Conditions For Local Stability Research Articles

Contribution Of Vaccination And Treatment In Reduction Of Re-Infection Of Hbv Disease:A Mathematical Model Analysis And Numerical Simulation

Hepatitis B is a globally infectious disease. HBV-reinfection after HBV-related liver transplantation is a frequent clinical problem. The risk of HBV-reinfection directly correlates with the viral load before liver transplantation. If reinfection does occur in most cases the course of the disease is enhanced compared to the situation before transplantation. The progression of HBVrelated liver disease is accelerated starting 2 months after transplantation and r in re-transplantation or death of the patients. Mathematical modeling of HBV transmission is an interesting research area. In this paper, we present characteristics of HBV virus transmission in the form of a mathematical model. We proposed and analyzed a compartmental nonlinear deterministic mathematical model SVEIRE for transmission dynamics and control of Hepatitis B virus disease. In this model, we used force infection which takes the contact rate of susceptible population and transmission probability into account. We proved that the solution of the considered dynamical system is positive and bounded. The model is studied qualitatively using the stability theory of differential equations and the basic reproductive number that represents the epidemic indicator is obtained from the largest eigenvalue of the next-generation matrix. Both local and global asymptotic stability conditions for disease-free and endemic equilibria are determined. Sensitive analysis results show that enhancing the vaccination rate for newborns and treatment for chronically infected individuals is very effective to stop the transmission of HBV. It is also shown that re-infection of HBV plays a great role in increasing the number of infected populations. Finally, this paper depicts combination of vaccination and treatment that will be the most useful way to control Hepatitis B virus infection.

On the analysis of stability, bifurcation, and chaos control of discrete-time predator-prey model with Allee effect on predator

In this paper, a discrete predator-prey model with Allee effect which is obtained by the forward Euler method has been investigated. The local stability conditions of the model at the fixed point have been discussed. In addition, it is shown that the model undergoes Neimark-Sacker bifurcation by using bifurcation theory. Then, the direction of Neimark-Sacker bifurcation has been given. The OGY method is applied in order to control chaos in considered model due to emergence of Neimark-Sacker bifurcation. Some numerical simulations such as phase portraits and bifurcation figures have been presented to support the theoretical results. Also, the chaotic features are justified numerically by computing Lyapunov exponents. Because of consistency with the biological facts, the parameter values have been taken from literature [Controlling chaos and Neimark-Sacker bifurcation discrete-time predator-prey system, Hacet. J. Math. Stat. 49 (5), 1761-1776, 2020].

Stability, cost-effectiveness, and global sensitivity analysis of COVID-19 model incorporating non-pharmaceutical interventions and indirect transmission

Covid-19 is an ongoing pandemic caused by SARS-CoV-2. Some interventions are implemented to control the spread of the disease. In Indonesia, there is a campaign related to non-pharmaceutical approach called 3M. This campaign is carried out so that people use masks, wash their hands, and keep their distance. In this paper, we propose a mathematical model considering non-pharmaceutical interventions and indirect transmission. The non-pharmaceutical interventions studied are the implementation of mask-wearing, handwashing, and social distancing. The model is presented as a system of first-order differential equations. The basic reproduction number is determined. The system has two equilibrium points, namely the disease-free equilibrium point and the endemic equilibrium point. The local stability condition of the disease-free equilibrium point is proved using the Lienard-Chipart criterion. Center manifold theory is used to prove the local stability condition of the endemic equilibrium point. We also study the optimal control strategy related to mask-wearing, handwashing, and social distancing. Furthermore, cost-effectiveness analysis of intervention strategies is also conducted by studying the average cost-effectiveness ratio of each intervention strategy. Our results show that the most effective strategy to control covid-19 spread is the combination of mask-wearing, handwashing, and social distancing. Moreover, the most cost-effective strategy is mask-wearing intervention. Global sensitivity analysis is performed by studying the partial rank correlation coefficient. The results show that mask-wearing intervention is the most influential intervention on basic reproduction number compared to social distancing and handwashing

Stability and Period-Doubling Bifurcation in a Modified Commensal Symbiosis Model with Allee Effect

In this article, the qualitative behaviour of discrete-time commensal symbiosis model which is obtained by implementing the forward Euler’s scheme is discussed in detail. Firstly, the local stability conditions of fixed points of the model are studied. It is proved that the considered model undergoes Period-Doubling bifurcation around coexistence fixed point with the help of bifurcation theory. In order to support the accuracy of obtained analytical finding, some parameter values have been determined and numerical simulations are carried out for these parameter values. Numerical simulations display new and rich nonlinear dynamical behaviours. More specifically, when the parameter 𝛿 is choosen as a bifurcation parameter, it is seen that the considered discrete-time commensal symbiosis model shows very rich nonlinear dynamical.

