Sufficient Conditions For Local Stability Research Articles

Dual CAR-T cell therapy for glioblastoma: strategies to cure tumour diseases based on a mathematical model

AbstractThe CAR-T cell immunotherapy entails the genetic reprogramming of T-lymphocytes, which then engage with cancer cells, triggering an anti-tumour immune response. While this treatment has gained approval for hematological cancers, addressing solid tumours presents new obstacles. Challenges include the heterogeneity of antigen expression within solid tumours, encompassing antigen-positive non-tumoural cells, the presence of immune inhibitory molecules, and the difficulty of CAR-T cell trafficking within the tumour microenvironment. In this article, we analytically study a generalisation of a mathematical model proposed by León-Triana et al. (Cancers 13(4):703, 2021a. https://doi.org/10.3390/cancers13040703, Commun Nonlinear Sci Numer Simul 94:105570). This model focuses on the dynamics of glioblastoma, the most aggressive brain tumour, and its response to CAR-T cell treatment. We study the basic properties of the model, the dynamics of the solutions of the model when the treatment is not sustained during the time, and finally we study analytically the model when the therapy is constant, periodic and/or impulsive. We derive sufficient conditions for global stability of tumour-free equilibrium, as well as necessary and sufficient conditions for local stability of the equilibrium obtaining conditions for an effective treatment. Finally, we perform different numerical simulations to find the strategies to keep the tumour under control. The obtained results are based on a combination of different analytical techniques in differential equations, dynamical systems and numerical simulations.

Dynamical Behaviours of Stage-Structured Fractional-Order Prey-Predator Model with Crowley–Martin Functional Response

In this paper, the dynamic behaviour of the stage-structure prey-predator fractional-order derivative system is considered and discussed. In this model, the Crowley–Martin functional response describes the interaction between mature preys with a predator. The existence, uniqueness, non-negativity, and the boundedness of solutions are proved. All possible equilibrium points of this system are investigated. The sufficient conditions of local stability of equilibrium points for the considered system are determined. Finally, numerical simulation results are carried out to confirm the theoretical results.

Dynamics of an Antitumour Model with Pulsed Radioimmunotherapy.

In this paper, an antitumour model for characterising radiotherapy and immunotherapy processes at different fixed times is proposed. The global attractiveness of the positive periodic solution for each corresponding subsystem is proved with the integral inequality technique. Then, based on the differentiability of the solutions with respect to the initial values, the eigenvalues of the Jacobian matrix at a fixed point corresponding to the tumour-free periodic solution are determined, resulting in a sufficient condition for local stability. The solutions to the ordinary differential equations are compared, the threshold condition for the global attractiveness of the tumour-free periodic solution is provided in terms of an indicator R0, and the permanence of a system with at least one tumour-present periodic solution is investigated. Furthermore, the effects of the death rate, effector cell injection dosage, therapeutic period, and effector cell activation rate on indicator R0 are determined through numerical simulations, and the results indicate that radioimmunotherapy is more effective than either radiotherapy or immunotherapy alone.

Security Analysis of Linearization for Nonlinear Networked Control Systems Under DoS

We explore a security analysis of nonlinear networked control systems under denial-of-service (DoS) attacks. In particular, we focus on the vulnerability of a linearization approach in a stabilization problem. When linearization-based control is used, DoS attacks can make the state leave the region of attraction. This situation can occur when the initial state lies outside a certain region around the equilibrium, the size of which depends on the strength of the attacks on the network. In this article, we first derive a sufficient condition for local stability of a nonlinear system under averagely constrained DoS attacks. Then, we investigate the region of attraction and derive an explicit relation between the initial states and the DoS parameters to guarantee convergence of the state trajectories. Furthermore, we investigate a destabilization problem from the perspective of malicious attackers and present an instability condition. By a numerical example, we demonstrate a new insight foattackers to induce undesired behaviors of the control system.

