basic reproduction number R0 < 1 and a globally asymptotically stable endemic equilibrium when R0 > 1. The global asymptotic stability of the disease-free equilibrium has been studied using the Castillo Chavez method. The Lyapunov function and the LaSalle invariance principle are used to prove the global asymptotic stability of the endemic equilibrium. To simulate the proposed model, a Matlab numerical code has been developed. Numerical simulations are performed using data of Burkina Faso. The obtained numerical results confirm analytical results as well as the evolution of hepatitis B in Burkina Faso. [ABSTRACT FROM AUTHOR]

The stability problem of stochastic networks with proportional delays and unsupervised Hebbian-type learning algorithms is studied. Applying the Lyapunov functional method, a stochastic analysis technique and the It o ^ formula, we obtain some sufficient conditions for global asymptotic stability. We also discuss the estimation of the second moment. The correctness of the main results is verified by two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the global asymptotic stability and projective lag synchronization of second‐order competitive neural networks with mixed time‐varying delays and uncertainties are studied without converting the original system into the usual first‐order system. Firstly, according to the Lyapunov functional method, inequality technique and the designed adaptive control strategy, the algebraic criteria of stability, and projective lag synchronization are derived by adjusting the control gain parameters in the controller. The obtained sufficient conditions are simple and easy to verify. Different from the traditional feedback controller, the adaptive controller can adjust its characteristics according to the system model, and external disturbance makes the system have better control performance. Besides, unlike the existing ones, this stability and synchronization problem is directly analyzed by constructing some new Lyapunov functionals with state variables and state variable derivatives. Finally, the effectiveness and practicability of the results are verified by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the global asymptotic stability (GAS) of continuous-time and discrete-time nonlinear impulsive switched positive systems (NISPS) are studied. For continuous-time and discrete-time NISPS, switching signals and impulse signals coexist. For both of these systems, using the multiple max-separable Lyapunov function method and average dwell-time (ADT) method, some sufficient conditions on GAS are given. Based on these, the GAS criteria are also given for continuous-time and discrete-time linear impulsive switched positive systems (LISPS). From our criteria, the stability of the systems can be judged directly from the characteristics of the system functions, switching signals and impulse signals of the systems. Finally, simulation examples verify the validity of the results. [ABSTRACT FROM AUTHOR]

Based on the spread of COVID-19, in the present paper, an imperfectly vaccinated SVEIR model for latent age is proposed. At first, the equilibrium points and the basic reproduction number of the model are calculated. Then, we discuss the asymptotic smoothness and uniform persistence of the semiflow generated by the solutions of the system and the existence of an attractor. Moreover, LaSalle's invariance principle and Volterra type Lyapunov functions are used to prove the global asymptotic stability of both the disease-free equilibrium and the endemic equilibrium of the model. The conclusion is that if the basic reproduction number R ρ is less than one, the disease will gradually disappear. However, if the number is greater than one, the disease will become endemic and persist. In addition, numerical simulations are also carried out to verify the result. Finally, suggestions are made on the measures to control the ongoing transmission of COVID-19. [ABSTRACT FROM AUTHOR]

We explore the dynamics of adhering to rational difference formula... where the initials are arbitrary nonzero real numbers. Specifically, we examine global asymptotically stability. Additionally, we provide examples and solutions graphs of some special cases. [ABSTRACT FROM AUTHOR]

The energy-intensive industrial economy results in frequent Fog–Haze weather which increases the risk of influenza spread and brings new challenges for the prevention and control of influenza. Therefore, the study of spread mechanics and controlling strategies of influenza based on Fog–Haze will be scientifically meaningful. Considering that when the concentration of Fog–Haze is low, Fog–Haze contributes to the number of infectious individuals and Fog–Haze suppresses the transmission of the influenza virus when the concentration of Fog–Haze is high, we establish the Fog–Haze dynamics model. Then we prove the global existence and boundedness of the solution, and the global asymptotic stability of the solution is given by constructing a suitable Lyapunov functional. Under the Fog–Haze weather, we study the influenza virus transmission model that incorporates the incidence rate to reflect the Fog–Haze-dependent saturation effect and investigate the effect of Fog–Haze pollution on the transmission of influenza. We show that the basic reproduction number R 0 determines the global dynamics of the system: if R 0 < 1 , the disease-free equilibrium is global asymptotically stable; the unique endemic equilibrium is global asymptotically stable if R 0 > 1. Simulations are carried out to validate the theoretical results. Our study provides further understanding of the dynamics of Fog–Haze and the effect of Fog–Haze pollution on the transmission of influenza. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a fractional-order tumor–immune interaction model with B-D function item and immunotherapy. First, the existence, uniqueness and nonnegativity of the solutions of the model are established. Second, the local and global asymptotic stability of some tumor-free equilibrium points and a unique positive equilibrium point are obtained. Finally, we use numerical simulation method to visualize and verify the theoretical conclusions. It is known that the fractional-order parameter β has a stabilization effect, and the tumor cells can be destroyed or controlled by using immunotherapy. [ABSTRACT FROM AUTHOR]