Stochastic evolutionary stability in matrix games with random payoffs.

Evolutionary game theory and the concept of an evolutionarily stable strategy have been not only extensively developed and successfully applied to explain the evolution of animal behavior, but also widely used in economics and social sciences. Recently, in order to reveal the stochastic dynamical properties of evolutionary games in randomly fluctuating environments, the concept of stochastic evolutionary stability based on conditions for stochastic local stability for a fixation state was developed in the context of a symmetric matrix game with two phenotypes and random payoffs in pairwise interactions [Zheng et al., Phys. Rev. E 96, 032414 (2017)2470-004510.1103/PhysRevE.96.032414]. In this paper, we extend this study to more general situations, namely, multiphenotype symmetric as well as asymmetric matrix games with random payoffs. Conditions for stochastic local stability and stochastic evolutionary stability are established. Conditions for a fixation state to be stochastically unstable and almost everywhere stochastically unstable are distinguished in a multiphenotype setting according to the initial population state. Our results provide some alternative perspective and a more general theoretical framework for a better understanding of the evolution of animal behavior in a stochastic environment.

Dynamic Analysis of a Delayed Fractional Infectious Disease Model with Saturated Incidence

This paper addresses a new fractional order infectious disease model with saturated incidence and time delay. In the new model, the isolated population and the asymptomatic infected population are considered. The invariant region and positive analysis of the solution of the model are established. Next, the basic reproduction number is obtained by the next-generation matrix method, and the sufficient conditions for local asymptotic stability for arbitrary time delays are proposed. Finally, the correctness of the theoretical results is verified by some numerical simulations.

Information geometry theory of bifurcations? A covariant formulation.

The conventional local bifurcation theory (CBT) fails to present a complete characterization of the stability and general aspects of complex phenomena. After all, the CBT only explores the behavior of nonlinear dynamical systems in the neighborhood of their fixed points. Thus, this limitation imposes the necessity of non-trivial global techniques and lengthy numerical solutions. In this article, we present an attempt to overcome these problems by including the Fisher information theory in the study of bifurcations. Here, we investigate a Riemannian metrical structure of local and global bifurcations described in the context of dynamical systems. The introduced metric is based on the concept of information distance. We examine five contrasting models in detail: saddle-node, transcritical, supercritical pitchfork, subcritical pitchfork, and homoclinic bifurcations. We found that the metric imposes a curvature scalar R on the parameter space. Also, we discovered that R diverges to infinity while approaching bifurcation points. We demonstrate that the local stability conditions are recovered from the interpretations of the curvature R, while global stability is inferred from the character of the Fisher metric. The results are a clear improvement over those of the conventional theory.

A Stage-Structured Fishery Model for African Catfish and Nile Tilapia Feeding on Two Food Resources with Harvesting

In this paper, a fishery model for African catfish and Nile tilapia is formulated. This model is used to compare financial profit and biomass outtakes in a two-species system versus single species systems. We consider a stage-structured fish population model consisting of the aforementioned fish species together with two food resources. The model dynamics include cannibalism, predator-prey, feeding, reproduction, maturation, development, mortality, and harvesting. We prove consistency of the model in the sense that the solutions will stay bounded and nonnegative over time. Conditions for local stability of fish-free equilibrium point are established. The simulation results reveal asymptotically stable solutions with coexistence of African catfish, Nile tilapia, and two food resources. The major conclusion from our findings is that fisheries should culture both species to maximize the biomass outtake and financial profit.

Impact of Hygiene on Malaria Transmission Dynamics: A Mathematical Model

Malaria continues to pose a major public health challenge, especially in developing countries, as 219 million cases of malaria were found in 89 countries. In this paper, a mathematical model using non-linear differential equations is formulated to describe the impact of hygiene on malaria transmission dynamics. The model is divided into seven compartments which includes five human compartments namely; unhygienic susceptible human population (Su), hygienic susceptible human population (Sn), unhygienic infected human population (Iu), hygienic infected human population (In) and the recovered human population (Rn) while the mosquito population is subdivided into susceptible mosquitoes (Sv) and infected mosquitoes Iv. The positivity of the solution shows that a domain exists where the model is biologically meaningful and mathematically well-posed. The Disease-Free Equilibrium (DFE) point of the model is obtained. Then, the basic reproduction number is computed using the next generation method and established the condition for local stability of the disease-free equilibrium. Thereafter the global stability of the disease-free equilibrium was obtained by constructing the Lyapunov function of the model system. Also, sensitivity analysis of the model system was carried out to identify the influence of the parameters on the basic reproduction number. The result shows that the natural death rate of the mosquitoes is most sensitive to the basic reproduction number.

