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Building upon our previous work on the Wilson-Cowan equations with distributed delays (Kaslik et al., 2022), we study the dynamic behavior in a system of two coupled Wilson-Cowan pairs. We focus in particular on understanding the mechanisms that govern the transitions in and out of oscillatory regimes associated with pathological behavior. We investigate these mechanisms under multiple coupling scenarios, and we compare the effects of using discrete delays versus a weak Gamma delay distribution. We found that, in order to trigger and stop oscillations, each kernel emphasizes different critical combinations of coupling weights and time delay, with the weak Gamma kernel restricting oscillations to a tighter locus of coupling strengths, and to a limited range of time delays. We finally illustrate the general analytical results with simulations for two particular applications: generation of beta-rhythms in the basal ganglia, and alpha oscillations in the prefrontal-limbic system. • We study long-term dynamics in coupled Wilson–Cowan systems with distributed delays. • Our analysis focuses on understanding the system's transitions in and out of stable oscillations. • We study how these transitions change under different connectivity schemes and strengths. • We explore the effects of different delay distributions and average delay values. • The analysis is illustrated with simulations in two different functional brain networks. [ABSTRACT FROM AUTHOR]

This study aims to present a complete Hopf bifurcation analysis of a model describing the relationship between prey and predator. A ratio-dependent reaction–diffusion system with two discrete time delays operating under Neumann boundary conditions governs the model that represents this competition. The bifurcation parameter for the analysis is a delay parameter that reflects the amount of time needed for the predator to be able to hunt. Bilazeroğlu and Merdan's algorithm (Bilazeroğlu et al., 2021), which is developed by using the center manifold theorem and normal form theory, is used to establish the existence of Hopf bifurcations and also the stability of the bifurcating periodic solutions. The same procedure is used to illustrate some specific bifurcation properties, such as direction, stability, and period. Furthermore, by examining a model with constant coefficients, we also analyze how diffusion and the amount of time needed for prey to mature impact the model's dynamics. To support the obtained analytical results, we also run some numerical simulations. The results indicate that the dynamic of the mathematical model is significantly influenced by diffusion, the amount of time needed for the predator to gain the capacity to hunt, and the amount of time required for prey to reach maturity that the predator can hunt. [ABSTRACT FROM AUTHOR]

• We provide an epidemic SIR model with long-range temporal memory governed by delay differential equations with fractional-order. • The existence of steady states and the asymptotic stability of the steady states are discussed. • The occurrence of Hopf bifurcation is captured when the time-delay passes through a critical value. • Theoretical results are validated numerically. In this paper, we provide an epidemic SIR model with long-range temporal memory. The model is governed by delay differential equations with fractional-order. We assume that the susceptible is obeying the logistic form in which the incidence term is of saturated form with the susceptible. Several theoretical results related to the existence of steady states and the asymptotic stability of the steady states are discussed. We use a suitable Lyapunov functional to formulate a new set of sufficient conditions that guarantee the global stability of the steady states. The occurrence of Hopf bifurcation is captured when the time-delay τ passes through a critical value τ *. Theoretical results are validated numerically by solving the governing system, using the modified Adams-Bashforth-Moulton predictor-corrector scheme. Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model. In some cases, the phase portrait gets stretched as the order of the derivative is reduced. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a mathematical model of tumor vaccine efficacy in the context of immunotherapy treatment. The model is governed by fractional-order delay differential equations with a control variable. Transforming growth factor beta (TGF- β) inhibition alone gives limited clinical advantages, but when combined with a tumor vaccine, it can dramatically increase the immune system's anti-tumor response. The mathematical model takes into account tumor growth dynamics, TGF- β concentrations, cytotoxic effector cells, and regulatory T-cells. The non-negativity of the solutions to such a model has been examined. The steady-state stability and Hopf bifurcation of tumor time delays τ are investigated. An optimal fractional-order control problem is derived and analyzed in the presence of immunotherapy treatments. Using Adams–Bashforth–Moulton predictor–corrector algorithms, we illustrate numerical examples that confirm the analytical findings. The optimal control treatment method reduces the load of tumor cells while increasing the effector cells. [ABSTRACT FROM AUTHOR]