The tenth Ebola outbreak in the Democratic Republic of Congo (DRC) that occurred from 2018 to 2020 was exacerbated by long-lasting conflicts and war in the region. We propose a deterministic model to investigate the impact of such disruptive events on the transmission dynamics of the Ebola virus disease. It is an extension of the classical susceptible-infectious-recovered model, enriched by an additional class of contaminated environment to account for indirect transmission as well as two classes of hospitalized individuals and patients who escape from the healthcare facility due to violence and attacks perpetrated by armed groups, rebels, etc. The model is formulated using two patches, namely Patch 1 consisting of the three affected eastern provinces in DRC and Patch 2, a war- and conflict-free area consisting of the go-to neighboring provinces for escaped patients. We introduce two key parameters, the escaping rate from hospitals and the destruction of hospitals, in terms of which the effect of war and conflicts is measured. The model is fitted and parameterized using the cumulative mortality data from the region. The basic reproduction number R 0 is computed and found to have a complex expression due to the high nonlinearity of the model. By using, not a Lyapunov function, but a decomposition theorem in Castillo-Chavez et al.(in Castillo-Chavez et al. (eds) Mathematical approaches for emerging and reemerging infectious diseases: an introduction, vol 126. Springer Science & Business Media, Berlin, 2002), it is shown that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 and unstable when R 0 > 1 . A nonstandard finite difference scheme which replicates the dynamics of the continuous model is designed. In particular, a discrete counterpart of the above-mentioned theorem on the global asymptotic stability of the disease-free equilibrium is investigated. Numerical experiments are presented to support the theoretical results. When R 0 > 1 , the numerical simulations suggest that there exists for the full model a unique globally asymptotically stable interior endemic equilibrium point, while it is shown theoretically and computationally that the model possesses at least a one Patch 1 and a one Patch 2 boundary equilibria (i.e., Patch 2 and Patch 1 disease-free equilibrium) points, which are locally asymptotically stable. Some recommendations to tackle Ebola in a conflict zone are stated. [ABSTRACT FROM AUTHOR]

For time-invariant finite-dimensional systems, it is known that global asymptotic stability (GAS) is equivalent to uniform GAS (UGAS), in which the decay rate and transient overshoot of solutions are requested to be uniform on bounded sets of initial states. This paper investigates this relationship for time-invariant delay systems. We show that UGAS and GAS are equivalent for this class of systems under the assumption of robust forward completeness, i.e., under the assumption that the reachable set from any bounded set of initial states on any finite time horizon is bounded. We also show that, if the state space is a space in a particular family of Sobolev or H\"older spaces, then GAS is equivalent to UGAS and that robust forward completeness holds. Based on these equivalences, we provide a novel Lyapunov characterization of GAS (and UGAS) in the aforementioned spaces. [ABSTRACT FROM AUTHOR]

The extensively studied fractional-order dynamical networks only contain a uniform fractional derivative, which restricts the scope of fractional-order investigation. One type of Takagi–Sugeno fuzzy second-fractional-order linear networks is established in this article. First, by defining a novel Lyapunov functional, a sufficient criterion is offered to ensure global asymptotic stability of the addressed networks with time delay. Second, a fuzzy state-feedback control scheme is designed for the Takagi–Sugeno fuzzy second-fractional-order delayed linear networks to achieve global stabilization. Third, considering that only a few scholars have worked on adaptive control of fractional-order networks, an adaptive control method is also designed to realize the global stabilization of the Takagi–Sugeno fuzzy second-fractional-order linear networks, which does not contain the sign function and chattering problem can be averted. All results, given as algebraic criteria, are proved directly from the Takagi–Sugeno fuzzy second-fractional-order networks without utilizing the reduced-order approach. Three simulation examples are conducted to show the validity of the theoretical results at last. [ABSTRACT FROM AUTHOR]