Stability and Hopf Bifurcation of Three-Species Prey-Predator System with Time Delays and Allee Effect

Allee effect is one of the important factors in ecology, and taking it into account can cause significant impacts in the system dynamics. In this paper, we study the dynamics of a two-prey one-predator system, where the growth of both prey populations is subject to Allee effects, and there is a direct competition between the two-prey species having a common predator. Two discrete time delays τ1 and τ2 are incorporated into the model to represent reaction time of predators. Sufficient conditions for local stability of positive interior equilibrium and existence of Hopf bifurcations in terms of threshold parameters τ1∗ and τ2∗ are obtained. A Lyapunov functional is deducted to investigate the global stability of positive interior equilibrium. Sensitivity analysis to evaluate the uncertainty of the state variables to small changes in the Allee parameters is also investigated. Presence of Allee effect and time delays in the model increases the complexity of the model and enriches the dynamics of the system. Some numerical simulations are provided to illustrate the effectiveness of the theoretical results. The model is highly sensitive to small changes in Allee parameters at the early stages and with low population densities, and this sensitivity decreases with time.

Analysis of a spatial memory model with nonlocal maturation delay and hostile boundary condition

In this paper, we propose and investigate a memory-based reaction-diffusion equation with nonlocal maturation delay and homogeneous Dirichlet boundary condition. We first study the existence of the spatially inhomogeneous steady state. By analyzing the associated characteristic equation, we obtain sufficient conditions for local stability and Hopf bifurcation of this inhomogeneous steady state, respectively. For the Hopf bifurcation analysis, a geometric method and prior estimation techniques are combined to find all bifurcation values because the characteristic equation includes a non-self-adjoint operator and two time delays. In addition, we provide an explicit formula to determine the crossing direction of the purely imaginary eigenvalues. The bifurcation analysis reveals that the diffusion with memory effect could induce spatiotemporal patterns which were never possessed by an equation without memory-based diffusion. Furthermore, these patterns are different from the ones of a spatial memory equation with Neumann boundary condition.

Hopf bifurcation of a delayed worm model with two latent periods

We investigate a delayed epidemic model for the propagation of worm in wireless sensor network with two latent periods. We derive sufficient conditions for local stability of the worm-induced equilibrium of the system and the existence of Hopf bifurcation by regarding different combination of two latent time delays as the bifurcation parameter and analyzing the distribution of roots of the associated characteristic equation. In particular, we investigate the direction and stability of the Hopf bifurcation by means of the normal form theory and center manifold theorem. To verify analytical results, we present numerical simulations. Also, the effect of some influential parameters of sensor network is properly executed so that the oscillations can be reduced and removed from the network.

On Matrix Norms, Stability and Stabilization of a Class of Discrete Takagi–Sugeno Fuzzy Systems

Based on a result of stability for linear time varying systems, a sufficient condition for global stability that does not use Lyapunov theory has been established for a class of discrete Takagi–Sugeno fuzzy systems. The condition involves testing the norms of products of the matrices of the consequences. The approach seeks to determine if at an instant of time the system becomes a contraction mapping. Because the computation burden associated with these tests may become prohibitive, a necessary and sufficient condition for local stability is derived. In a way, this decreases the computational burden and allows to customize the test. A two-step iterative stabilization algorithm is then introduced. The problem involves norm minimization and a posteriori stability test. Numerical examples are presented to demonstrate the applicability of the approach.

Stability and Performance of Compound TCP With a Proportional Integral Queue Policy

The increase in queueing delays in the Internet motivates the study of transmission control protocol (TCP) and queue management policies. This paper studies Compound TCP with a Proportional–Integral (PI) policy for queue management at the Internet routers. A nonlinear fluid model for the Compound TCP-PI system is considered. For this model, a sufficient condition for local stability, which yields some design guidelines, is derived. Fragility analysis of this model highlights that even marginal variations in the PI parameters may induce instability. Next, a regime where the integral component, of the PI policy, operates over a small time-scale is considered. For a fluid model befitting this regime, the necessary and sufficient condition for local stability is derived. It is explicitly shown that, when this condition is violated, the system undergoes a Hopf bifurcation, which would lead to limit cycles. Furthermore, a detailed local bifurcation analysis is conducted to characterize the type of the Hopf bifurcation and determine the orbital stability of the limit cycles. Packet-level simulations corroborate the analytical insight. The analysis and simulations demonstrate some drawbacks of the PI policy. Following this, a simple threshold-based queue policy, which has desirable stability properties, is proposed for queue management at routers. Furthermore, a simulation-based performance evaluation reveals that this policy outperforms PI. Through a combination of theory and simulations, it is shown that the threshold-based policy can ensure system stability, while ensuring reduced queueing delays.

