Narrow Results

In this paper, we consider the optimal boundary control problem of a two-species competitive system with time delay and size structure in a polluted environment. First, the well posedness of the system is studied by using the characteristic line method and the fixed point principle. Second, the necessary conditions for optimal boundary control are obtained by conjugate system and normal cone property. Finally, the existence and uniqueness of the optimal strategy are proved by Ekeland variational principle.

The dynamical behaviors for a delay-diffusion housefly equation with two kinds of Dirichlet boundary conditions are considered in this paper. The existence and uniqueness of the steady state solutions are investigated, and the stability of the constant steady state solutions is obtained by using qualitative theory. The existence of Hopf bifurcation near the positive constant steady state solution is discussed and the expressions which can identify the bifurcation properties, including the stability of the bifurcating periodic solution and the bifurcation direction, are presented.

In this work, we propose and investigate a human immunodeficiency virus (HIV) infection model that considers CD4${}^{+}$ T cell homeostasis and the cytotoxic T lymphocyte (CTL) response. The local stability of the disease-free and endemic equilibrium point is established. Further, the global stability of the disease-free and endemic equilibrium points is investigated. Under specific parametric conditions, it is shown that the model exhibits backward, forward (transcritical) bifurcation, Hopf and Hopf–Hopf bifurcation. We have further considered a time lag in the model to represent the time delay between CD4${}^{+}$ T cell infection and the viral particle production and performed the stability and bifurcation analysis for the delay model. We conduct comprehensive numerical experiments to visualize the dynamical behavior of the HIV model and validate our findings.

In this paper, a delayed fractional-order epidemic model with general incidence rate and incubation period is proposed for the Corona Virus Disease 2019 (COVID-19) pandemic. The corresponding sufficient conditions are established to analyze the existence and stability of disease-free equilibrium and endemic equilibrium of the proposed model. The conditions for the existence of Hopf bifurcation are obtained by selecting the time delay as the bifurcation parameter. The control strategies for the COVID-19 pandemic are designed, and the corresponding delay fractional order optimal control problem (DFOCP) is analyzed based on the generalized Euler–Lagrange equation. The parameters of the model are identified based on the data of multiple types of the COVID-19 pandemic. Further, the effectiveness of the model in describing the trend of the COVID-19 pandemic is verified. Based on the results of parameter identification, the influence of incubation period on the COVID-19 pandemic is discussed. The forward–backward sweep method (FBSM) is adopted to numerically solve DFOCP, and the control effects under different control measures are analyzed.

In this paper, a mathematical model for a solid spherically symmetric vascular tumor growth with nutrient periodic supply and time delays is studied. Compared to the apoptosis process of tumor cells, there is a time delay in the process of tumor cell division. The cells inside the tumor obtain nutrient $\sigma (r,t)$ through blood vessels, and the tumor attracts blood vessels at a rate proportional to $\alpha (t)$. So, the boundary value condition

Considering the food diversity of natural enemy species and the habitat complexity of prey populations, a pest-natural enemy model with non-monotonic functional response is proposed for biological management of Bemisia tabaci. The dynamic characteristics of the proposed model are analyzed. In addition, considering that the conversion from prey to predator has a certain time lag rather than instantaneous, a time delay is introduced into this model, and it is shown that the Hopf bifurcation occurs at the interior equilibrium when the time delay is used as the bifurcation parameter. Furthermore, the values of the parameters that determine the direction of the Hopf bifurcation as well as the stability of the periodic solution are calculated. In order to illustrate the theoretical analysis results, numerical simulations and validation are carried out to demonstrate the effects of non-monotonic functional response, additional food supply and habitat complexity.