This article investigates the periodically intermittent event‐triggered control (PIEC) for delayed nonlinear systems. The PIEC strategy, by means of introducing event‐triggered strategy into periodically intermittent controller, is proposed to reduce the amount of communications while still preserving a satisfactory closed‐loop performance. Some sufficient conditions are derived, which are incorporated into the controller design to determine the controller gain and the triggered parameters in a co‐design case. The overall strategy ensures a global asymptotic stability for the closed‐loop system and the existence of positive dwell time between two consecutive events. Therefore, Zeno behavior is excluded. Finally, the simulation example of Lu oscillator is given to verify the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a predatorâ€"prey system with nonlinear prey-taxis under Neumann boundary condition. For a class of chemotactic sensitive functions, we obtain the existence and boundedness of global classical solutions for initial boundary value problems in arbitrary dimensional space. In addition, we also study the local stability of the constant steady state solution, and obtain the global asymptotic stability of the steady state solution under different predation intensity by constructing appropriate Lyapunov functions. Furthermore, the steady state bifurcation, Hopf bifurcation and fold-Hopf Singularity are analysed in detail by using Lyapunovâ€"Schmidt reduction method. [ABSTRACT FROM AUTHOR]

In this paper, the spatial-temporal dynamics of a delayed reaction-diffusion Hopfield's neural network model (HNNM) under a Neumann boundary condition is considered. Our main concern is the absolute, also called the delay-independent, global or local stability of the trivial and nontrivial steady states of the HNNM with non-monotone activation functions. The dissipativity of the semiflow generated by HNNM is first proved and then the absolute attractivity of both the trivial and the nontrivial steady states is obtained by adopting the idea of connecting the spatial-temporal dynamics of the HNNM with the asymptotic behaviors of a finite-dimensional discrete dynamical system. Specifically, it is shown that the strong attractors of the finite-dimensional discrete dynamical system generated by the nonlinear activation function of the HNNM are indeed the attractors of the corresponding delayed reaction-diffusion HNNM. Numerical simulations are also conducted at last to illustrate the effectiveness of our established results. [ABSTRACT FROM AUTHOR]

In this paper, we construct the solutions expressions for a system of three‐dimensional nonlinear difference equations Rn+1=a1Tn−1Sn−1Rn−1+Sn−1+Tn−1,Sn+1=a2Tn−1Rn−1Rn−1+Sn−1+Tn−1,Tn+1=a3Rn−1Sn−1Rn−1+Sn−1+Tn−1.In addition, we prove some properties of the proposed system such as boundedness and periodicity of the positive solutions and the global asymptotic stability of the equilibrium points. Finally, we present some numerical examples in order to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we have investigated global dynamics of a two-species food chain model with the Holling type III functional response that includes linear harvesting for the prey and nonlinear harvesting for the predator. The long-time continued existence of both species is discussed using uniform persistence theory. Stability of various equilibrium points is described in terms of model parameters. The local asymptotic stability of non-hyperbolic equilibrium points is determined with the help of center manifold theorem. Global behavior of solutions of the model system when both species are present is determined by considering the global properties of the coexistence equilibrium. Here, we have taken a comprehensive view by considering different bifurcations of co-dimension one and two and have discussed the importance of various model parameters on the system dynamics. The model system shows much more complex and realistic behavior compared to a model system without any harvesting, with constant harvesting or linear-yield harvesting of either or both of the species. Numerical simulations have been conducted to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