A Delayed Epidemic Model for Propagation of Malicious Codes in Wireless Sensor Network

In this paper, we investigate a delayed SEIQRS-V epidemic model for propagation of malicious codes in a wireless sensor network. The communication radius and distributed density of nodes is considered in the proposed model. With this model, first we find a feasible region which is invariant and where the solutions of our model are positive. To show that the system is locally asymptotically stable, a Lyapunov function is constructed. After that, sufficient conditions for local stability and existence of Hopf bifurcation are derived by analyzing the distribution of the roots of the corresponding characteristic equation. Finally, numerical simulations are presented to verify the obtained theoretical results and to analyze the effects of some parameters on the dynamical behavior of the proposed model in the paper.

A mathematical model of viral oncology as an immuno-oncology instigator.

We develop and analyse a mathematical model of tumour-immune interaction that explicitly incorporates heterogeneity in tumour cell cycle duration by using a distributed delay differential equation. We derive a necessary and sufficient condition for local stability of the cancer-free equilibrium in which the amount of tumour-immune interaction completely characterizes disease progression. Consistent with the immunoediting hypothesis, we show that decreasing tumour-immune interaction leads to tumour expansion. Finally, by simulating the mathematical model, we show that the strength of tumour-immune interaction determines the long-term success or failure of viral therapy.

Hopf bifurcation of a heroin model with time delay and saturated treatment function

In this paper, local stability and Hopf bifurcation of a delayed heroin model with saturated treatment function are discussed. First of all, sufficient conditions for local stability and existence of Hopf bifurcation are obtained by regarding the time delay as a bifurcation parameter and analyzing the distribution of the roots of the associated characteristic equation. Directly afterward, properties of the Hopf bifurcation, such as the direction and stability, are investigated with the aid of the normal form theory and the manifold center theorem. Finally, numerical simulations are presented to justify the obtained theoretical results, and some suggestions are offered for controlling heroin abuse in populations.

Stability, convergence and Hopf bifurcation analyses of the classical car-following model

Reaction delays play an important role in determining the qualitative dynamical properties of a platoon of vehicles traversing a straight road. In this paper, we investigate the impact of delayed feedback on the dynamics of the Classical Car-Following Model (CCFM). Specifically, we analyze the CCFM in no delay, small delay and arbitrary delay regimes. First, we derive a sufficient condition for local stability of the CCFM in no-delay and small-delay regimes using. Next, we derive the necessary and sufficient condition for local stability of the CCFM for an arbitrary delay. We then demonstrate that the transition of traffic flow from the locally stable to the unstable regime occurs via a Hopf bifurcation, thus resulting in limit cycles in system dynamics. Physically, these limit cycles manifest as back-propagating congestion waves on highways. In the context of human-driven vehicles, our work provides phenomenological insight into the impact of reaction delays on the emergence and evolution of traffic congestion. In the context of self-driven vehicles, our work has the potential to provide design guidelines for control algorithms running in self-driven cars to avoid undesirable phenomena. Specifically, designing control algorithms that avoid jerky vehicular movements is essential. Hence, we derive the necessary and sufficient condition for non-oscillatory convergence of the CCFM. Next, we characterize the rate of convergence of the CCFM, and bring forth the interplay between local stability, non-oscillatory convergence and the rate of convergence of the CCFM. Further, to better understand the oscillations in the system dynamics, we characterize the type of the Hopf bifurcation and the asymptotic orbital stability of the limit cycles using Poincare normal forms and the center manifold theory. The analysis is complemented with stability charts, bifurcation diagrams and MATLAB simulations.

Permanence and Hopf bifurcation of a delayed eco-epidemic model with Leslie–Gower Holling type III functional response

This paper is concerned with a delayed eco-epidemic model with a Leslie–Gower Holling type III functional response. The main results are given in terms of permanence and Hopf bifurcation. First of all, sufficient conditions for permanence of the model are established. Directly afterward, sufficient conditions for local stability and existence of Hopf bifurcation are obtained by regarding the delay as bifurcation parameter. Finally, properties of the Hopf bifurcation are investigated with the aid of the normal form theory and centre manifold theorem. Numerical simulations are carried out to verify the obtained theoretical results.