We propose single-species population models with the psychological effects and time delay in a toxicant environment in this study, of which the concentrations of toxicant nonlinearly affect the density of adults, linearly affect the density of juveniles. The models, consisting of a system of stochastic differential equations, govern the dynamics of juveniles and adults, as well as the concentrations of toxicant in the environment and organisms. First of all, the existence and uniqueness of the global solution to the models are derived, the sufficient conditions of the extinction and the time that the densities of adults and juveniles approach zero are investigated. Further, the sufficient conditions for the weak persistence in the mean are obtained around the pollution-free equilibrium point, and the stochastic permanence of adults and juveniles occurs around the pollution equilibrium points under moderate conditions. As a consequence, the corresponding numerical simulations reveal that higher psychological effects and less time delay create larger densities of juveniles and adults in the sense of weak persistence and stochastic permanence, and that less fluctuations of white noises and less psychological effects produce the earlier extinction time for adults and juveniles.

In this paper, we study an age-structured population model with non-local diffusion and distributed delay. By using the non-densely defined operators and extended phase spaces, we first rewrite the model into an abstract ordinary differential equation. Then we prove the existence of the solution of the model by using the operator semigroup theory. Finally, we study the spectrum of the non-densely defined operators and analyze the asymptotic behavior via asynchronous exponential growth. Our results extend the results for the age-structured population models without time delay.

In this paper, the effect of time delay is investigated on the system dynamics of a glucose-insulin model incorporating obesity. Treating the time delay as a bifurcation parameter, the stability switching on the positive equilibrium with global bifurcation is obtained. With the method of normal forms and central manifold theory, the direction and stability of limit cycles arising from Hopf bifurcation are analyzed. Using the method of multiple time scales, the normal form associated with non-resonant double Hopf bifurcation is derived. Moreover, the bifurcations are classified in the two-dimensional parameter plane near the critical point, and numerical simulations are presented to demonstrate the applicability of the theoretical results. Our results indicate that time delay in the glucose-insulin model can not only induce Hopf bifurcation and double Hopf bifurcation but also generate multiple stable periodic solutions. These results may help to understand the dynamical mechanism of glucose-insulin metabolic regulation systems, and to design control strategies for regulating and mitigating the occurrence of related diseases.

Syphilis is one of the top three chronic infectious diseases in the world. To investigate the effect of media coverage and time delay on the prevalence of syphilis, we develop a model of syphilis infection incorporating the incubation delay of primary syphilis and the effect of control measures. We prove that when $R_c$ exceeds unity, the model has a unique endemic equilibrium $E^{\ast }$, while the disease-free equilibrium $E_0$ is consistently present. Meanwhile, by the Lyapunov function method, the globally asymptotically stable of the two equilibria is derived. And then, sensitivity analysis via the partial rank correlation coefficient (PRCC) method reveals that the control reproduction number is most sensitive to the infection rate $\beta$. In addition, the numerical results show that (a) when $R_c>1$, the peak size of the primary and secondary syphilis infectious decreases as the time delay increases, respectively, and the peak time of infection is postponed; (b) the peak size of the total infected individuals reduces by 4.7% when the primary syphilis treatment rate ${\alpha }_1$ enhances from 3.1 to 3.5, and while the secondary syphilis treatment rate ${\alpha }_2$ increases from 2.1 to 2.5, the peak size of the total infectious reduces by 10.7%; (c) contemporaneously, when the effect of media coverage $f$ augments from 0.1 to 0.5, the peak size of total infectious decreases by 17.3%. In short, improving the treatment rates and intensifying the effect of media coverage to enhance public awareness can significantly reduce the peak size of infections and prevent the spread of the disease.

In this work, we propose a stochastic Human Papillomavirus (HPV) epidemic model with two kinds of delays and media influences. These two time delays are the delay time caused by media receiving the disease information and the delay time of public feedback after the media coverage. In addition, media coverage not only has a negative impact on the infection rate, but it also has a positive impact on the vaccination rate of disease. We discuss the existence and uniqueness of the positive solution for the HPV epidemic model, and then put forward a positively invariant set. The sufficient conditions of the extinction and persistence for the HPV epidemic are given. For the optimal control problem of the HPV epidemic, we obtain an optimal strategy. Our numerical simulations validate the theoretical results of this paper, showing that appropriate media coverage can help control the development of the disease.