Motivated by the recent paper [M.R.S. Kulenović, M. Nurkanović, and A.A. Yakubu, Asymptotic behaviour of a discrete-time density-dependent SI epidemic model with constant recruitment, J. Appl. Math. Comput. 67 (2021), pp. 733–753. DOI:], in this paper, we consider the class of the SI epidemic models with recruitment where the Poisson function, a decreasing exponential function of the population of infectious individuals, is replaced by a general probability function that satisfies certain conditions. We compute the basic reproduction number R 0. We establish the global asymptotic stability of the disease-free equilibrium (GAS) for R 0 < 1. We use the Lyapunov function method developed in [P. van den Driessche and A.-A. Yakubu, Disease extinction versus persistence in discrete-time epidemic models, Bull. Math. Biol. 81 (2019), pp. 4412–4446], to demonstrate the GAS of the disease-free equilibrium and uniform persistence of the considered class of models. We show that the considered type of model is permanent for R 0 > 1. For R 0 = 1 , the transcritical bifurcation appears. For R 0 > 1 , we prove the global attractivity result for endemic equilibrium and instability of the disease-free equilibrium. We apply theoretical results to specific escape functions of the susceptibles from infectious individuals. For each case, we compute the basic reproduction number R 0 . [ABSTRACT FROM AUTHOR]

The fear response is an important anti-predator adaptation that can significantly reduce prey's reproduction by inducing many physiological and psycho- logical changes in the prey. Recent studies in behavioral sciences reveal this fact. Other than terrestrial vertebrates, aquatic vertebrates also exhibit fear responses. Many mathematical studies have been done on the mass mortality of pelican birds in the Salton Sea in Southern California and New Mexico in recent years. Still, no one has investigated the scenario incorporating the fear effect. This work investigates how the mass mortality of pelican birds (predator) gets in uenced by the fear response in tilapia fish (prey). For novelty, we investigate a modified fractional-order eco-epidemiological model by incorporating fear response in the prey population in the Caputo-fractional derivative sense. The fundamental mathematical requisites like existence, uniqueness, non-negativity and boundedness of the system's solutions are analyzed. Local and global asymptotic stability of the system at all the possible steady states are investigated. Routh-Hurwitz criterion is used to analyze the local stability of the endemic equilibrium. Fractional Lyapunov functions are constructed to determine the global asymptotic stability of the disease-free and endemic equilibrium. Finally, numerical simulations are conducted with the help of some biologically plausible parameter values to compare the theoretical findings. The order α of the fractional derivative is determined using Matignon's theorem, above which the system loses its stability via a Hopf bifurcation. It is observed that an increase in the fear coeffcient above a threshold value desta- bilizes the system. The mortality rate of the infected prey population has a stabilization eect on the system dynamics that helps in the coexistence of all the populations. Moreover, it can be concluded that the fractional-order may help to control the coexistence of all the populations. [ABSTRACT FROM AUTHOR]

The stability of fractional-order complex-valued neural networks (FOCVNNs) with probabilistic time-varying delays is investigated in this paper. By constructing suitable Lyapunov–Krasovskii functional and utilizing inequality technique, a complex-valued linear matrix inequality (LMI) criterion guaranteeing the global asymptotic stability of the proposed FOCVNNs is deduced. A numerical example with simulations is provided to demonstrate the feasibility and availability of the obtained theoretical result. [ABSTRACT FROM AUTHOR]

The problem of making the output of a linear system with polytopic uncertainties and discontinuities in the state evolution totally insensitive to an unknown disturbance input by state feedback is investigated. Suitable geometric notions are introduced and used to provide a structural, constructive solvability condition. The requirement of achieving global robust asymptotic stability of the compensated dynamics is then added and further solvability conditions are provided by requiring that the time instants at which discontinuities in the state evolution, or jumps, occur are sufficiently far from each other. [ABSTRACT FROM AUTHOR]

In this paper, we consider a viral infection dynamics model with immune impairment and time delay in immune expansion. By calculation, it is shown that the model has three equilibria: infection-free equilibrium, immunity-inactivated infection equilibrium, and immunity-activated infection equilibrium. By analyzing the distributions of roots of corresponding characteristic equations, the local stability of the infection-free equilibrium and the immunity-inactivated infection equilibrium is established. Furthermore, we discuss the existence of Hopf bifurcation at the immunity-activated infection equilibrium. Sufficient conditions are obtained for the global asymptotic stability of each feasible equilibria of the model by using LaSalle's invariance principle and iteration technique, respectively. Numerical simulations are carried out to illustrate the main theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we analyze various stability (from stability to global uniform asymptotic stability) of Caputo fractional-order nonautonomous systems (CFONSs). With f (t , x) being only continuous, it is proven that the Caputo fractional derivative (CFD) of a general Lyapunov function V (t , x (t)) along solutions is continuous and moreover, t 0 C D t α V (t , x (t)) ≤ ∂ V (t , x) ∂ t (t , x (t)) t 0 C D t α t + ∂ V (t , x) ∂ x ⋅ f (t , x) (t , x (t)) , on the maximal interval of existence (MIE) of x (t). This together with the continuation of solutions suffices to prove the various stability theorems for CFONSs that are as general as those for integer-order systems, and make them practically applicable. The work reduces the assumption on vector field functions f for stability analysis from continuously differentiable (CD) to only continuous, which advances existing results to a large extent. Finally, some derived results are applied to real examples with numerical simulations. [ABSTRACT FROM AUTHOR]