一类具有时滞的云杉蚜虫种群模型的Hopf分岔分析

The dynamic behavior of a population model with stage structure for spruce budworms with time delay was investigated. Firstly, existence of a unique positive equilibrium of the model was discussed and sufficient conditions for local stability of the positive equilibrium and Hopf bifurcation occurrence were obtained. Next, the direction of the Hopf bifurcation and the stability of the periodic bifurcation solutions were analyzed with the normal form method combined with the center manifold theorem. Finally, some numerical simulations to verify the theoretical results were also conducted. The work provides an applicable reference for control of spruce budworms.

Stability and Hopf bifurcation analysis of lac Operon model with distributed delay and nonlinear degradation rate.

We propose a simple model of lac operon that describes the expression of B-galactosidase from lac Z gene in Escherichia coli, through the interaction among several identical mRNA. Our goal is to explore the complex dynamics (i.e. the oscillation phenomenon) of this architecture mediated by this interaction. This model was theoretically and numerically investigated using distributed time delay. We considered the average delay as a bifurcation parameter and the nonlinear degradation rate as a control parameter. Sufficient conditions for local stability were gained by using the Routh-Hurwitz criterion in the case of a weak delay kernel. Then we proved that Hopf bifurcation happened and the direction of the periodic solution was determined using multiple time scale technique. Our results suggest that the interaction among several identical mRNA plays the main role in gene regulation.

Existence and Stability of Equilibrium of DC Microgrid With Constant Power Loads

The problem of existence and stability of equilibria of DC microgirds with constant power loads methods is addressed in this paper. Constant power loads (CPLs) often cause instability due to its negative impedance characteristics. What is the worse, CPLs may cause that the system admits no equilibria for its nonlinearity. The purpose of paper can be summarized as: A) designing a controller to overcome CPLs instability; B) obtaining the sufficient conditions to guarantee the existence of the equilibria; C) Under the conditions of equilibria existing, obtaining the sufficient conditions of local stability. For the first objective, the method based on virtual resistance and virtual inductance is proposed. For the second objective, we transform the problem of solvability of quadratic equations into the existence of the fixed point of a mapping. Thus, the analytical sufficient conditions is obtained based on Tarski fixed point theorem. Moreover, a numerical method is also provided to reduce conservatism. For the third objective, the small-signal model is established to predict the system qualitative behavior around equilibria. The stability conditions are obtained by using quadratic eigenvalue problem (GEP) theorem. These explicit conditions are obtained as a function of system parameters and it lead to build reliable microgrid.

Hopf Bifurcation of a Delayed Ecoepidemic Model with Ratio-Dependent Transmission Rate

A delayed ecoepidemic model with ratio-dependent transmission rate has been proposed in this paper. Effects of the time delay due to the gestation of the predator are the main focus of our work. Sufficient conditions for local stability and existence of a Hopf bifurcation of the model are derived by regarding the time delay as the bifurcation parameter. Furthermore, properties of the Hopf bifurcation are investigated by using the normal form theory and the center manifold theorem. Finally, numerical simulations are carried out in order to validate our obtained theoretical results.

Stability Analysis of Integrated Pest Management with Impulsive Biological Control

The aim of the present work is to study the dynamics of stage-structured pest control model including biological control, i.e. by releasing of natural enemies and infected pests periodically. It is assumed that only immature susceptible pests are attacked by natural enemies admitting Beddington DeAngelis functional response and mature susceptible pests are contacted by infected pests with bilinear incidence rate and become exposed. The sufficient condition for local stability of pest extinction periodic solution is derived by making use of Floquet’s theory and small amplitude perturbation technique. The global attractivity of pest extinction periodic solution is also established by applying comparison principle of impulsive differential equations.

Sufficient Stability Conditions of Nonlinear Differential Systems Under Impulsive Control With State-Dependent Delay

In this technical note we study the delayed impulsive control of nonlinear differential systems, where the impulsive control involves the delayed state of the system for which the delay is state-dependent. Since the state dependence of the delay makes the impulsive transients dependent on the historical information of the states, which means that it is hard to know exactly a priori how far in the history the information is needed, the main challenge is how to determine the historical states. We resolve this challenge and establish some sufficient conditions for local stability of nonlinear differential systems with state-dependent delayed impulsive control based on impulsive control theory. Two examples are given to show the effectiveness of the proposed approach.