An Arbitrary High-Order DGTD Method With Local Time-Stepping for Nonlinear Field-Circuit Cosimulation

In this article, an efficient transient analysis method with arbitrary high-order accuracy for nonlinear field-circuit cosimulation problems is proposed. The proposed method combines a local time-stepping (LTS) technique with the discontinuous Galerkin time-domain (DGTD) method based on an arbitrary high-order derivatives (ADER) time integration scheme. The stability of the proposed method is not limited by the global time step size, allowing nonlinear and linear elements to update themselves with individual and optimal time intervals in accordance with the local stability condition. It is shown that the efficiency of the proposed method can be significantly improved when the nonlinear effects increase, which requires a smaller time interval to ensure system convergence. Several field-circuit coupling structures are simulated to demonstrate the accuracy and efficiency of the proposed method, confirming that it represents an effective tool for transient analysis of nonlinear field-circuit structures.

Population Dynamic Study of Prey‐Predator Interactions with Weak Allee Effect, Fear Effect, and Delay

In this study, a predator‐prey model with the Allee effect and fear effect is established. We use the comparison principle to prove boundedness. The zero equilibrium point and nonzero equilibrium point of the model are calculated, and the local stability conditions are obtained. Next, according to the Sotomayor theorem, the cross‐sectional conditions of transcritical bifurcation and Hopf bifurcation are obtained. The conditions for the Hopf bifurcation to be supercritical or subcritical can be calculated by the normal form theory. Then, to make the model more realistic, we introduce the gestation delay in the proposed mathematical model. Stability and Hopf bifurcation are also analyzed. Finally, several numerical simulations are presented to verify the conclusions. Our results demonstrate that the Allee effect, fear effect, and delay play significant roles in population dynamics. The Allee effect and delay destabilize the originally stable model, after which Hopf bifurcation occurs. However, the fear effect can enhance stable coexistence.

Nonlinear Dynamics and Multistability in a Cobweb Model

This paper studies the dynamic characteristics of an economic cobweb model whose producers adopt a gradient‐based mechanism. The model’s producers are taken to be boundedly rational, and then the model is studied as a 1D and 2D discrete time dynamical system. Due to the lack of information in the market regarding the function of demand, the producers depend on the estimation of the marginal profit variations so as to update their prices next period. The equilibrium price point is calculated, and its local stability conditions are discussed through analytical and numerical investigations. In the 2D model, where the memory factor is introduced, the equilibrium price is obtained, and it is shown that it can be destabilized through chaotic behaviors, which are formed due to period doubling and Neimark‐Sacker bifurcation. Furthermore, we study the symmetric case when the producers update their prices based on the marginal profit in the past two time periods, and therefore, we investigate the influence of the speed of adjustment parameter on the stability of equilibrium price.

Stability and Bifurcation Analysis of Fractional-Order Delayed Prey–Predator System and the Effect of Diffusion

Most biological systems have long-range temporal memory. Such systems can be modeled using fractional-order differential equations. The combination of fractional-order derivative and time delay provides the system more consistency with the reality of the interactions and higher degree of freedom. A fractional-order delayed prey–predator system with the fear effect has been proposed in this work. The time delay is considered in the cost of fear; therefore, there are no dynamical changes observed in the system due to time delay in the absence of fear. The existence and uniqueness of the solutions of the proposed system are studied along with non-negativity and boundedness. The existence of biologically relevant equilibria is discussed, and the conditions for local asymptotic stability are derived. Hopf bifurcation occurs in the system with respect to delay parameter. Further, a spatially extended system is proposed and analyzed. Hopf bifurcation also occurs in the extended system due to the delay parameter. Numerical examples are provided in support of analytical findings. Fractional-order derivative improves the stability and damps the oscillatory behaviors of the solutions of the system. Bistability behavior of the system admits stable dynamics by decreasing the fractional-order. Also, chaotic behavior is destroyed by decreasing fractional-order.

Stability Analysis of Systems with Recurrent Neural Network Controllers

The increasing utilization of recurrent neural networks (RNNs) in feedback control and the complexity of RNNs motivate us to provide a method for verifying the stability of RNN-controlled systems. To circumvent the nonlinearities in RNNs, we describe them using local integral quadratic constraints. A sufficient condition based on linear matrix inequalities is achieved for local asymptotic stability of feedback interconnections between a linear time-invariant plant and an RNN controller. This condition is employed as a constraint to construct semidefinite programming for finding a large ellipsoidal inner approximation to the region of attraction. A bisection algorithm is proposed to enlarge the inner approximation further. A necessary condition for local asymptotic stability is derived from linearization, which can be exploited to determine initial values for the bisection algorithm. Illustrative examples complete the presentation and show the improvement by our work.