Spruce budworm is a major defoliator of forests in North America and its periodic outbreak can cause severe economic growth loss. In 2008, Vaidya and Wu proposed a delayed spruce budworm population model and studied the impact of physiological structure on outbreak control (Vaidya and Wu, 2008 [27]). In this paper, we revisit this model and study its global dynamics that has not been analyzed previously. By carefully dealing with the major difficulty caused jointly by the time delay and the nonlinearity, we give a rather complete stability analysis for the model, including global asymptotic stability, bistability, semi-stability and Hopf bifurcation. Moreover, we show that the time delay can induce interesting dynamics including multitype bistability and long transients. [ABSTRACT FROM AUTHOR]

This paper presents a new asymmetric Lyapunov–Krasovskii functional method for investigating the asymptotic stability of time‐delay systems. Two resulting delay‐dependent asymptotic stability conditions in the form of linear matrix inequalities are given. The advantages of the new approaches are verified theoretically and numerically. [ABSTRACT FROM AUTHOR]

Corona virus disease which has caused frustration in the human community remains the concern of the globe as every government struggles to defeat the pandemic. In this manuscript, we have extensively studied a new deterministic SEQIHR model of deadly Corona virus pandemic governed by nonlinear ordinary differential equations to provide deep insight into the dynamics of the disease. The purpose is to perform a complete mathematical analysis and the design of an optimal control strategy for the developed deterministic model by utilizing the preventive measures of quarantine and hospitalization. Some comprehensive mathematical techniques are employed to demonstrate the positivity and boundedness of solutions. Two main equilibrium points of the pandemic model are stated. To handle the future dynamical behavior of the pandemic, the threshold parameter value is computed using the next-generation method. The local and global asymptotic stability conditions of both the equilibria are obtained successfully. A numerical analysis to observe the effectiveness of quarantine and hospitalization strategies is performed and illustrated by graphs. We implemented the well known Non-Standard Finite Difference (NSFD) method to find the numerical solution of model. Numerical simulations are conducted to support our analytic results. The developed model is subjected to sensitivity analysis and the most sensitive parameters are identified. The bifurcation nature of the developed model is examined. An optimal control problem is introduced for the proposed model to determine the best controls for the implemented hospitalization and quarantine strategies. With the attention of reducing the number of exposed and infected persons, an optimum control problem and its derived associated optimality conditions of Pontryagin type are explored. An important feature of this study is to employ NSFD method backward in time for the first time to solve optimal control problem instead of other standard methods. The extremals are obtained numerically. Moreover, we have shown the effectiveness of optimal controls on the human community to view more features of the state variables in the proposed model. • We develop a Covid-19 SEQIHR pandemic model and derive analytically several essential properties. • Necessary and sufficient conditions for the stability of equilibrium points are established. • The impact of quarantine and hospitalization on human community is demonstrated numerically. • An optimal control problem is introduced to determine the best controls for both the strategies. • Numerical simulations using an efficient NSFD scheme confirm the derived analytical results. [ABSTRACT FROM AUTHOR]

This paper investigates the global asymptotic stability problem of recurrent neural networks with multiple time-varying delays. First, to make full use of the relationship between all delayed states x (t - τ i) (i = 1 , ... , N) and current state x (t) , an augmented delays-dependent region partitioning (ADRP) approach is proposed. Then combining Wirtinger-based integral inequality and the reciprocally convex approach, two delay-dependent stability criteria with less conservatism are developed. Finally, two numerical simulations are shown to verify the effectiveness and the feasibility. [ABSTRACT FROM AUTHOR]