Understanding the impact of information-induced behavioral responses on the public, as well as precise forecasting of hospital bed demand, is critical during infectious disease epidemics to prevent managing healthcare facilities. Hence to study the impact of information-induced behavioral response in the public and the reinfection of diseases on the disease dynamics, we created a nonlinear SIHRZ mathematical model. We calculated the basic reproduction numbers and used mesh and contour plots to investigate the effect of various parameters on disease dynamics. It is observed that even if $\widetilde{R_0}<1$, the disease cannot be eradicated because of reinfection. The most sensitive parameters expected to affect the disease’s endemicity are found by computing the sensitivity indices. The dynamic system has an endemic equilibrium point, which is stable while $\widetilde{R_0}>1$ and unstable when $\widetilde{R_0}<1$. Using the Routh–Hurwitz criterion and the construction of the Lyapunov function, the equilibrium point’s local and global stability is examined. We have further examined the model system for the population’s time lag in immunity loss as a result of the efficacy of medicines, vaccination, self-defense, etc. Due to this delay, an oscillatory nature of the population is obtained. We determined the existence and direction of the Hopf bifurcation, as well as the stability of the equilibrium point, using the delay as a bifurcation parameter. Comprehensive numerical experiments are conducted to explore and validate qualitative results, providing valuable biological insights. This research highlights the critical role that information, treatment intensity, the overall number of hospital beds available, and the occupancy rate of those beds have in determining the behavioral reaction of susceptibles. The model also evaluated cases of fading immunity to look for epidemic peaks. By raising immunization and vaccine effectiveness rates, this peak can be lowered. Moreover, our results suggest that the oscillations that cause problems in managing disease outbreaks would make it extremely difficult to determine the real data of hospitalized and infected individuals. Hence, the WHO, governmental organizations, health policymakers, etc. cannot accurately estimate the scope of an epidemic. As a result, information provided by health authorities and the government regarding disease outbreaks must be kept up to date to limit the disease burden, which is also dependent on funding availability and policymaker decisions.

We consider time delay and symmetrized time delay (defined in terms of sojourn times) for quantum scattering pairs {H₀ = h(P), H}, where h(P) is a dispersive operator of hypoelliptic-type. For instance, h(P) can be one of the usual elliptic operators such as the Schrödinger operator h(P) = P² or the square-root Klein–Gordon operator . We show under general conditions that the symmetrized time delay exists for all smooth even localization functions. It is equal to the Eisenbud–Wigner time delay plus a contribution due to the non-radial component of the localization function. If the scattering operator S commutes with some function of the velocity operator ∇h(P), then the time delay also exists and is equal to the symmetrized time delay. As an illustration of our results, we consider the case of a one-dimensional Friedrichs Hamiltonian perturbed by a finite rank potential.

Our study puts into evidence an integral formula relating the operator of differentiation with respect to the kinetic energy h(P) to the time evolution of localization operators.

We define, prove the existence and obtain explicit expressions for classical time delay defined in terms of sojourn times for abstract scattering pairs (H₀, H) on a symplectic manifold. As a by-product, we establish a classical version of the Eisenbud–Wigner formula of quantum mechanics. Using recent results of Buslaev and Pushnitski on the scattering matrix in Hamiltonian mechanics, we also obtain an explicit expression for the derivative of the Calabi invariant of the Poincaré scattering map. Our results are applied to dispersive Hamiltonians, to a classical particle in a tube and to Hamiltonians on the Poincaré ball.

We review the analytic results for the phase shifts δ_l(k) in nonrelativistic scattering from a spherical well. The conditions for the existence of resonances are established in terms of time-delays. Resonances are shown to exist for p-waves (and higher angular momenta) but not for s-waves. These resonances occur when the potential is not quite strong enough to support a bound p-wave of zero energy. We then examine relativistic scattering by spherical wells and barriers in the Dirac equation. In contrast to the nonrelativistic situation, s-waves are now seen to possess resonances in scattering from both wells and barriers. When s-wave resonances occur for scattering from a well, the potential is not quite strong enough to support a zero momentum s-wave solution at E=m. Resonances resulting from scattering from a barrier can be explained in terms of the "crossing" theorem linking s-wave scattering from barriers to p-wave scattering from wells. A numerical procedure to extract phase shifts for general short range potentials is introduced and illustrated by considering relativistic scattering from a Gaussian potential well and barrier.

