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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.

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.

In this paper, we introduce and study Holling type II functional response predator–prey system with digest delay and impulsive harvesting on the prey, which contains with periodically pulsed on the prey and time delay on the predator. We investigate the existence and global attractivity of the predator-extinction periodic solutions of the system. By using the theory on delay functional and impulsive differential equation, we obtain the sufficient condition with time delay and impulsive perturbations for the permanence of the system.

A predator–prey model with time delay and Holling type-II functional response is investigated. By choosing time delay as the bifurcation parameter and analyzing the associated characteristic equation of the linearized system, the local stability of the system is investigated and Hopf bifurcations are established. The formulae determining the direction of bifurcations and the stability of bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results.

First, we obtain a new result for the permanence of a well known delayed discrete-time model of single species. Second, based on this new condition, we discuss the permanence of a delayed discrete-time predator–prey model in which the prey disperses in two patches with biased dispersion. The biological implications of the results are briefly discussed.

Introducing discrete time delay into the model for producing 1, 3-propanediol by microbial continuous fermentation, the stability and Hopf bifurcation of a delay differential system for microorganisms in continuous culture are considered in this paper, including the changing regularity of bifurcation value and oscillating period. Algebraic criteria for absolute stability, as well as the transversality condition for Hopf bifurcation of this kind system are obtained. Explicit algorithm for determining the direction of Hopf bifurcation and the stability of periodic solution is derived, using the theory of normal form and center manifold. Finally, numerical simulations show the effectiveness of our results. The pictures of periodic solutions and phase planes with specified parameters suggest that our results can qualitatively describe oscillatory phenomena occurring in experiments.

Dynamical behavior of a vivax malaria model is provided and regular recurrences of the symptoms of the tertian fever are described in the human body. We calculate the basic reproduction number R₀ and explain the connection between the basic reproduction number and the parasite-threshold. If R₀ < 1, then plasmodium vivax will be eliminated. If R₀ > 1, then malarial parasites are survivable and there is a so called parasite-threshold. When the value of the parasites is larger than this parasite-threshold the symptoms of the tertian fever appear; otherwise, if the value of the parasites is less than this parasite-threshold the tertian fever cannot give signs of the symptoms suddenly in vivo. We illustrate that the gravity of infected baby is worse than that of infected grownup and also explain that the advancing of the vivax malaria can be arrested by eliminating malarial parasites in erythrocyte stage with clinical treatment by numerical simulations.

A delayed ratio-dependent one-predator and two-prey system with Michaelis–Menten type functional response is investigated. We show the existence of nonnegative equilibria under some appropriated conditions. Criteria for local stability, instability of nonnegative equilibria are obtained. The existence of Hopf bifurcations at the endemic equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the normal form and the center manifold theory. At last, some numerical simulations to support the analytical conclusions are carried out.

A chemostat model with time delay, variable yield and ratio-dependent functional response is investigated. By analyzing the corresponding characteristic equations, the local stability of a boundary equilibrium and a positive equilibrium is discussed and the existence of Hopf bifurcation is established. By using the comparison arguments, sufficient conditions are obtained for the global stability of the boundary equilibrium. By constructing a suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the positive equilibrium. Finally, numerical simulations are carried out to illustrate the theoretical results.

In this paper, two sunflower equations are considered. Using delay τ as a parameter and applying the global Hopf bifurcation theorem, we investigate the existence of global Hopf bifurcation for the sunflower equation. Furthermore, we analyze the local Hopf bifurcation of the modified equation with nonlinear relation about stem's increase, including the occurrence, the bifurcation direction, the stability and the approximation expression of the bifurcating periodic solution using the theory of normal form and center manifold. Finally, the obtained results of these two equations are compared, which finds that the result about the period of their bifurcating periodic solutions is obviously different, while the bifurcation direction and stability are identical.

In this paper, we consider the discrete Hematopoiesis model with a time delay:

In this paper, a time-delayed predator-prey system is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.

The combined effects of harvesting and time delay on predator-prey systems with Beddington–DeAngelis functional response are studied. The region of stability in model with harvesting of the predator, local stability of equilibria and the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation due to the two-parameter geometric criteria developed by Ma, Feng and Lu [Discrete Contin. Dyn. Syst. Ser B9 (2008) 397–413]. The global stability of the positive equilibrium is investigated by the comparison theorem. Furthermore, local stability of steady states and the existence of Hopf bifurcation for prey harvesting are also considered. Numerical simulations are given to illustrate our theoretical findings.

Ratio-dependent predator–prey models are favored by many animal ecologists recently as more suitable ones for predator–prey interactions where predation involves searching process. In this paper, a ratio-dependent predator–prey model with stage structure and time delay for prey is proposed and analyzed. In this model, we only consider the stage structure of immature and mature prey species and not consider the stage structure of predator species. We assume that the predator only feed on the mature prey and the time for prey from birth to maturity represented by a constant time delay. At first, we investigate the permanence and existence of the proposed model and sufficient conditions are derived. Then the global stability of the nonnegative equilibria are derived. We also get the sufficient criteria for stability switch of the positive equilibrium. Finally, some numerical simulations are carried out for supporting the analytic results.

A dynamical model for toxin producing phytoplankton and zooplankton has been formulated and analyzed. Due to gestation of prey, a discrete time delay is incorporated in the predator dynamics. The stability of the delay model is discussed and Hopf bifurcation to a periodic orbit is established. Stability and direction of bifurcating periodic orbits are investigated using normal form theory and center manifold arguments. Global existence of periodic orbits is also established. To substantiate analytical findings, numerical simulations are performed. The system shows rich dynamic behavior including chaos and limit cycles. The influence of seasonality in intrinsic growth parameter of the phytoplankton population is also investigated. Seasonality leads to complexity in the system.

A competitive Lotka–Volterra reaction-diffusion system with two delays subject to Neumann boundary conditions is considered. It is well known that the positive constant steady state of the system is globally asymptotically stable if the interspecies competition is weaker than the intraspecies one and is unstable if the interspecies competition dominates over the intraspecies one. If the latter holds, then we show that Hopf bifurcation can occur as the parameters (delays) in the system cross some critical values. In particular, we prove that these Hopf bifurcations are all spatially homogeneous if the diffusive rates are suitably large, which has the same properties as Hopf bifurcation of the corresponding delayed system without diffusion. However, if the diffusive rates are suitably small, then the system generates the spatially nonhomogeneous Hopf bifurcation. Furthermore, we derive conditions for determining the direction of spatially nonhomogeneous Hopf bifurcations and the stability of bifurcating periodic solutions. These results indicate that the diffusion plays an important role for deriving the complex spatiotemporal dynamics.