Dengue fever is a typical mosquito-borne infectious disease, and four strains of it are currently found. Clinical medical research has shown that the infected person can provide life-long immunity against the strain after recovering from infection with one strain, but only provide partial and temporary immunity against other strains. On the basis of the complexity of transmission and the diversity of pathogens, in this paper, a multi-strain dengue transmission model with latency age and cross immunity age is proposed. We discuss the well-posedness of this model and give the terms of the basic reproduction number R0 = max{R1, R2}, where Ri is the basic reproduction number of strain i (i = 1, 2). Particularly, we obtain that the model always has a unique diseasefree equilibrium P0 which is locally stable for R0 < 1. And same time, an explicit condition of the global asymptotic stability of P0 is obtained by constructing a suitable Lyapunov functional. Furthermore, we also shown that if Ri > 1, the strain-i dominant equilibrium Pi is locally stable for Rj < Ri* i (i, j = 1, 2, i 6= j). Additionally, the threshold criteria on the uniformly persistence, the existence and global asymptotically stability of coexistence equilibrium are also obtained. Finally, these theoretical results and interesting conclusions are illustrated with some numerical simulations. [ABSTRACT FROM AUTHOR]

This paper presents criteria for the convergence of trajectories of time-varying nonlinear systems in terms of Lyapunov densities. The results are provided without assuming local stability and forward completeness of trajectories. As well as general time-varying nonlinear systems, periodically time-varying systems are also considered in this paper, where a weaker criterion is proposed for periodically time-varying systems. Also the existence of Lyapunov densities is proved for general and periodic time-varying nonlinear systems under the asymptotic stability of the equilibrium. [ABSTRACT FROM AUTHOR]

A non-linear SEIR mathematical model for coronavirus disease in India has been proposed, by incorporating the saturated incidence rate on the occurrence of new infections. In the model, the threshold quantity known as the reproduction number is evaluated which determines the stability of disease-free equilibrium and the endemic equilibrium points. The disease-free equilibrium point becomes globally asymptotically stable when the corresponding reproduction number is less than unity, whereas, if it is greater than unity then the endemic equilibrium point comes into existence, which is locally asymptotically stable under certain restrictions on the parameters value in the model. The impact of various parameters on the threshold quantity is signified by the sensitivity analysis. Numerical results imply that by implementing and strictly following the prevention measures a rapid reduction in the reproduction number for COVID-19 can be observed, through which the coronavirus disease can be controlled. [ABSTRACT FROM AUTHOR]

Summary: In this article, the stability of quaternion‐valued neural networks (QVNNs) with hybrid impulses is investigated, which contain stabilizing impulses and destabilizing impulses simultaneously. Through constructing the relationship between impulses and quaternions, this article has obtained a unified criterion for the global exponential stability of the system. First, a new form of impulsive representation is constructed, and the established systems can be decomposed into four different real parts. According to the characteristics of the impulses, the system divides the average interval of impulses (AII) into two cases, that is, Tα<∞ and Tα=∞, which can process hybrid impulses simultaneously. Second, for QVNNs, by using the method of matrix p‐norm and the inequality technology, the criterion of each interval is derived. Considering the impulsive effects on the overall stability of QVNNs, the uniform criterion for the global exponential stability can be obtained. In addition, the convergence rate of QVNNs with hybrid impulses is also discussed. Finally, in order to verify the theoretical conclusions, some numerical simulations are given. [ABSTRACT FROM AUTHOR]

This paper revisits the problem of global asymptotic positioning of uncertain motion systems subject to actuator constraint and friction. A simple model-free saturated control is proposed by incorporating a relay term driven by position error into proportional-derivative (PD) methodology. Lyapunov's direct method is employed to prove global asymptotic positioning stability. The appealing advantages of the proposed approach are that it is conceived within the framework of saturated PD (SPD) control methodology with intuitive structure and absence of modeling parameter and embeds the whole control action within a single saturation function. Benefitted from these advantages, the proposed approach omits the complicated discrimination of the terms that shall be bounded in several saturation functions of the commonly used design and permits easy implementation with an improved performance. An additive feature is that the proposed control has the ability to ensure that the actuator constraint is not breached and assures global asymptotic positioning stability in the presence of unknown friction. Numerical simulations and experimental validations demonstrate the effectiveness and improved performance of the proposed approach. The proposed approach provides a model-free solution for fast transient and high-precision steady-state positioning of uncertain motion systems subject to unknown friction and actuator constraint. [ABSTRACT FROM AUTHOR]