Highlights • A network with wireless access links with a delay differential equation (DDE) model and input delay is considered. • Integral sliding mode control is assumed as the nonlinear robust active queue management in the network. • Hopf bifurcation analysis is investigated with the transmission delay considered as the Hopf bifurcation parameter. • The direction and the stability of the bifurcation periodic solutions are determined. • Simulation results are provided to illustrate the system behavior with different parameters. Abstract Transmission control protocol (TCP), which is usually implemented in the transport layer of a communication network, can detect the congestion after its occurrence. But congestion control can be more complicated in networks with wireless access links due to time-varying fading and packet error rate (PER). Thus, it is essential to use robust active queue management (RAQM) to prevent congestion with certain tolerance against serious disturbance of the wireless environment. As the stability of a TCP/RAQM is tremendously related to the transmission delay, this paper focuses on Hopf bifurcation analysis of a network governed by nonlinear delayed differential equations and an integral sliding mode control as a nonlinear RAQM. The transmission delay is assumed as the bifurcation parameter. It is shown that if the delay passes through a critical value, the closed loop system would not remain stable and the Hopf bifurcation occurs. Moreover, based on the normal form theory and the center manifold theorem, the direction and the stability of the bifurcation periodic solutions are determined. Finally, simulation results are given to confirm the analytical achievements and to show the effectiveness of the procedure compared to some recent protocols. [ABSTRACT FROM AUTHOR]

As a widely used tool modeling some processes and systems in a variety of fields, fractional delay differential equations (FDDEs) with higher order have attracted much attention of the scientific community for years. Motivated by Yao et al. (2022), in which a single term has been done, we are much more interested in the stability analysis for multi-term FDDEs. In addition to the widely used Laplace transform method and decoupling technique for the characteristic equation, a region embedding technique is first introduced to handle the multiple fractional exponents. The existing results are generalized to multi-term FDDEs with higher order and the damping term of the classical integer-order delay differential equation is extended to fractional calculus. Numerical simulations for FDDEs and time-fractional telegraph equations with time delay are presented to illustrate the efficiency and validity of our results. • A novel condition for the stability is expressed in an algebraical criterion. • The Laplace transform method and a method of region embedding are used. • Our results extend the stability results in Yao et al. (2022) to the multi-term FDDEs. • The attenuation phenomenon of the damping term is generalized to FDDEs. [ABSTRACT FROM AUTHOR]

In this study, we are concerned with the dynamics of a new established fractional-order delayed zooplankton–phytoplankton system. The existence and uniqueness of the solution are proved via Banach fixed point theorem. Non-negativeness of the solution is studied by mathematical inequality technique. The boundedness of the solution is analyzed by virtue of constructing an appropriate function. A novel delay-independent sufficient condition ensuring the stability and the onset of Hopf bifurcation for the established fractional-order delayed zooplankton–phytoplankton system is derived by means of Laplace transform, stability criterion and bifurcation knowledge of fractional-order differential equation. The global stability condition for the involved fractional-order delayed zooplankton–phytoplankton system is built by using a suitable positive definite function. Taking advantage of hybrid control tactics, we effectively control the time of occurrence of Hopf bifurcation for the established fractional-order delayed zooplankton–phytoplankton system. The study manifests that delay plays a vital role in controlling the stability and the time of occurrence of Hopf bifurcation for the involved fractional-order delayed zooplankton–phytoplankton system and the fractional-order controlled zooplankton–phytoplankton system involving delays. To verify the correctness of established chief results, computer simulation figures are distinctly displayed. The derived conclusions of this research are entirely new and possess potential theoretical value in preserving the balance of biological population. Up to now, there are few publications on detailed and comprehensive dynamic analysis on fractional-order delayed zooplankton–phytoplankton system via various exploration ways. [ABSTRACT FROM AUTHOR]

Fractional derivative and delay are important tools in modeling memory properties in the natural system. This work deals with the stability analysis of a fractional order delay differential equation D α x (t) = δ x (t − τ) − ϵ x (t − τ) 3 − p x (t) 2 + q x (t). We provide linearization of this system in a neighborhood of equilibrium points and propose linearized stability conditions. To discuss the stability of equilibrium points, we propose various conditions on the parameters δ , ϵ , p , q and τ. Even though there are five parameters involved in the system, we are able to provide the stable region sketch in the q δ − plane for any positive ϵ and p. This provides the complete analysis of stability of the system. Further, we investigate chaos in the proposed model. This system exhibits chaos for a wide range of delay parameter. • Delay differential equation with cubic nonlinearity with fractional order is discussed. • Stability analysis is provided by using the linearization near equilibrium points. • Stable region is sketched in the parameter plane q – δ. • Stability results are valid for any positive parameters p and ϵ. • Chaos is observed for a wide range of parameters. [ABSTRACT FROM AUTHOR]