This paper studies the evolution of a fish stock that is exploited by an n-country oligopoly. A feature of the economic structure is that the countries exploiting the fish stock experience time lags in obtaining and implementing information on the fish stock. The local asymptotic behavior of the equilibrium is analyzed, including asymptotic stability, instability, and cyclical behavior. Under the assumption of symmetric countries, two special cases are examined in detail. In the first case identical time delays are assumed, and in the second case it is assumed that one country has a different time delay from the others. This semi-symmetric case gives some insight into the consequence of asymmetry of the countries on the asymptotic behavior of the fish stock.

A stochastic version of the SIR model is investigated in this paper. The stability in probability of the steady state of the system is proved under suitable conditions on the white noise perturbations. Linearizations of the systems both with and without delay are given and their exponentially mean square stabilities are studied.

In this work, we establish the linear convergence estimate for the gradient descent involving the delay $\tau \in \mathbb{N}$ when the cost function is $\mu$-strongly convex and $L$-smooth. This result improves upon the well-known estimates in [Y. Arjevani, O. Shamir and N. Srebro, A tight convergence analysis for stochastic gradient descent with delayed updates, Proc. Mach. Learn. Res. 117 (2020) 111–132; S. U. Stich and S. P. Karimireddy, The error-feedback framework: Better rates for SGD with delayed gradients and compressed updates, J. Mach. Learn. Res. 21(1) (2020) 9613–9648] in the sense that it is non-ergodic and is still established in spite of weaker constraint of cost function. Also, the range of learning rate $\eta$ can be extended from $\eta \le 1/(10L\tau )$ to $\eta \le 1/(4L\tau )$ for $\tau =1$ and $\eta \le 3/(10L\tau )$ for $\tau \ge 2$, where $L>0$ is the Lipschitz continuity constant of the gradient of cost function. In a further research, we show the linear convergence of cost function under the Polyak–Łojasiewicz(PL) condition, for which the available choice of learning rate is further improved as $\eta \le 9/(10L\tau )$ for the large delay $\tau$. The framework of the proof for this result is also extended to the stochastic gradient descent with time-varying delay under the PL condition. Finally, some numerical experiments are provided in order to confirm the reliability of the analyzed results.

Active magnetic bearing (AMB) can actively control the vibration of the rotor, and its control law plays a key role. In this study, the vibration control of a rotor-AMB system with time delay is studied. The nonlinear vibration equation is derived based on the Newton law considering the time-varying stiffness (TVS) control. The method of multiple scales is applied to obtain the approximate solutions in the case of the primary parametric resonance and 1:1 internal resonance. The results show that the vibration amplitude and stable region are both periodic with respect to time delay. Their period is exactly equal to the rotating period of rotor. Additionally, the eccentricity-response curves are constructed to illustrate the vibration control of the time-varying stiffness. The results show that the time-varying stiffness control suppresses the vibration effectively. At the same time, it is also found that the time-varying stiffness control may cause an increase in vibration amplitude or even instability in small range of unbalance eccentricity. The influence of time delay on the time-varying stiffness control is investigated. It is found that the time-varying stiffness controller exhibits different performance depending on the time delay.

In this work, we consider a delayed stage-structured variable coefficients predator-prey system with impulsive perturbations on predators. By using the discrete dynamical system determined by stroboscopic map and the standard comparison theorem, we obtain the sufficient conditions which guarantee the global attractivity of prey-extinction periodic solution of the investigated system. We also prove that all solutions of the system are uniformly ultimately bounded. Our results provide reliable tactic basis for the practical pest management.