A deterministic model for the transmission dynamics of melioidosis disease in human population is designed and analyzed. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the basic reproduction number ℛ 0 is less than one. It is further shown that the backward bifurcation dynamics is caused by the reinfection of individuals who recovered from the disease and relapse. The existence of backward bifurcation implies that bringing down ℛ 0 to less than unity is not enough for disease eradication. In the absence of backward bifurcation, the global asymptotic stability of the disease-free equilibrium is shown whenever ℛ 0 < 1. For ℛ 0 > 1 , the existence of at least one locally asymptotically stable endemic equilibrium is shown. Sensitivity analysis of the model, using the parameters relevant to the transmission dynamics of the melioidosis disease, is discussed. Numerical experiments are presented to support the theoretical analysis of the model. In the numerical experimentations, it has been observed that screening and treating individuals in the exposed class has a significant impact on the disease dynamics. [ABSTRACT FROM AUTHOR]

This paper deals with the global dynamics of a tuberculosis (TB) model with age-structure and delay. We perform some rigorous analyses for the model, including presenting an explicit formula for the basic reproduction number of the model, addressing the persistence of the solution semi-flow and the existence of the global attractor. Based on these analyses, we establish some results on stability and instability of equilibrium of the system. Finally, some numerical examples are provided to illustrate our obtained results. [ABSTRACT FROM AUTHOR]

This paper deals with a stage-structured predator–prey system which incorporates cannibalism in the predator population and harvesting in both population. The predator population is categorized into two divisions; adult predator and juvenile predator. The adult predator and prey species are harvested via hypothesis of catch-per-unit-effort, whereas juveniles are safe from being harvested. Mathematically, the dynamic behavior of the system such as existing conditions of equilibria with their stability is studied. The global asymptotic stability of prey-free equilibrium point and nonzero equilibrium point, if they exist, is proved by considering respective Lyapunov functions. The system undergoes transcritical and Hopf–Andronov bifurcations. The impacts of predator harvesting rate and prey harvesting rate on the system are analyzed by taking them as bifurcation parameters. The route to chaos is discussed by showing maximum Lyapunov exponent to be positive with sensitivity dependence on the initial conditions. The chaotic behavior of the system is confirmed by positive maximum Lyapunov exponent and non-integer Kaplan–Yorke dimension. Numerical simulations are executed to probe our theoretic findings. Also, the optimal harvesting policy is studied by applying Pontryagin's maximum principle. Harvesting effort being an emphatic control instrument is considered to protect prey–predator population, and preserve them also through an optimal level. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying a stochastic patch structure Nicholson's blowflies system with mixed delays which is a new model for the generalization of classic Nicholson's blowflies system. We examine stochastically ultimate boundedness and global asymptotic stability for the considered model by stochastic analysis technique. Finally, numerical simulations verify theoretical results of the present paper. [ABSTRACT FROM AUTHOR]

This paper deals with the global dynamics for a SLIS epidemic model with infection age. In our model, we also consider the time delay in the progression from the latent individuals to becoming infectious individuals. We verify the well posedness of the model by changing it into an abstract nondensely defined Cauchy problem and find conditions for the existence of disease free equilibrium and endemic equilibrium. The theoretic analysis shows that the disease-free equilibrium is globally asymptotically stable as the basic reproduction number R 0 is less than unity and that the endemic equilibrium is locally asymptotically stable and the system is uniformly persistent as R 0 is greater than unity. The numerical simulations illustrate that the endemic equilibrium may be asymptotically stable as R 0 > 1 . [ABSTRACT FROM AUTHOR]

Motivated by the ubiquitous sampled-data setup in applied control, we examine the stability of a class of difference equations that arises by sampling a right- or left-invariant flow on a solvable matrix Lie group. The map defining such a difference equation has three key properties that facilitate our analysis: (1) its Lie series expansion enjoys a type of strong convergence; (2) the origin is an equilibrium; (3) the algebraic ideals enumerated in the lower central series of the Lie algebra are dynamically invariant. We show that certain global stability properties are implied by stability of the Jacobian linearization of the dynamics at the origin, in particular, global asymptotic stability. If the Lie algebra is nilpotent, then the origin enjoys semiglobal exponential stability. [ABSTRACT FROM AUTHOR]