Among many other factors that affect the preventive interventions to any infectious disease, not reporting timely in a hospital is also one of the catastrophic behavior of human beings in any society. Similarly, masses who do not report make it difficult for healthcare researchers to measure the actual data and develop prevention strategies, accordingly. Therefore, there is a critical need to structure a potential epidemic model with the unreported class of individuals. This novel idea is deliberated in this paper to study the profiles of the epidemic model of virulent diseases due to the individuals that report timely and those who don't report in hospitals for any reason. Mathematically, a system of seven equations is taken into consideration, which describes the susceptible individuals, exposed, people who do not report to the hospital and those who report to the hospital, and individuals who are quarantined, infected, and recovered. So, with the consideration of new compartments, the conventional SIR epidemic model expands to SEU R R P QIR. The innovative design is made more realistic by utilizing proportional fractional-order differential equations with time delay. A special simplified expansion of this derivative reduces its computational cost and produces the results with fractional index, which helps to predict each fractional change. In addition, an optimal control methodology is also carried out to analyze the effectiveness of the awareness campaign in shifting the unreported individuals to the reported class, with optimal cost function for the unreported cases. Discussions are supported through the very recent deadly pandemic as an example to conclude the practical advantage of the model. The sensitivity analysis of basic reproduction numbers based on effective awareness campaigns is also the part of this study, which infers public awareness campaigns may be devised to motivate and guide such individuals to approach any healthcare center or a hospital. • Compartments for unreported and reported cases in the epidemic model. • Analysis through system of fractional order differential equations with time lags. • Realistic profiles of disease dynamics at different states. • Awareness campaign as optimal control approach to restrain unreported class. [ABSTRACT FROM AUTHOR]

• A novel necessary and sufficient condition for the stability is expressed in an easily verified algebraical criterion. • The Laplace transform method and a method of imbedding the characteristic equation are used. • Our results yield a delay independent stability condition for discrete delays, which coincides with the well-known real-valued differential equations. • For fractional-order complex-valued integro-differential equation, the stability criterion is delay-dependent. In this paper, we investigate the global asymptotic stability of fractional-order complex-valued differential equations with distributed delays. Based on the Laplace transform method, a novel necessary and sufficient condition for the stability is established by imbedding the characteristic equation into two-dimensional complex system. The algebraical criterion is expressed by the fractional exponent, coefficients and the delay. Finally, two numerical examples are given to show the feasibility and effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

• In this article we have studied the asymptotic stability and mean square stability of fractional-order (1,2] stochastic differential equations (SDEs). • We have considered the family of SDEs with variable delay in state. • For obtaining main results we used Banach fixed point theorem and imposed Lipschitz condition on nonlinearity. • Results are advanced and weighted enough as a contribution to stability theory of fractional SDEs. In this article, we discuss the asymptotic stability and mean square stability of stochastic differential equations of fractional-order 1 < α ≤ 2. We have considered the family of stochastic differential equations with variable delay in the state. For proving our main results, we apply the Banach fixed point theorem and imposed the Lipschitz condition on nonlinearity. Finally, we present an example to illustrate the obtained theory. [ABSTRACT FROM AUTHOR]

• The first contribution of this research is Hopf bifurcation analysis of a class of system of reaction-diffusion equations involving two discrete time delays under Newmann boundary conditions. The research presents the condition on parameters at which a periodic solution arises locally which is important for the control of a system governed by this class of equations. • The second contribution is determining some properties of existing periodic solutions, including its stability and period. These properties are important to understand the behavior of solutions of a system belonging to this class of equations. • The third contribution is the derivation of an algorithm that will be used for the existence and direction analysis of Hopf bifurcation of a system of equations which lie in that class by simply utilizing the cooefficients in the characteristic equation of its linearized form. This algorithm reduces the complex and time consuming computations to several steps through a systematic schema. We analyze Hopf bifurcation and its properties of a class of system of reaction-diffusion equations involving two discrete time delays. First, we discuss the existence of periodic solutions of this class under Neumann boundary conditions, and determine the required conditions on parameters of the system at which Hopf bifurcation arises near equilibrium point. Bifurcation analysis is carried out by choosing one of the delay parameter as a bifurcation parameter and fixing the other in its stability interval. Second, some properties of periodic solutions such as direction of Hopf bifurcation and stability of bifurcating periodic solution are studied through the normal form theory and the center manifold reduction for functional partial differential equations. Moreover, an algorithm is developed in order to determine the existence of Hopf bifurcation (and its properties) of variety of system of reaction-diffusion equations that lie in the same class. The benefit of this algorithm is that it puts a very complex and long computations of existence of Hopf bifurcation for each equation in that class into a systematic schema. In other words, this algorithm consists of the conditions and formulae that are useful for completing the existence analysis of Hopf bifurcation by only using coefficients in the characteristic equation of the linearized system. Similarly, it is also useful for determining the direction analysis of the Hopf bifurcation merely by using the coefficients of the second degree Taylor polynomials of functions in the right hand side of the system. Finally, the existence of Hopf bifurcation for three different problems whose governing equations stay in that class is given by utilizing the algorithm derived, and thus the feasibility of the algorithm is presented. [ABSTRACT FROM AUTHOR]