A mathematical model analysis of marriage divorce

In this paper, a deterministic model for the marriage divorce in a population is proposed and analysed qualitatively using the stability theory of differential equations. The basic reproduction number with respect to the divorce free equilibrium was obtained using next generation matrix approach. The conditions for local and global asymptotic stability of divorce free and endemic equilibria were established. The model exhibits backward bifurcation and the sensitivity indices of the parameters with respect to eradicating or spreading divorce in marriage was determined. Numerical simulation was performed and displayed graphically to justify the analytical results.

A competition model in the chemostat with allelopathy and substrate inhibition

<p style='text-indent:20px;'>A model of two microbial species in a chemostat competing for a single resource is considered, where one of the competitors that produces a toxin, which is lethal to the other competitor (allelopathic inhibition), is itself inhibited by the substrate. Using general growth rate functions of the species, necessary and sufficient conditions of existence and local stability of all equilibria of the four-dimensional system are determined according to the operating parameters represented by the dilution rate and the input concentration of the substrate. With Michaelis-Menten or Monod growth functions, it is well known that the model can have a unique positive equilibrium which is unstable as long as it exists. If a non monotonic growth rate is considered (which is the case when there is substrate inhibition), it is shown that a new positive equilibrium point exists which can be stable according to the operating parameters of the system. We describe its operating diagram, which is the bifurcation diagram giving the behavior of the system with respect to the operating parameters. By means of this bifurcation diagram, we show that the general model presents a set of fifteen possible behaviors: washout, competitive exclusion of one species, coexistence, multi-stability, occurrence of stable limit cycles through a super-critical Hopf bifurcations, homoclinic bifurcations and flip bifurcation. This diagram is very useful to understand the model from both the mathematical and biological points of view.</p>

Dissipative Deep Neural Dynamical Systems

In this paper, we provide sufficient conditions for dissipativity and local asymptotic stability of discrete-time dynamical systems parametrized by deep neural networks. We leverage the representation of neural networks as pointwise affine maps, thus exposing their local linear operators and making them accessible to classical system analytic and design methods. This allows us to “crack open the black box” of the neural dynamical system's behavior by evaluating their dissipativity, and estimating their stationary points and state-space partitioning. We relate the norms of these local linear operators to the energy stored in the dissipative system with supply rates represented by their aggregate bias terms. Empirically, we analyze the variance in dynamical behavior and eigenvalue spectra of these local linear operators with varying weight factorizations, activation functions, bias terms, and depths.

Spatiotemporal Evolution Characteristics of Time‐Delay Ecological Competition Systems with Food‐Limited and Diffusion

In this paper, we put forward a time‐delay ecological competition system with food restriction and diffusion terms under Neumann boundary conditions. For the case without delay, the conditions for local asymptotic stability and Turing instability are constructed. For the case with delay, the existence of Hopf bifurcation is demonstrated by analyzing the root distribution of the corresponding characteristic equations. Furthermore, by using the normal form theory and the center manifold reduction of partial functional differential equations, explicit formulas are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Finally, some simulation examples are provided to substantiate our analysis.

Dynamic System Analysis of Investment in Both Production and R&D with Time Delay

This paper, according to the process of capital return, establishes a differential dynamics model of investment with two time delays. When both time delays are zero, it is proved that the model is positively invariant, uniformly bounded, and globally asymptotically stable by using the comparison principle and Bendixson–Dulac theorem. When at least one time delay is not zero, according to Hopf bifurcation theorem, the conditions of local asymptotic stability and existence of periodic solution of investment model are obtained. By using the normal form theory and the center manifold theory, the discriminant formula of periodic solution property of investment model is given. Under the condition of controlled time delay, the model is numerically simulated to verify the correctness of relevant analytical conclusions. Therefore, the investment model describes the dynamic process and development trend of project investment quite closely.

The dynamics and analysis of an SIS disease of a stage-structured prey-predator model with a prey refuge

In this paper, the dynamical behaviour of an {epidemiological system} has been investigated. A stage-structured prey-predator model includes harvest and refuge for only prey, the disease of type (SIS) is just in the immature of the prey and the disease is spread by contact and by external source has been studied. transmission of infectious diseases in the prey populations has been described by the linear type. While Lotka-Volterra functional response is used to describe the predation process of the whole prey population. This model has been represented by a set of nonlinear differential equations. solution's existence, uniqueness and boundedness have been studied. The local and global stability conditions of all the equilibrium points have been confirmed. As a final point, numerical simulation has been used to study the global dynamics of the model.